Given that for every $4^\circ$ rise in temperature, the volume of a certain gas expands by $3$ cubic centimeters, and when the temperature is $30^\circ$, the volume of the gas is $40$ cubic centimeters, determine the volume of the gas when the temperature drops first to $22^\circ$ and then further to $14^\circ$.


For how many positive integer values of $n$ are both $\frac{n}{4}$ and $4n$ three-digit whole numbers?


Four fair dice with 8 faces each (numbered 1 through 8) are tossed. What is the probability that the numbers on four of the dice can be arranged to form an arithmetic progression with a common difference of two?


The base three representation of $x$ is $1122001_3$. Determine the first digit (on the left) of the base nine representation of $x$.


Given that $x^2 + 2bx + c = 0$ and $x^2 + 2cx + b = 0$ do not have two distinct real solutions, calculate the number of ordered pairs of positive integers $(b, c)$.


If $4(5y + 3\pi) = Q$, then express the value of $8(10y + 6\pi + 2\sqrt{3})$ in terms of $Q$.

Solution: \boxed{4Q + 16\sqrt{3}}


Given that Crystal runs due north for 1 mile, then due east for 2 miles, and then due south for 1 mile, determine the distance of the last portion of her run that takes her on a straight line back to where she started in miles.


Let X be the set of the 3000 smallest positive multiples of 5, and let Y be the set of the 3000 smallest positive multiples of 8. Find the number of elements common to sets X and Y.


Given soda is sold in packs of 6, 12, 24, and 30 cans, find the minimum number of packs needed to buy exactly 120 cans of soda.


The number of revolutions of a wheel, with a fixed center and an outside diameter of $8$ feet, required to cause a point on the rim to go two miles can be calculated.


Given the list of numbers $8-3\sqrt{10}$, $3\sqrt{10}-8$, $23-6\sqrt{15}$, $58-12\sqrt{30}$, $12\sqrt{30}-58$, identify the smallest positive number in the list.

Solution: \boxed{3\sqrt{10} - 8}


For how many integers $x$ is the number $x^4 - 53x^2 + 150$ negative?


Given Clara's correct scores were doubled and summed, but one score of 15 was mistakenly tripled instead of doubled, calculate the difference between the correct sum and the incorrect sum of her scores.


A circle of radius 5 is centered at point $B$. An equilateral triangle with a vertex at $B$ has a side length of 10. Find the difference between the area of the region that lies inside the circle but outside the triangle and the area of the region that lies inside the triangle but outside the circle.

Solution: \boxed{25(\pi - \sqrt{3})}


If the ratio of $3x-2y$ to $2x+y$ is $\frac{4}{5}$, calculate the ratio of $x$ to $y$.


Find the least common positive period p for any function f that satisfies f(x+5) + f(x-5) = f(x) for all real x.


Four runners start simultaneously from the same point on a 600-meter circular track and run clockwise. Their constant speeds are 4.2, 4.5, 4.8, and 5.1 meters per second respectively. Find the time in seconds when all runners are together again at some point on the track.


Given a set of $2n$ numbers where $n > 1$, of which one is $1 + \frac{1}{n}$ and all the others are $1$, calculate the arithmetic mean of these $2n$ numbers.

Solution: \boxed{1 + \frac{1}{2n^2}}


Determine the difference between the sum of the first one hundred positive even integers and the sum of the first one hundred positive multiples of 3.


Given a square piece of paper is folded so that point (0,4) is matched with (4,0) using a fold that also involves a $45^\circ$ rotation, find the aligned coordinates of point $(8,6)$ and determine the sum of its coordinates.


A four-digit palindrome is chosen at random. What is the probability that it is divisible by 11?
A) $\dfrac{1}{10}$
B) $\dfrac{1}{9}$
C) $\dfrac{1}{2}$
D) $\dfrac{1}{5}$
E) $1$


A particle begins at a point P on the parabola y = x^2 - 2x - 8 where the y-coordinate is 8. It rolls along the parabola to the nearest point Q where the y-coordinate is -8. Calculate the horizontal distance traveled by the particle, defined as the absolute difference between the x-coordinates of P and Q.

Solution: \boxed{\sqrt{17} - 1}


Given that a rhombus $PQRST$ has side length $5$ and $\angle Q = 90^{\circ}$, find the area of region $W$ that consists of all points inside the rhombus that are closer to vertex $Q$ than any of the other three vertices.


Given a three-digit number with digits $x, y, z$, where $x$ is the first digit, $z$ is the third digit, and $y$ is the second digit, the number is not divisible by $5$, has digits that sum to less than $15$, and $x=z>y$, calculate the number of such numbers.


Lara in her car notices Leo on his motorcycle $1$ mile ahead moving in the same direction. After she overtakes him, she can still see him in her rearview mirror until he is $1$ mile behind her. Lara drives at a constant speed of $60$ miles per hour, and Leo cruises at a constant speed of $40$ miles per hour. Calculate the time in minutes that Lara can see Leo.


The number of eighth graders and fifth graders who bought a pencil can be represented as variables x and y, respectively. Given that some eighth graders each bought a pencil and paid a total of $2.34 dollars, and some of the $50$ fifth graders each bought a pencil and paid a total of $3.25 dollars, determine the difference in the number of fifth graders and eighth graders who bought a pencil.


Given a right triangle with legs $a$ and $b$, and the hypotenuse $c$, where $a:b = 2:5$, find the ratio of $r$ to $s$ if a perpendicular is dropped from the right-angle vertex to the hypotenuse, dividing it into segments $r$ and $s$.


Given that Mary is $25\%$ older than Sally, and Sally is $30\%$ younger than Danielle, and the sum of their ages is $36$ years, determine Mary's age on her next birthday.


If the digits $12$ are placed after a three-digit number whose hundreds' digit is $h$, tens' digit is $t$, and units' digit is $u$, express the new formed number as an algebraic expression.

Solution: \boxed{10000h + 1000t + 100u + 12}


What is the probability that a three-digit number is formed from 5 pieces of paper labeled with the digits 2, 3, 4, 5, and 6, such that the number is even?


Carl has a collection of $6$-pound rocks worth $$16$ each, $3$-pound rocks worth $$9$ each, and $2$-pound rocks worth $$3$ each, with at least $30$ of each size. Determine the maximum value, in dollars, of the rocks he can carry out of the cave, given that he can carry at most $24$ pounds and no more than $4$ rocks of any size.


What is the hundreds digit of the expression $(25! - 20! + 10!)$?


Suppose the estimated €25 billion (Euros) cost to send a person to the planet Mars is shared equally by the 300 million people in a consortium of countries. Given the exchange rate of 1 Euro = 1.2 dollars, calculate each person's share in dollars.


The original price of the article in terms of p and q can be expressed as $\frac{10000}{10000 + 100(p - q) - pq}$.

Solution: \boxed{\frac{10000}{10000 + 100(p - q) - pq}}


Given that Alex received $150$ points in a $60$-question multiple choice math contest, where $5$ points are awarded for a correct answer, $0$ points for an answer left blank, and $-2$ points for an incorrect answer, find the maximum number of questions that Alex could have answered correctly.


How many ways are there to write $3060$ as the sum of twos and threes, ignoring order?


Given a convex polygon with the sum of its interior angles equal to $3140^\circ$, calculate the degree measure of the missing angle.


Given the sum of the first twenty terms of an arithmetic progression is five times the sum of the first ten terms, determine the ratio of the first term to the common difference.


Find the product of the three smallest prime factors of 180.


Given real numbers $a$ and $b$, $\frac{\frac{1}{a} + \frac{1}{b}}{\frac{1}{a} - \frac{1}{b}} = 1001.$ Find the value of $\frac{a+b}{a-b}$.


Let \( p, q, r, s, \) and \( t \) be distinct integers such that \((8-p)(8-q)(8-r)(8-s)(8-t) = 120\). Calculate the value of \( p+q+r+s+t\).


If 104 is divided into three parts which are proportional to 1, 1/2, and 1/3, calculate the value of the middle part.


Triangle $ABC$ has a right angle at $B$. Point $D$ is the foot of the altitude from $B$, $AD=5$, and $DC=12$. Calculate the area of $\triangle ABC$.


For every dollar Ben spent on coffee, David spent $0.5 less. Ben paid $15 more than David. Determine the total amount they spent in the coffee shop together.


Consider a scenario where a cubical die is labeled with the numbers $2$, $3$, $3$, $4$, $4$, and $5$. Another die is marked with $1$, $2$, $3$, $5$, $6$, and $7$. Calculate the probability that rolling these two dice simultaneously results in a sum on their top faces of either $6$, $8$, or $10$.


Given Sean's current test scores of 82, 76, 88, 94, 79, and 85, determine the minimum test score he would need to achieve to increase his overall average by exactly 5 points and not drop below his current lowest score.


Given Timothy's house has $4$ bedrooms, each one is $15$ feet long, $12$ feet wide, and $10$ feet high, and the doorways and windows in each bedroom occupy $75$ square feet, calculate the total square feet of walls that must be painted.


Given that bricklayer Alice can build a wall alone in 8 hours, and bricklayer Bob can build it alone in 12 hours, and they complete the wall in 6 hours when working together with a 15-brick-per-hour decrease in productivity, determine the number of bricks in the wall.


Jack walks to a park 1.5 miles away at a speed of 3 miles per hour, while Jill roller skates at a speed of 8 miles per hour. Calculate the time difference in minutes between their arrivals at the park.


Rectangle ABCD and right triangle AEF share side AD and have the same area. Side AD = 8, and side AB = 7. If EF, which is perpendicular to AD, is denoted as x, determine the length of hypotenuse AF.


A sphere with center O has a radius of 8. A triangle with sides of lengths 13, 13, 10 is situated such that all of its sides are tangent to the sphere. Calculate the distance from O to the plane determined by the triangle.


Given the equations $4x + ay + d = 0$ and $dx - 3y + 15 = 0$, determine the number of pairs of values of $a$ and $d$ such that these two equations have the same graph.


Jamar bought some pencils costing more than a penny each at the school bookstore and paid $\textdollar 2.16$. Sharona bought some of the same pencils and paid $\textdollar 2.72$. Determine the number of more pencils Sharona bought than Jamar.


What is the probability that a randomly drawn positive factor of $72$ is less than $8$?


In a science fair, 64 students are wearing green shirts, and another 68 students are wearing red shirts. The 132 students are grouped into 66 pairs. In exactly 28 of these pairs, both students are wearing green shirts. Determine the number of pairs in which both students are wearing red shirts.


An equilateral triangle ABC has a side length of 4. A right isosceles triangle DBE, where $DB=EB=1$ and angle $D\hat{B}E = 90^\circ$, is cut from triangle ABC. Calculate the perimeter of the remaining quadrilateral.

Solution: \boxed{10 + \sqrt{2}}


Given that Ahn chooses a two-digit integer, subtracts it from 300, multiplies the result by 2/3, and doubles the whole result, find the largest number Ahn can get.


If \(x\) is real and positive and grows beyond all bounds, calculate the limit as \(x\) approaches infinity of \(\log_4{(8x-3)}-\log_4{(5x+2)}\).

Solution: \boxed{\log_4{\left(\frac{8}{5}\right)}}


Calculate the value of $\text{rem}(x, y)$ where $x = \frac{5}{7}$ and $y = \frac{3}{4}$, and then multiply the result by 2.


Given that Mrs. Martinez teaches math to $20$ students, and that the average grade for the class was $75$ when she graded everyone's test except Leah's and Jake's, was $76$ after grading Leah's, and $77$ after grading Jake's, determine Jake's score on the test.


In multiplying two positive integers $a$ and $b$, Ron mistakenly reversed the digits of the two-digit number $b$. His erroneous product was $143.$ From this, find the correct value of the product of $a$ and $b$.


Given a store offering a promotion where all items are priced at "one-third off" the original price, and a special coupon grants an extra 25% discount on the promotional prices, determine the percentage off the original price a customer ultimately pays.


A box contains 30 red balls, 22 green balls, 18 yellow balls, 15 blue balls, 10 white balls, and 6 black balls. Calculate the minimum number of balls that must be drawn from the box without replacement to guarantee that at least 20 balls of a single color will be drawn.


Given that the track has a width of 8 meters, the longer it takes Keiko to walk around the outer edge of the track compared to the inner edge is 48 seconds, and each straight section of the track has a length of 100 meters, calculate Keiko's speed in meters per second.


Given the numbers $5, 7, 10, 20,$ and $x$, find the sum of all real numbers $x$ for which the median of these numbers is equal to the mean of the numbers.


Given a number is called flippy if its digits alternate between two distinct digits from the set {4, 6}, calculate the number of four-digit flippy numbers that are divisible by 4.


A straight line passing through the point $(2,5)$ is perpendicular to the line $x-2y-8=0$. Find its equation.

Solution: \boxed{y + 2x - 9 = 0}


Given the equation $(\log_b a)^3=\log_b(a^3)$, where $a$ is a positive real number and $b$ is an integer between $3$ and $300$, inclusive, determine the number of pairs $(a,b)$.


Given that the new fort is designed to be $20$ feet long, $15$ feet wide, and $8$ feet high, with walls that are two feet thick and the floor one foot thick, calculate the total number of one-foot cubical blocks needed for the fort.


What is the value of $\dfrac{13!-12!}{10!}$?


A cube with a volume of 1 cubic foot is divided into three slabs by making two cuts parallel to the top face of the cube. The first cut is made $\frac{1}{4}$ foot from the top, and the second cut is $\frac{1}{6}$ foot below the first cut. Determine the total surface area of the resulting solid assembly.


Three runners start simultaneously from the same point on a 600-meter circular track, running clockwise at constant speeds of 4.4 m/s, 4.9 m/s, and 5.1 m/s. Determine the time it takes for them to meet again somewhere on the track.


If 1 pint of paint is needed to paint a statue 6 ft. high, calculate the number of pints it will take to paint 1080 statues similar to the original but only 2 ft. high.


The prime factorization of the denominator of the fraction $\frac{987654321}{2^{30} \cdot 5^3}$ is $2^{30} \cdot 5^3$. Therefore, to determine the minimum number of digits to the right of the decimal point, find the minimum number of factors of 2 and 5 in the numerator, 987654321.


Given that $f(x+5)=4x^3 + 5x^2 + 9x + 6$ and $f(x)=ax^3 + bx^2 + cx + d$, find the value of $a+b+c+d$.


Determine how many perfect cubes exist between \(3^6 + 1\) and \(3^{12} + 1\), inclusive.


At Hilltop High, the ratio of sophomore students to freshman students involved in a yearly science fair is 7:4 and the ratio of junior students to sophomore students is 6:7. Determine the smallest number of students that could be participating in this fair from these three grades.


Two circles with centers at points A and B are externally tangent and have radii 7 and 4, respectively. A line tangent to both circles intersects ray AB at point C. Calculate the length of segment BC.


Determine the number of minutes before Jack arrives at the park that Jill arrives at the park, given that they are 2 miles apart, Jill cycles at a constant speed of 12 miles per hour, and Jack jogs at a constant speed of 5 miles per hour.


Given the function $f(n) =\begin{cases}\log_{10}{n}, &\text{if }\log_{10}{n}\text{ is rational,}\\ 1, &\text{if }\log_{10}{n}\text{ is irrational.}\end{cases}$, calculate the value of $\sum_{n = 1}^{256}{f(n)}$.


Two 8-sided dice are rolled once. The sum of the numbers rolled determines the diameter of a circle, but only if this sum is at least 5. Find the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference.


Parallelogram ABCD has an area of 80 square meters. Points E and G are the midpoints of sides AB and CD respectively. Point F is the intersection of line segment EG and diagonal BD. Find the area of triangle BFG in square meters.


Given the set $\{-10, -7, -5, 0, 4, 6, 9\}$, find the minimum possible product of three different numbers from this set.


An automobile travels $2a/5$ feet in $2r$ seconds. If this rate is maintained for $5$ minutes, how many yards does it travel in $5$ minutes?


Evaluate the expression $\sqrt{7+4\sqrt{3}} - \sqrt{7-4\sqrt{3}}$.


Eight points on a circle are given. Four of the chords joining pairs of the eight points are selected at random. Find the probability that the four chords form a convex quadrilateral.


A store owner purchased $1800$ pencils at $0.15$ each. If he sells them for $0.30$ each, how many of them must he sell to make a profit of exactly $150.00$?


Given the lighthouse is 60 meters tall and the conical light container has a volume of 150,000 liters, determine the height of the model lighthouse if the volume of the light container for the model is 0.15 liters.


Given a number ten times as large as y is decreased by three, then find one third of the result equals what expression?

Solution: \boxed{\frac{10y}{3} - 1}


The smallest possible even five-digit number is formed using the digits 1, 2, 3, 5, and 8. Determine the digit that must be in the tens place.


Determine the greatest number of consecutive integers whose sum is $36$.


Jo and Blair take turns counting numbers, starting from $3$. Jo begins the sequence, and each subsequent number said by Blair or Jo is two more than the last number said by the other person. Find the $20^\text{th}$ number said.


At noon on a certain day, Minneapolis is $N$ degrees warmer than St. Louis, and by $6{:}00$ PM, the temperature in Minneapolis has fallen by $7$ degrees while the temperature in St. Louis has risen by $4$ degrees. The temperatures in the two cities differ by $1$ degree at that time. What is the product of all possible values of $N?$


Given Professor Lee has a collection of ten different books lined up on a shelf: three Arabic, three German, and four Spanish, calculate the number of arrangements of the ten books on the shelf so that the Arabic books are always together and the German books are always together.


Determine the number of perfect cubic divisors in the product $1! \cdot 2! \cdot 3! \cdot \ldots \cdot 6!$.


Given a $25$-quart radiator is filled with water, and the described process of removing and replacing quarts of the mixture with pure antifreeze is repeated five times, calculate the fractional part of the final mixture that is water.


For how many positive integer values of $n$ are both $\frac{n}{4}$ and $4n$ three-digit whole numbers?


Given $\theta$ is an acute angle, and $\cos 2\theta = a$, then find $\sin\theta \cos\theta$.


Determine the number of distinct points common to the curves $x^2 + 9y^2 = 9$ and $9x^2 + y^2 = 9$.


A square hallway is covered entirely with tiles in a repeating pattern. Each corner of the hallway displays a similar arrangement of tiles. What fraction of the tiled hallway is made of darker tiles if the pattern repeats every $8 \times 8$ tiles, and in each top-left $4 \times 4$ square of these tiles there are 9 dark tiles?


If $a \star b = \frac{a \times b}{a+b-3}$ for $a,b$ positive integers where $a+b>3$, calculate the value of $6 \star 9$.


Given a group of 60 girls, each of whom is either blonde or brunette and has either blue eyes or brown eyes, and if there are 20 blue-eyed blondes, 35 brunettes, and 22 brown-eyed girls, calculate the number of brown-eyed brunettes.


Given that (x,y) are real numbers, what is the least possible value of (xy+1)^2 + (x+y+1)^2 ?


In four consecutive soccer matches of a season, a player scored $18$, $12$, $15$, and $14$ goals, respectively. His goals-per-game average was higher after these four games compared to the previous three games. If his total number of goals after seven matches is at least $100$, determine the minimum number of goals he must have scored in the three previous matches.


Susie buys 5 muffins and 4 bananas. Calvin spends three times as much as Susie buying 3 muffins and 20 bananas. Determine the ratio of the price of one muffin to the price of one banana.


In $\triangle ABC$, $AB = 6$, $BC = 8$, $AC = 10$, and $D$ is on $\overline{AC}$ with $BD = 6$. Calculate the ratio of $AD:DC$.


Given that the base of the number system is changed to six, count the twentieth number in this base.


For how many positive integer values of $N$ is the expression $\dfrac{49}{N+3}$ an integer, and this resulting integer is odd?


Lily is riding her bicycle at a constant rate of 15 miles per hour and Leo jogs at a constant rate of 9 miles per hour. If Lily initially sees Leo 0.75 miles in front of her and later sees him 0.75 miles behind her, determine the duration of time, in minutes, that she can see Leo.


A sphere with center O has a radius of 5. A triangle with sides of length 13, 13, and 10 is situated in space so that each of its sides is tangent to the sphere. Find the distance between O and the plane determined by the triangle.


Given $S$ is the set of the 1000 smallest positive multiples of $5$, and $T$ is the set of the 1000 smallest positive multiples of $9$, determine the number of elements common to both sets $S$ and $T$.


Given a rhombus with diagonals such that one diagonal is three times the length of the other, express the side of the rhombus in terms of $K$, where $K$ is the area of the rhombus in square inches.


A rectangular metal plate measuring \(10\) cm by \(8\) cm has a circular piece of maximum size cut out, followed by cutting a rectangular piece of maximum size from the circular piece. Calculate the total metal wasted in this process.


Let M be the third smallest positive integer that is divisible by every positive integer less than 9. What is the sum of the digits of M?


A wooden cube with side length $n$ units is painted green on all six faces and then cut into $n^3$ unit cubes. Exactly one-third of the total number of faces of the unit cubes are green. What is $n$?


Consider the set $\{2,3,4,5,6,7,8,9,10\}$. Find the number of subsets of three different numbers, one of which is $7$, that sum to $18$.


A 7' × 11' table sits in the corner of a square room. The table is to be rotated so that the side formerly 7' now lies along what was previously the end side of the longer dimension. Determine the smallest integer value of the side S of the room needed to accommodate this move.


Given that the average daily running time for third graders is $14$ minutes, fourth graders is $17$ minutes, and fifth graders is $12$ minutes, and there are three times as many third graders as fifth graders, and one and a half times as many fourth graders as fifth graders, find the average number of minutes run per day by all students.


Javier the painter initially had enough paint for 40 identically sized rooms. On his way to the painting site, he accidentally loses 4 cans of paint, leaving him with enough paint for only 30 rooms. Determine the number of cans he used for these 30 rooms.


Given Sarah earns $2$ dollars for doing her tasks, $4$ dollars for doing them well, or $6$ dollars for doing them exceptionally over a period of $15$ days, and she accumulated a total of $70$ dollars, determine the number of days on which Sarah did her tasks exceptionally well.


Given a ferry starts transporting tourists to an island at 9 AM and continues every hour until the last trip at 4 PM, and the number of tourists on each trip decreases by 2 compared to the previous trip, starting with 120 tourists on the first trip, calculate the total number of tourists transported to the island that day.


Given the line y = (1/2)x + 3 and a line K that is parallel to the given line but 5 units away from it, find the equation of line K.

Solution: \boxed{y = \dfrac{1}{2}x + \left(3 - \dfrac{5\sqrt{5}}{2}\right)}


Given that 20% of the students scored 65 points, 40% scored 75 points, 25% scored 85 points, and the rest scored 95 points, calculate the difference between the mean and median score of the students' scores on this test.


Given that 20% of students in a class are juniors and the remaining 80% are seniors, the average score of the class on the test was 84, and the average score of seniors was 82, determine the score of each junior.


Consider the quadratic function $f(x) = px^2 + qx + r$ where $p > 0$ and the discriminant $q^2 - 4pr < 0$. Calculate the least value of this function.

Solution: \boxed{\frac{4pr - q^2}{4p}}


Consider the sequence where the n-th term is given by $(-1)^n \cdot n$. Determine the average of the first $200$ terms of the sequence.


Samantha has \(3q + 2\) quarters, while Bob has \(2q + 8\) quarters. Determine the difference in their amount of money in dimes.

Solution: \boxed{2.5q - 15}


Let’s consider Nubia has an $8 \times 10$ index card. She decides to reduce the length of one side of this card by $2$ inches, resulting in an area of $64$ square inches. Determine the area of the card in square inches if instead she reduces the length of the other side by $2$ inches.


Points $P$ and $Q$ are 12 units apart, points $Q$ and $R$ are 7 units apart, and points $R$ and $S$ are 5 units apart. If $P$ and $S$ are as close as possible, find the distance between them.


If $\log_k x \cdot \log_5 (k^2) = 3$, calculate the value of $x$.


What is the greatest number of consecutive integers whose sum is $136$?


Given that the product $\dfrac{5}{3}\cdot \dfrac{6}{5}\cdot \dfrac{7}{6}\cdot \dfrac{8}{7}\cdot \ldots\cdot \dfrac{a}{b} = 16$, calculate the sum of $a$ and $b$.


Let $M = 72^5 + 5\cdot72^4 + 10\cdot72^3 + 10\cdot72^2 + 5\cdot72 + 1$. Find the number of positive integers that are factors of $M$.


A rectangular floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers with $b > a$. An artist paints a rectangle in the center of the floor with the sides of the rectangle parallel to the floor's sides. The unpainted part of the floor forms a border of width $2$ feet around the painted rectangle and occupies two-thirds of the area of the entire floor. Determine the number of possible ordered pairs $(a, b)$.


Real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $(0,1)$. Find the probability that $\lfloor\log_3x\rfloor=\lfloor\log_3y\rfloor$.


What is the maximum number of balls of clay of radius $3$ that can completely fit inside a cube of side length $9$ assuming the balls can be reshaped but not compressed before they are packed in the cube?


The area of the triangle formed by the lines $y = 7$, $y = 2x + 3$, and $y = -2x + 3$.


Given that $1024 = 2^{10}$, find the number of positive integers $b$ for which $\log_b 1024$ is a positive integer.


Points $G$, $H$, $I$, and $J$ are collinear on a line with $GH = HI = IJ = 2$. Points $K$ and $L$ rest on another line, parallel to the first, and distanced such that $KL = 2$. Find the number of distinct possible values for the area of the triangle formed by any three of the six points.


The base of a new rectangle is twice the product of the smaller side and the larger side of a given rectangle, while the altitude of the new rectangle is half the product of the diagonal and the smaller side of the given rectangle. Calculate the area of the new rectangle.

Solution: \boxed{a^2b \sqrt{a^2 + b^2}}


The number of significant digits in the measurement of the side of a square whose computed area is $2.4896$ square inches to the nearest ten-thousandth of a square inch.


Determine the number of lattice points on the line segment whose endpoints are $(15, 35)$ and $(75, 515)$.


Let $n$ be the smallest positive integer such that $n$ is divisible by $30$, $n^2$ is a perfect fourth power, and $n^4$ is a perfect cube. Determine the number of digits of $n$.


Given a box containing 30 red balls, 25 green balls, 20 yellow balls, 15 blue balls, 10 white balls, and 5 black balls, determine the minimum number of balls that must be drawn without replacement to guarantee that at least 18 balls of a single color will be drawn.


Given the hexagons grow by adding subsequent layers of hexagonal bands of dots, with each new layer having a side length equal to the number of the layer, calculate how many dots are in the hexagon that adds the fifth layer, assuming the first hexagon has only 1 dot.


Determine the volume of the original cube given that one dimension is increased by $3$, another is decreased by $2$, and the third is left unchanged, and the volume of the resulting rectangular solid is $6$ more than that of the original cube.

Solution: \boxed{(3 + \sqrt{15})^3}


At Pine Lake Summer Camp, $70\%$ of the children play soccer, $50\%$ of the children swim, and $30\%$ of the soccer players swim. Find the percent of the non-swimmers that play soccer.


For how many integers $x$ does a triangle with side lengths $12, 24$ and $x$ have all its angles acute?


A point is randomly chosen within a square with side length 4040 units. The probability that the point is within d units of a lattice point is 3/5. Calculate d to the nearest tenth.


Given \(1+3+5+\cdots+2023+2025-2-4-6-\cdots-2022-2024\), evaluate the value of the expression.


Given that \(x\) is a perfect square, find an expression that represents the second larger perfect square after \(x\).

Solution: \boxed{x + 4\sqrt{x} + 4}


A gumball machine contains $10$ red, $12$ white, $9$ blue, and $11$ green gumballs. Calculate the least number of gumballs a person must buy to be sure of getting four gumballs of the same color.


Given Chloe chooses a real number uniformly at random from the interval $[0,1000]$ and Laurent chooses a real number uniformly at random from the interval $[0,3000]$, find the probability that Laurent's number is greater than Chloe's number.


An iterative average of the numbers 6, 7, and 8 is computed as follows: Arrange the three numbers in some order. Firstly, find the mean of the first two numbers, and then compute the mean of that result with the third number. Find the difference between the largest and smallest possible values that can be obtained using this procedure.


What is the largest number of solid \(1\text{-in} \times 3\text{-in} \times 2\text{-in}\) blocks that can fit in a \(4\text{-in} \times 3\text{-in} \times 5\text{-in}\) box?


Determine the value of $r$ if the line $x - 2y = r$ is tangent to the parabola $y = x^2 - r$.


Consider a large semicircle with diameter $D$ and $N$ congruent semicircles fitting exactly on its diameter. Let $A$ be the combined area of these $N$ small semicircles, and $B$ be the area of the large semicircle that is not covered by the small semicircles. Given that the ratio $A:B$ is $1:10$, determine the value of $N$.


Given that Jeff, Maria, and Lee paid $90, $150, and $210 respectively, find j - m where Jeff gave Lee $j dollars and Maria gave Lee $m dollars to settle the debts such that everyone paid equally.


A truck travels $\dfrac{2b}{7}$ feet every $2t$ seconds. There are $3$ feet in a yard. Given this information, calculate the distance the truck travels in $4$ minutes.


Given that three different numbers are selected from the set $\{-5, -4, -1, 6, 7\}$, find the largest possible product.


If Shauna takes five tests, each worth a maximum of $120$ points, and her scores on the first three tests are $86$, $112$, and $91$, determine the lowest score she could earn on one of the other two tests in order to average $95$ for all five tests.


Given that two tangents are drawn to a circle from an exterior point $A$, and a third tangent intersects segment $AB$ at point $P$ and segment $AC$ at point $R$, and point $Q$ is exactly midway between points $B$ and $C$ on the circle, where $AB=18$, find the perimeter of $\triangle APR$.


Given that Erin the ant starts at one vertex of a regular tetrahedron and crawls along exactly 3 edges in such a way that she visits every vertex exactly once without revisiting any vertex or repeating any edge, determine the number of distinct paths Erin can take.


Given that distinct points $A$, $B$, $C$, and $D$ lie on a line, with $AB=1$, $BC=2$, and $CD=3$, and points $E$ and $F$ lie on a second line, parallel to the first, with $EF=2$, determine the number of possible values for the area of a triangle with positive area formed from three of the six points.


A merchant acquires a gadget for 30 dollars less 15%. He desires to sell the gadget at a gain of 40% on his cost price after providing a 25% discount on his marked price. What should be the marked price of the gadget in dollars?


Calculate $\left(50 - \left(4050 - 450\right)\right) \times \left(4050 - \left(450 - 50\right)\right)$.


Let $a_1, a_2, \ldots$ and $b_1, b_2, \ldots$ be arithmetic progressions such that $a_1 = 10, b_1 = 90$, and $a_{50} + b_{50} = 200$. Find the sum of the first fifty terms of the progression $a_1 + b_1, a_2 + b_2, \ldots$.


A right-angled isosceles triangle $DBE$ with legs $DB=EB=1$ is cut from an equilateral triangle $ABC$ of side length $4$. Calculate the perimeter of the remaining quadrilateral $ACED$.

Solution: \boxed{10 + \sqrt{2}}


Determine the number of pairs of values from the set {$-3, -2, -1, 0, 1, 2, 3$} such that $b^2 - 4c \geq 0$, where $c$ is non-negative.


Carlos took $60\%$ of a whole pie, and Jessica took one fourth of the remainder. Determine the portion of the whole pie that was left.


A pentagon is inscribed in a circle. Calculate the sum of the angles inscribed in the five arcs cut off by the sides of the pentagon.


The mean of the remaining numbers is 5. How many subsets of two elements can be removed from the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9\}$?


Given that 63 students are wearing blue shirts, 69 students are wearing yellow shirts, and 132 students are assigned into 66 pairs, where 27 of these pairs consist of two blue shirts, determine the number of pairs where both students are wearing yellow shirts.


Given that there are 7-pound rocks worth $20 each, 3-pound rocks worth $10 each, and 2-pound rocks worth $4 each, and Carl can carry at most 21 pounds, find the maximum value, in dollars, of the rocks he can carry out of the situation.


Given Ahn chooses a three-digit integer, subtracts it from 500, subtracts another 50, then doubles the result, calculate the largest number Ahn can obtain.


Given that YZ and XZ form the legs of a right triangle XYZ, where YZ = 3 and XZ = d, and a new right triangle XYW is constructed externally sharing hypotenuse XY, with XW = 4, find the length of YW.

Solution: \boxed{\sqrt{d^2 - 7}}


Suppose that $a$ cows give $b$ gallons of milk in $c$ days during the summer. If the rate of milk production doubles during the winter, calculate the number of gallons of milk that $d$ cows will produce in $e$ days of summer followed by $e$ days of winter.


Given that $0, z,$ and $z^2,$ when represented by points in the complex plane, form an equilateral triangle, determine the number of nonzero complex numbers $z$ that satisfy this condition.


A cube with $4$-inch edges is to be constructed from $64$ smaller cubes with $1$-inch edges. Forty-eight of the cubes are colored blue, and $16$ are colored yellow. If the $4$-inch cube is constructed to have the smallest possible yellow surface area showing, calculate the fraction of the surface area that is yellow.


Seven test scores have a mean of 85, a median of 88, and a mode of 90. Calculate the sum of the three lowest test scores.


Josh writes the numbers $2,4,6,\dots,198,200$. He marks out $2$, skips $4$, marks out $6$ and continues this pattern of skipping one number and marking the next until he reaches the end of the list. He then returns to the beginning and repeats this pattern on the new list of remaining numbers, continuing until only one number remains. What is that number?


A triangular section with side lengths DB=EB=2 is cut from an equilateral triangle ABC with a side length of 5. Calculate the perimeter of the remaining quadrilateral.


Given a circle with endpoints of its diameter at $(-2,0)$ and $(6,4)$, find the x-coordinate of the intersection point on the x-axis.


A lattice point is a point in the plane with integer coordinates. Determine the number of lattice points on the line segment whose endpoints are $(8, 34)$ and $(73, 430)$.


If parallelogram ABCD has area 48 square meters, and E and F are the midpoints of sides AB and CD respectively, and G and H are the midpoints of sides BC and DA respectively, calculate the area of the quadrilateral EFGH in square meters.


A 4x4x4 cube is made of 64 normal dice. Each die's opposite sides sum to 7. Find the smallest possible sum of all the values visible on the 6 faces of the large cube.


Let $2^a \cdot 3^b \cdot 5^c \cdot 11^d = 14850$, and calculate the value of $3a + 2b + 4c + 6d$.


Tom's age is currently $T$ years. The sum of the ages of his four children is $T$. Tom's age $N$ years ago was three times the sum of their ages at that time, find the value of $T/N$.


Given Mr. A owns a home worth $500,000, determine the final net gain or loss for Mr. A, after selling the house to Mr. B at a 15% profit, having Mr. B sell it back at a 5% loss, having Mr. A sell it again at a 10% profit, and having Mr. B sell it back again at a 10% loss.


What is the largest number of solid $3\text{-in} \times 2\text{-in} \times 1\text{-in}$ blocks that can fit in a $4\text{-in} \times 3\text{-in}\times3\text{-in}$ box?


Given that the first container is $\tfrac{3}{5}$ full of water, and the second container is $\tfrac{1}{4}$ full, after transferring the water, the second container becomes $\tfrac{4}{5}$ full. Find the ratio of the volume of the first container to the volume of the second container.


Divide the polynomial $x^4 + x^3 - 4x + 1$ by $x^3 - 1$. Find the remainder.

Solution: \boxed{-3x + 2}


It takes Clea 90 seconds to walk up an escalator when it is not moving, and 45 seconds when it is moving up. Determine the time it would take Clea to ride the escalator up when she is not walking.


Solution: \boxed{3^{128} - 2^{128}}


The centers of two circles are $50$ inches apart. One circle has a radius of $7$ inches, while the other has a radius of $10$ inches. Calculate the length of the common internal tangent between these two circles.

Solution: \boxed{\sqrt{2211}\text{ inches}}


On hypotenuse AB of a right triangle ABC, a second right triangle ABD is constructed with hypotenuse AB. If BC = 3, AC = a, and AD = 1, calculate the length of BD.

Solution: \boxed{x = \sqrt{a^2 + 8}}


The new price of a product must be increased by what percentage to restore it to its original price after a 15% reduction.


A wooden prism with dimensions 4 units, 5 units, and 6 units is painted blue all over, then cut into unit-sized blocks. Determine the total number of blocks that either have exactly one face painted or are completely unpainted.


Given that the six digits are $1, 2, 3, 4, 5, 6$, and there are six boxes in the addition problem, find the largest sum of two $3$-digit numbers that can be formed by placing each of these digits in the boxes.


A square with side length $2$ has an isosceles right triangle placed at each of its corners, with the legs of each triangle lying along the sides of the square. The sum of the areas of the four isosceles right triangles is equal to the area of the square. Determine the length of the hypotenuse of one of the isosceles right triangles.


Determine the number of solutions for x in the equation \[\frac{3x^2 - 15x}{x^2 - 7x + 10} = x - 4.\]


A cell phone plan costs $20 each month, plus $5 cents per text message sent, plus $20 cents for each minute used over 20 hours. Determine the total cost for Michelle in January, where she sent 150 text messages and talked for 22 hours.


How many integer values of $x$ satisfy $|x-1| < 4\pi$?


Let $x=-3000$. Calculate the value of $\bigg|$ $||x| + x| - 2|x|$ $\bigg|$ $+x$.


Given the digits $1, 3, 7, 8, 9$, find the smallest difference that can be achieved in the subtraction problem
\[\begin{tabular}[t]{cccc}  & \boxed{} & \boxed{} & \boxed{} \\ - & & \boxed{} & \boxed{} \\ \hline \end{tabular}\]


A cube has six faces numbered with consecutive integers. The sum of the numbers on each pair of opposite faces is the same. If the smallest number is 15, calculate the sum of the numbers on this cube.


Compute the value of the expression $3 + \sqrt{3} + \frac{1}{3 + \sqrt{3}} + \frac{1}{\sqrt{3} - 3}$.

Solution: \boxed{3 + \frac{2\sqrt{3}}{3}}


Given that EF = 40 units, FG = 30 units, and one diagonal EH = 50 units, calculate the perimeter of parallelogram EFGH.


Cassie leaves Escanaba at 8:30 AM heading for Marquette on her bike at a uniform rate of 14 miles per hour, while Brian leaves Marquette at 9:00 AM heading for Escanaba on his bike at a uniform rate of 14 miles per hour. They both bike on the same 72-mile route between Escanaba and Marquette. Determine the time in the morning when they meet.


Let a sequence $\{v_n\}$ be defined by $v_1=3$ and the relationship $v_{n+1}-v_n=5+6(n-1), n=1,2,3,\dots.$ Express $v_n$ as a polynomial in $n$ and find the algebraic sum of its coefficients.


What is the largest number of solid $1.5\text{-in} \times 2\text{-in} \times 2\text{-in}$ blocks that can fit in a $4\text{-in} \times 6\text{-in} \times 3\text{-in}$ box?


Monica decides to tile the floor of her 15-foot by 20-foot dining room. She plans to create a two-foot-wide border using one-foot by one-foot square tiles around the edges of the room and fill in the rest of the floor with three-foot by three-foot square tiles. Calculate the total number of tiles she will use.


Mary begins peeling a pile of 60 potatoes at the rate of 4 potatoes per minute. After peeling alone for 5 minutes, Tom joins her and peels at the rate of 6 potatoes per minute. Calculate the total number of potatoes peeled by Tom by the time they finish.


While on a long drive, John's odometer initially showed $27372$ miles, a palindrome. After $4$ hours, the odometer showed the next palindrome, determine John's average speed during these $4$ hours.


A circle of radius 7 is inscribed in a rectangle. The ratio of the length of the rectangle to its width is 3:1. Calculate the area of the rectangle.


Given the set $\{31, 47, 58, 62, 89, 132, 164\}$, how many subsets containing three different numbers can be selected so that the sum of the three numbers is even?


Given that the sum of two natural numbers is $22,500$, one of the numbers is divisible by $5$, and if the units digit of that number is erased, the other number is obtained, find the difference between these two numbers.


A rectangular playground that is 12 feet wide and 25 feet long is tiled with 300 two-foot square tiles. A bug starts walking from one corner of the rectangle to the diagonally opposite corner in a straight line. Including both the starting and ending tiles, calculate the number of tiles the bug visits.


Let $ABCD$ be a rectangle where $AB = 2 \times AD$. Point $E$ is the midpoint of side $\overline{AB}$, and $\overline{DE}$ meets diagonal $\overline{AC}$ at point $F$. The area of quadrilateral $BFED$ is $50$. Calculate the area of rectangle $ABCD$.


Calculate the number of terms with rational coefficients in the expansion of $(x\sqrt[4]{2} + y\sqrt{5})^{1200}$.


The acute angles of a right triangle are $a^\circ$ and $b^\circ$, where $a>b$ and both $a$ and $b$ are Fibonacci numbers. Find the least possible value of $b$.


A 4x4x4 cube is made of 64 small dice. Each die has the properties that opposite sides sum to 7. Calculate the smallest possible sum of all of the values visible on the 6 faces of the large cube.


Given the arithmetic expression of the first $2n$ perfect squares, calculate its mean.


Solution: \boxed{5^{64} + 3^{5^{16}} + 4^{3^{125}}}


In a class, $3/7$ of the students are wearing glasses, and $5/6$ of the students are wearing scarves. Find the minimum number of students in the class wearing both glasses and a scarf.


Given that $\triangle XYZ$ is right-angled at $Y$ and $XW = WZ$, $XY = YW$, calculate the magnitude of $\angle XZY$.


Given a circle of radius 6 inches, a half-sector of the circle along with its interior is rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown, find the volume of the cone in cubic inches.

Solution: \boxed{9\pi \sqrt{3}}


If a number five times as large as $x$ is increased by three, and then one is subtracted from the result, calculate one fourth of this final value.

Solution: \boxed{\frac{5x}{4} + \frac{1}{2}}


Given the expression $b^5 - b^{-5}$, express it using a similar type of factorization.

Solution: \boxed{\left(b-\frac{1}{b}\right)\left(b^4 + b^2 + 1 + \frac{1}{b^2} + \frac{1}{b^4}\right)}


Given the curve defined by:
\[ f(x) =
  \begin{cases}
   x & \text{for } 0 \le x \le 4, \\
   3x - 8 & \text{for } 4 \le x \le 7, \\
   x^2 - 14x + 49 & \text{for } 7 \le x \le 10
  \end{cases}
\]
find the area bounded by the $x$-axis, the line $x=10$, and the curve.


Given that point B is the vertex of three angles: $\angle \text{ABC}$, $\angle \text{ABD}$, and $\angle \text{CBD}$, where $\angle \text{CBD}$ is a right angle and the sum of the angles around point B is $270^\circ$, and $\angle \text{ABD} = 110^\circ$, calculate the measure of $\angle \text{ABC}$.


Given two or more consecutive positive integers with a sum of 30, determine the number of sets.


Given that the positive factors of $90$ are to be considered, find the probability that a randomly drawn factor is less than $10$.


Determine the number of days required for $C$ to do the job alone, given that $A$ and $B$ together can complete a job in $3$ days, $B$ and $C$ can do it in $3$ days, and $A$ and $C$ can complete it in $1.5$ days.


Given that seven circles are tangent to each other consecutively and to two parallel lines $L_1$ and $L_2$, if the radius of the largest circle is 24 and the radius of the smallest circle is 6, find the radius of the fourth circle.


Last week, a bundle containing one can of soup and one loaf of bread cost $7.50 for 3 cans of soup and $5 for 2 loaves of bread, so calculate the price per item for each good. Then, this week, a bundle containing one can of soup and one loaf of bread costs $8 for 4 cans of soup and $6 for 3 loaves of bread, so calculate the price per item for each good. Calculate the percent decrease in price per item on average for a bundle containing one can of soup and one loaf of bread.


Estimate the population of Island X in the year 2045, given that the population doubles every 15 years and the population in 2020 was 500.


The first two terms of a geometric progression are $4^{1/2}$ and $4^{1/3}$. The exponent in each subsequent term decreases by $1/6$ for the first two steps and then by $1/12$ thereafter. What is the fifth term of this geometric progression?


Given a number $m$ is randomly selected from the set $\{11,13,15,17,19\}$, and a number $n$ is randomly selected from $\{2000,2001,\ldots,2009\}$, calculate the probability that $m^n$ has a units digit of $1$.


In the spring of 2020, a community garden had 1200 volunteers. Each year, the number of volunteers is expected to increase by 30%. However, due to an unexpected road construction near the garden in 2021, the increase for that year drops by half. Calculate the expected number of volunteers by spring 2022.


Given Lucy starts with an initial term of 8 in her sequence, where each subsequent term is generated by either doubling the previous term and subtracting 2 if a coin lands on heads, or halving the previous term and subtracting 2 if a coin lands on tails, determine the probability that the fourth term in Lucy's sequence is an integer.


A retailer buys a product for $30, less a discount of 15%. He aims to sell the product at a profit of 40% based on his cost after providing a 25% discount on the marked price. What should be the marked price of the product in dollars?


Evaluate the sum of the infinite series $\frac{2}{5} + \frac{3}{5^2} + \frac{4}{5^3} + \dots$, where the $n$th term is given by $\frac{n+1}{5^n}$.


If the area of $\triangle ABC$ is $81$ square units and the geometric mean between sides $AB$ and $AC$ is $15$ inches, calculate $\sin A$.


Given an equilateral triangle $DEF$ with $DE = DF = EF = 8$ units and a circle with radius $4$ units tangent to line $DE$ at $E$ and line $DF$ at $F$, calculate the area of the circle passing through vertices $D$, $E$, and $F$.


Let $g(x) = 2 - 3x^2$ and $f(g(x)) = \frac{2-3x^2}{2x^2}$ when $x \neq 0$. Evaluate the expression $f(-1)$.


Given the sequence $\{v_n\}$ defined by $v_1 = 7$ and $v_{n+1} - v_n = 5 + 6(n-1)$ for $n=1,2,3,\dots$, express $v_n$ as a polynomial in $n$ and calculate the algebraic sum of its coefficients.


Solution: \boxed{3 \log_{10} 2 + 3 \log_{10} 3}


Consider the diagram where $ABCD$ and $DEFG$ are squares, and $CDE$ is a right triangle with $\angle CDE = 90^\circ$. Determine the measure of $\angle GDA$.


Given that Marla has a large white cube with an edge of 12 feet and enough green paint to cover 432 square feet, find the area of one of the white circles created on each face, surrounded by a green border.


Consider the expression $\left(\frac{(x+2)^3(x^2-2x+4)^3}{(x^3+8)^3}\right)^4 \cdot \left(\frac{(x-2)^3(x^2+2x+4)^3}{(x^3-8)^3}\right)^4$. Simplify this expression.


How many ordered pairs (m, n) of positive integers, with m ≥ n, have the property that their squares differ by 72?


Simplify the expression \( 1 - \frac{1}{1 + \frac{x}{x^2 - 1}} \) and find its simplest form.

Solution: \boxed{\frac{x}{x^2 + x - 1}}


The base of isosceles $\triangle XYZ$ is 30 units and its area is 60 square units.


Let $k > 1$, $\sin x > 0$, $\cos x > 0$, and $\log_k \csc x = d$, where $\csc x = \frac{1}{\sin x}$. Determine $\log_k \sec x$, where $\sec x = \frac{1}{\cos x}$.

Solution: \boxed{-\frac{1}{2} \log_k \left(1 - k^{-2d}\right)}


Given that a scale inaccurately measures weights below 50 pounds or over 200 pounds, and the weights of three parcels, obtained by weighing them in pairs, are 132 pounds, 145 pounds, and 150 pounds, calculate the total weight of the three parcels.


The taxi fare in Riverdale City is $3.00 for the first 0.75 mile and additional mileage charged at the rate of $0.25 for each additional 0.1 mile. You plan to give the driver a $3 tip. Calculate the total distance you can ride for $15.


Consider distinct nonzero real numbers \( x \) and \( y \) such that \( x + \frac{3}{x} = y + \frac{3}{y} \). Find the product \( xy \).


Mark places six ounces of tea into a ten-ounce cup and eight ounces of milk into a twelve-ounce cup. He then pours one-third of the tea from the first cup into the second cup, stirs thoroughly, and then transfers one-fourth of the mixed liquid from the second cup back to the first cup. What fraction of the liquid in the first cup is now milk?
A) $\frac{1}{15}$
B) $\frac{3}{13}$
C) $\frac{2}{7}$
D) $\frac{1}{6}$
E) $\frac{2}{6.5}$


Evaluate the expression $-20 + 15 \times (4^{\div -1} \times 2)$.


Points P and Q lie in a plane with PQ = 10. Determine the number of locations for point R in this plane such that the triangle with vertices P, Q, and R is a right triangle with an area of 15 square units.


Jo climbs a flight of 8 stairs every day but is never allowed to take a 3-step when on any even-numbered step. Jo can take the stairs 1, 2, or 3 steps at a time, if permissible, under the new restriction. Find the number of ways Jo can climb these eight stairs.


Given square PQRS with side 10 feet. A circle is drawn through vertices P and S and tangent to side PQ. Find the radius of this circle, in feet.


Integers less than $4010$ but greater than $3000$ have the property that their units digit is the sum of the other digits and also the full number is divisible by 3. How many such integers exist?


A vehicle with six tires (including two full-sized spares) travels a total of 48,000 miles, and tires are rotated such that each tire is used for an equal amount of miles with only four tires being used at any one time. Calculate the number of miles each tire was used.


Given Chelsea leads by 60 points halfway through a 120-shot archery tournament, scores at least 5 points per shot, and scores at least 10 points for each of her next n shots, determine the minimum number of shots, n, she must get as bullseyes to guarantee her victory.


Team $A$ and team $B$ are playing in a basketball league where each game results in a win for one team and a loss for the other team. Team $A$ has won $\tfrac{2}{3}$ of its games and team $B$ has won $\tfrac{3}{5}$ of its games. Also, team $B$ has won $6$ more games and lost $6$ more games than team $A.$ Determine the number of games that team $A$ has played.


Shauna takes six tests, each worth a maximum of 100 points. Her scores on the first three tests are 82, 90, and 88. In order to average 85 for all six tests, and knowing that she wants one of her remaining tests to be between 70 and 85 points, determine the lowest score she could earn on one of the other tests.


Given that $3/7$ of the students are using laptops and $5/6$ of the students are using tablets in the classroom, calculate the minimum number of students in the classroom using both a laptop and a tablet.


Given a medium-sized bottle of shampoo holds 80 milliliters and a very large bottle holds 1200 milliliters, determine the minimum number of medium-sized bottles needed to fill the very large bottle and have no more than 5 additional bottles remaining.


Letters $A, B, C,$ and $D$ represent four different digits selected from $0,1,2,\ldots,9.$ If $(A+B)/(C+D)$ is a nonzero integer minimally achievable, determine the value of $A+B$.


A 4x4x4 cube is made of $64$ normal dice. Each die's opposite sides sum to $7$. Calculate the smallest possible sum of all of the values visible on the $6$ faces of the large cube.


If a total distance of $25$ is covered in seven steps, find the number $z$ that you reach after taking four steps from $0$.


If $n = \frac{2abc}{c-a}$, solve for $c$.

Solution: \boxed{\frac{na}{n - 2ab}}


Compute the number of handshakes at a meeting of 40 people, where 25 individuals are acquainted with each other and 15 are unfamiliar.


Given that one fifth of Ellie's erasers are pink, one sixth of them are orange, and ten of them are purple, determine the smallest number of white erasers that Ellie could have.


Find the least positive integer $n$ for which $\frac{n-15}{3n+4}$ is a non-zero reducible fraction.


Given that the age of Jack and Bill are two-digit numbers with digits reversed, and In ten years, Jack will be three times as old as Bill will be then, find the difference in their current ages.


Set $A$ has 25 elements, and set $B$ has 18 elements. Assume at least 10 elements of $B$ are not in $A$. Determine the smallest possible number of elements in the union of sets $A$ and $B$.


Logan decides to create a more elaborate scaled model of a different town's water tower. This tower is 80 meters high and the top portion is a sphere holding 200,000 liters of water. Logan's new miniature models holds 0.2 liters. Determine the height, in meters, of the new model tower.


Given points P(-2,-3) and Q(5,3) in the xy-plane; point R(x,m) is such that x=2 and PR+RQ is a minimum. Find m.


Find the sum of the squares of all real numbers satisfying the equation $x^{512} - 16^{128} = 0$.


Two numbers have their difference, their sum, and their product in the ratio of $1 : 8 : 15$. If $xy = 15(x-y)$ and $x+y=8(x-y)$, find the product of these two numbers.


In an isosceles right triangle $PQR$, with $PQ=PR=10$ and angle $QPR$ being a right angle, a circle is inscribed. Find the radius of this circle.

Solution: \boxed{10 - 5\sqrt{2}}


Eight spheres, each of radius $2$, are placed one per octant, and are each tangent to the coordinate planes. Determine the radius of the smallest sphere, centered at the origin, that contains these eight spheres.

Solution: \boxed{2\sqrt{3} + 2}


If the digit 5 is placed after a two-digit number whose tens' digit is t, and units' digit is u, determine the mathematical expression for the new number.

Solution: \boxed{100t + 10u + 5}


Cagney can frost a cupcake every 15 seconds, Lacey can frost a cupcake every 25 seconds, and Hardy can frost a cupcake every 50 seconds. Calculate the number of cupcakes that Cagney, Lacey, and Hardy can frost together in 6 minutes.


What is the greatest number of consecutive integers whose sum is $50$?


Given that Sofia has a $5 \times 7$ index card, if she shortens the length of one side by $2$ inches and the card has an area of $21$ square inches, find the area of the card in square inches if instead she shortens the length of the other side by $1$ inch.


The base of a triangle is 18 inches. A line is drawn parallel to the base, terminating in the other two sides, and dividing the triangle into two equal areas. Determine the length of the parallel line.


Given the coordinates of $A$, $B$, and $C$ are $(4,6)$, $(3,0)$, and $(k,0)$ respectively, find the value of $k$ that makes $\overline{AC} + \overline{BC}$ as small as possible.


Given the digits $4$, $4$, $4$, $7$, $7$, $1$, $1$, determine the number of $7$-digit palindromes that can be formed.


Calculate the sum of the numerical coefficients in the expansion of the binomial \((a-b)^8\).


A rope of length 1 meter is randomly cut into two pieces. Find the probability that the length of the longer piece is at most 3 times the length of the shorter piece.


Three flower beds overlap as described. Bed A contains 600 plants, bed B contains 550 plants, and bed C contains 400 plants. Beds A and B share 60 plants, beds A and C share 110 plants, and beds B and C share 90 plants, with 30 plants being common to all three beds. Calculate the total number of unique plants across all three beds.


Four unit squares are arranged to form a larger square. If $A$, $B$, and $C$ are vertices such that $A$ is at the top left corner of the top left square, $B$ is at the top right corner of the top right square, and $C$ is at the bottom right corner of the bottom right square, calculate the area of $\triangle ABC$.


A flower bouquet contains pink roses, red roses, pink carnations, red carnations, and yellow tulips. Two fifths of the pink flowers are roses, half of the red flowers are carnations, and red and pink flowers each make up forty percent of the bouquet. Yellow flowers make up the remaining twenty percent. Calculate the percentage of the flowers that are carnations.


Given the areas $A$ and $B$ of two triangles with side lengths $20, 20, 24$ and $20, 20, 32$ respectively, determine the relationship between $A$ and $B$.

Solution:
\boxed{1. Calculate the area of the first triangle using Heron's formula. Heron's formula: $A = \sqrt{s(s - a)(s - b)(s - c)}$. The semiperimeter is $s = \frac{20 + 20 + 24}{2} = 32$. Area $A$:
     A = \sqrt{32(32 - 20)(32 - 20)(32 - 24)} = \sqrt{32 \times 12 \times 12 \times 8}
     Further calculation:
     A = \sqrt{32 \times 144 \times 8} = \sqrt{36864} = 192

2. Calculate the area of the second triangle using Heron's formula. The semiperimeter is $s = \frac{20 + 20 + 32}{2} = 36$. Area $B$:
     B = \sqrt{36(36 - 20)(36 - 20)(36 - 32)} = \sqrt{36 \times 16 \times 16 \times 4}
     Further calculation:
     B = \sqrt{36 \times 256 \times 4} = \sqrt{36864} = 192

3. Compare the areas $A$ and $B$: Both $A$ and $B$ are 192, so: A = B

Conclusion: A = B}


Originally, Sam had enough paint to decorate 40 rooms, but after a mishap, he lost four cans of paint, leaving him with enough paint for only 30 rooms. How many cans of paint did he use for the 30 rooms?


Given a box contains a total of 180 marbles, 25% are silver, 20% are gold, 15% are bronze, 10% are sapphire, and 10% are ruby, and the remainder are diamond marbles. If 10% of the gold marbles are removed, calculate the number of marbles left in the box.


Given that the Green Park Middle School chess team consists of three boys and four girls, and a girl at each end and the three boys and one girl alternating in the middle, determine the number of possible arrangements.


A rectangular grazing area is fenced on three sides using part of a 150 meter rock wall as the fourth side. The area needs to be 50 m by 70 m. Calculate the fewest number of fence posts required.


Given a trapezoid $ABCD$, where $AB$ and $CD$ are the bases, with $AB = 40$ units and $CD = 70$ units, and the non-parallel sides $AD$ and $BC$ are equal in length. If the height of the trapezoid from $D$ to base $AB$ is $24$ units, calculate the perimeter of trapezoid $ABCD$.

Solution: \boxed{110 + 2\sqrt{801}}


Five friends do yardwork for their neighbors over the weekend, earning $12, $18, $24, $30, and $45 respectively. Calculate the amount the friend who earned $45 will give to the others.


A month with $30$ days has the same number of Tuesdays and Fridays. How many of the seven days of the week could be the first day of this month?


Fox doubles his money every time he crosses the bridge by Trickster Rabbit's house and pays a $50 coin toll after each crossing. If after the fourth crossing, Fox has lost all his money, determine the initial amount of coins Fox had.


A biased coin lands heads with a probability of $\frac{3}{4}$ and tails with a probability of $\frac{1}{4}$. Each toss is independent. For Game A, calculate the probability that the player wins by tossing the coin three times and obtaining all the same outcome, and for Game C, calculate the probability that the player wins by tossing the coin four times and obtaining all the same outcome.


Given the expression $3 \left(\frac{x + 3}{x - 2}\right)$, replace each $x$ with $\frac{x + 3}{x - 2}$ and evaluate the resulting expression when $x=3$.


In a chemistry exam, 15% of the students scored 65 points, 40% scored 75 points, 20% scored 85 points, and the rest scored 95 points. Calculate the difference between the mean and the median score of the students' scores on this exam.


The line $20x + 3y = 60$ forms a triangle with the coordinate axes, calculate the sum of the lengths of the altitudes of this triangle.

Solution: \boxed{23 + \frac{60}{\sqrt{409}}}


Consider a simplified grid of size $2 \times 2$. Frieda the frog starts in the top-left corner and hops randomly choosing one of three possible movements with equal probability: moving right, moving down, or staying in the same cell (no diagonal moves). She must make exactly three hops. Calculate the probability that she ends up in the same corner where she started.


Given the integers $k$ such that $2 \le k \le 13$, find the difference between the two smallest integers greater than $1$ which, when divided by any $k$, leave a remainder of $1$.


If a number is selected at random from the set of all five-digit numbers in which the sum of the digits is equal to 44, find the probability that this number will be divisible by 5.


Given $x, y$ and $3x + \frac{y}{3}$ are not zero, evaluate $\left( 3x + \frac{y}{3} \right)^{-1} \left[(3x)^{-1} + \left( \frac{y}{3} \right)^{-1} \right]$.


Find a value of \( x \) that satisfies the equation \( x^2 + 4b^2 = (2a - x)^2 \).

Solution: \boxed{\frac{a^2 - b^2}{a}}


Two integers have a sum of $28$. When two more integers are added to the first two, the sum is $45$. Finally, when two more integers are added to the sum of the previous $4$ integers, the sum is $60$. Determine the minimum number of odd integers among the $6$ integers.


Let $M = 24 \cdot 36 \cdot 49 \cdot 125$. Find the ratio of the sum of the odd divisors of $M$ to the sum of the even divisors of $M$.


Given that the probability of getting heads in a single flip is \(\frac{1}{3}\), and that four individuals flip coins until they get their first head and stop, calculate the probability that all four individuals flip their coins the same number of times.


Let $x$ be the number of juniors and $y$ be the number of seniors in the class. It is given that $\frac{x}{x+y}=0.2$ and $\frac{y}{x+y}=0.8$. The sum of the scores of the juniors and seniors is $86(x+y)$, while the sum of the scores of the seniors is $85y$. If $J$ represents the score of each junior, then find the value of $J$.


Given that a circle is divided into 15 equal sectors with central angles forming an arithmetic sequence, determine the degree measure of the smallest possible sector angle.


Given that $(w^2+x^2+y^2+z^2)^2 \leq n(w^4 + x^4 + y^4 + z^4)$ for all real numbers $w, x, y, z$, find the smallest integer $n$.


Alicia earns $25 per hour, and $2.5% of her wage is deducted for local taxes. Calculate the number of cents per hour of Alicia's wages used to pay local taxes and her after-tax earnings per hour in cents.


Given that the player scored 18, 15, 13, 17, and 19 points in the first five games, and 14, 20, 12, and 21 points respectively in the subsequent four games, calculate the minimum number of points she must score in the tenth game to have an average points per game greater than 17 after ten games.


How many positive even multiples of $5$ less than $2500$ are perfect squares?


Given triangle $XYZ$ where $XY=30$, $XZ=15$, the area of the triangle is $90$. Let $M$ be the midpoint of $\overline{XY}$, and let $N$ be the midpoint of $\overline{XZ}$. The angle bisector of $\angle XYZ$ intersects $\overline{MN}$ and $\overline{YZ}$ at $P$ and $Q$, respectively. Calculate the area of quadrilateral $MPQY$.


A pair of standard 6-sided dice is rolled once. The sum of the numbers rolled determines one side of a square. What is the probability that the numerical value of the area of the square is less than the perimeter?


In a classroom, $20\%$ of the students are juniors and the remaining $80\%$ are seniors. The combined average score on a test was $80$. All juniors scored the same on the test, while the average score of the seniors was $78$. Find the score received by each junior.


The coefficient of $x^5$ in the expansion of $\left(\frac{x^2}{3}-\frac{3}{x}\right)^7$ is $-\frac{35}{3}$.


Segments $AD = 14$, $BE = 8$, and $CF = 26$ are drawn from the vertices of triangle $ABC$, each perpendicular to a straight line $RS$, not intersecting the triangle. Points $D$, $E$, and $F$ are the intersection points of $RS$ with the perpendiculars. Find the length of the perpendicular segment $GH$ drawn to $RS$ from the intersection point $G$ of the medians of the triangle.


If the product $\dfrac{4}{3}\cdot \dfrac{5}{4}\cdot \dfrac{6}{5}\cdot \dfrac{7}{6}\cdot \ldots\cdot \dfrac{c}{d} = 12$, determine the sum of $c$ and $d$.


Given a rectangular grid constructed with toothpicks of equal length, with a height of 15 toothpicks and a width of 12 toothpicks, calculate the total number of toothpicks required to build the grid.


The highest power of 5 that divides the expression $150! + 151! + 152!$.


A gumball machine contains $12$ purple, $6$ orange, $8$ green, and $5$ yellow gumballs. Determine the minimum number of gumballs a person must purchase to be guaranteed of getting four gumballs of the same color.


The boys and girls must sit alternately, and there are 3 boys. The number of such arrangements is the product of the number of ways to choose 3 positions out of a total of 7, and the number of ways to arrange the girls for the remaining spots.


Andrew writes down one number three times and another number four times. Together, these seven numbers add up to 161. If one of the numbers is 17, calculate the other number.


Given $2^{10}+1$ and $2^{20}+1$, inclusive, find the number of integers that are both cubes and squares.


Given that $x$ is $p$ percent more than $y$, derive an expression for $p$ in terms of $x$ and $y$.

Solution: \boxed{100(\frac{x}{y} - 1)}


If $a,b>0$ and the triangle in the first quadrant bounded by the coordinate axes and the graph of $ax+by=8$ has an area of 8, determine the value of $ab$.


Two spinners are spun once where Spinner A has sectors numbered 1, 4, 6 and Spinner B has sectors numbered 1, 3, 5, 7. What is the probability that the sum of the numbers in the two sectors is an odd number?


Points $B$ and $C$ lie on line segment $\overline{AD}$. The length of $\overline{AB}$ is $3$ times the length of $\overline{BD}$, and the length of $\overline{AC}$ is $7$ times the length of $\overline{CD}$. Determine the fraction of the length of $\overline{AD}$ that is the length of $\overline{BC}$.


Given a pan of brownies measuring $24$ inches by $15$ inches cut into pieces that are $3$ inches by $2$ inches, calculate the total number of pieces of brownies the pan contains.


The number of revolutions of a wheel, with fixed center and with an outside diameter of $10$ feet, required to cause a point on the rim to go half a mile is calculated.


Given the expression $2-(-3)-4-(-5)-6-(-7)-8$, evaluate the expression.


A grocer stacks oranges in a pyramid-like stack with a rectangular base of $6$ oranges by $9$ oranges. Each orange above the first level rests in a pocket formed by four oranges below, and the stack completes with a single row of oranges. Calculate the total number of oranges in the stack.


Given two numbers with a difference, a sum, and a product in the ratio $1:8:40$, find the product of these two numbers.


A rectangular prism with dimensions 10 feet, 5 feet, and 3 feet had a cube with dimensions 2 feet by 2 feet by 2 feet removed. The cube was at a corner such that three of its faces contributed to the external surface of the prism. Calculate the increase or decrease in the surface area of the prism after the removal of the cube.


If four times the larger of two numbers is nine times the smaller and the difference between the numbers is 12, find the larger of the two numbers.


The sum to infinity of an infinite geometric series is 10, and the sum of the first three terms is 9. Find the first term of the progression.


If 1 pint of paint is needed to completely coat the surface of a cube 4 ft. on each side, determine the number of pints of paint required to coat 125 similar cubes each 2 ft. on each side.


Given that a boy took a shortcut along the diagonal of a rectangular field and saved a distance equal to $\frac{1}{3}$ the longer side, determine the ratio of the shorter side of the rectangle to the longer side.


Find the value of $x$ if $\log_8 x = 1.75$.


Given that $x$ and $y$ are distinct nonzero real numbers such that $x + \frac{3}{x} = y + \frac{3}{y}$, determine the value of $xy$.


Given a set of $n > 2$ numbers, where two of them are $1 - \frac{1}{n}$ and the remaining $n-2$ are $1$, calculate the arithmetic mean of these $n$ numbers.

Solution: \boxed{1 - \frac{2}{n^2}}


Paul owes Paula $45$ cents and has a pocket full of $5$-cent coins, $10$-cent coins, and $25$-cent coins that he can use to pay her. Find the difference between the largest and smallest number of coins he can use to pay her.


Lucas is constructing a rectangular shelter using one-foot cubical blocks. The shelter measures 14 feet in length, 12 feet in width, and 6 feet in height. The shelter includes a floor and four walls that are all one foot thick, and it also includes a ceiling made of one-foot thick blocks. Calculate the total number of blocks used in the construction of this shelter.


Given $f(x)=\frac{x(x+3)}{2}$, find the expression for $f(x-1)$.

Solution: \boxed{\frac{x^2 + x - 2}{2}}


Given the numbers 4, 5, 6, 7, and 8, compute the largest and smallest possible values that can be obtained from an iterative averaging process, starting by averaging the first three numbers and continuing by adding the next numbers one by one, and calculate the difference between these maximal and minimal values.


How many ways can a student schedule $4$ science courses – physics, chemistry, biology, and earth science – in an $8$-period day if no two science courses can be taken in consecutive periods?


A ferry boat shuttles tourists to an island every hour starting at 10 AM until its last trip, which starts at 2 PM. On the 10 AM trip, there were 85 tourists on the ferry boat, and on each successive trip, the number of tourists was 3 more than on the previous trip. Calculate the total number of tourists the ferry took to the island that day.


Find the number of distinct points in the $xy$-plane where the graphs of $(x+2y-8)(3x-y+6)=0$ and $(2x-3y+2)(4x+y-16)=0$ intersect.


Given the product of the digits of a 3-digit positive integer equals 36, determine the number of such integers.


Find the fifth term in the expansion of \((\frac{a}{x} + \frac{x}{a^3})^8\).


A modified sign pyramid with five levels, where a cell gets a "+" if the two cells below it have the same sign, and it gets a "-" if the two cells below it have different signs. If a "-" is to be at the top of the pyramid, calculate the number of possible ways to fill the five cells in the bottom row.


An iterative process is used to find an average of the numbers -1, 0, 5, 10, and 15. Arrange the five numbers in a certain sequence. Find the average of the first two numbers, then the average of the result with the third number, and so on until the fifth number is included. What is the difference between the largest and smallest possible final results of this iterative average process?


Given the equation $m+n = mn - 1$, determine the number of pairs $(m,n)$ of integers that satisfy this equation.


Ike and Mike enter a bakery with a total of $50.00 to spend. Sandwiches cost $6.00 each and pastries cost $1.50 each. They decide to buy as many sandwiches as they can with their money, then spend the remainder on pastries. Calculate the total number of items (sandwiches and pastries) they will buy.


Given a box with dimensions $4\text{-in} \times 3\text{-in} \times 6\text{-in}$ and blocks with dimensions $3\text{-in} \times 1\text{-in} \times 2\text{-in}$, calculate the maximum number of blocks that can fit in the box.


Given Professor Chang has ten different language books, with two Arabic, four German, and four Spanish, calculate the number of arrangements of the books on the shelf while keeping the Arabic books together and the Spanish books together.


What is the least possible value of the expression (x+1)(x+2)(x+3)(x+4) + 2021 where x is a real number?


If the product $\dfrac{4}{3}\cdot \dfrac{5}{4}\cdot \dfrac{6}{5}\cdot \ldots\cdot \dfrac{a}{b} = 16$, calculate the sum of $a$ and $b$.


Given that A can finish a task in 12 days and B is 75% more efficient than A, determine the number of days it takes B to complete the same task.


A rug is made with three differently colored regions, and their areas form an arithmetic progression. The inner rectangle is two feet wide, and each of the two shaded regions is $2$ feet wide on all four sides. Determine the length in feet of the inner rectangle.


The Red Robin High School chess team consists of three boys and four girls. A photographer wants to take a picture of the team for a school magazine. She decides to have them sit in a row with a boy at each end and the remaining team members in the middle. Determine the number of arrangements possible.


A symposium began at 3:00 p.m. and lasted for 1500 minutes. What time did the symposium end?

Solution: \boxed{4:00 \text{ p.m.}}


Given that in a 60-question multiple choice math contest, students receive 5 points for a correct answer, 0 points for an answer left blank, and -2 points for an incorrect answer, and Evelyn's total score on the contest was 150, determine the maximum number of questions that Evelyn could have answered correctly.


Consider the graphs of $y=3\log{x}$ and $y=\log{3x}$. Determine the value(s) of $x$ where they intersect.


Given the sequence where $x_{k+1} = x_k + \frac{1}{3}$ for $k = 1, 2, \dots, n-1$ and $x_1 = 2$, find the sum $x_1 + x_2 + \dots + x_n$.


Queen Middle School has 1500 students. Each student takes 6 classes a day. Each teacher teaches 5 classes. Each class has 25 students and 1 teacher. Find the number of teachers at Queen Middle School.


A bakery prepares a giant cookie sheet measuring 30 inches by 24 inches. The cookies are intended to be 3 inches by 4 inches each. Calculate the number of full cookies that can be made from the sheet.


Given the ratio of $w$ to $x$ is $4:3$, the ratio of $y$ to $z$ is $5:3$, and the ratio of $z$ to $x$ is $1:6$, determine the ratio of $w$ to $y$.


Let P be the total number of pages in the novel. On the first day, Paul read \(\frac{1}{6}\) of the pages plus 10 more. On the second day, he read \(\frac{1}{5}\) of the remaining pages plus 14 pages, and on the third day, he read \(\frac{1}{4}\) of the remaining pages plus 16 pages. After these readings, exactly 48 pages were left. Determine the total number of pages in the novel.


Given the polynomial expression \( x^4 - 61x^2 + 60 \), for how many integers \( x \) is the expression negative.


Calculate the sum of the numbers in the four corners of a 9x9 checkerboard extended with numbers 10 to 90.


A right rectangular prism whose edge lengths are $\log_{5}x, \log_{6}x,$ and $\log_{7}x$ and whose surface area is twice its volume. Find the value of $x$.


Given a rectangle divided into a 2x4 grid of equally spaced points, calculate the total number of distinct triangles that can be formed using three of these points as vertices.


A month has 30 days in which the number of Mondays and Fridays are the same. How many of the seven days of the week could be the first day of this month?


A set of 36 square blocks is arranged into a 6 × 6 square. Calculate the number of different combinations of 4 blocks that can be selected from that set so that no two are in the same row or column.


Given Zara has 5 marbles: an Aggie, a Bumblebee, a Steelie, a Tiger, and a Cat's Eye, find the number of ways to arrange them in a row on her shelf such that the Steelie and the Tiger are not next to each other and the Aggie and the Bumblebee are not adjacent.


The circle having (0,0) and (10,10) as the endpoints of a diameter intersects the x-axis at a point. Find the x-coordinate of this point.


Given the numbers $1357, 2468, 3579, 4680, 5791$, calculate their sum.


Given points A and B are 12 units apart in a plane, find the number of points C such that the perimeter of triangle ABC is 60 units and the area of triangle ABC is 150 square units.


Find the value of $x$ and $y$ that satisfies the equation: $5^{-6} = \frac{5^{56/x}}{5^{32/y} \cdot 25^{22/x}}$.

Solution: \boxed{x = 3, y = 4}


A store owner buys a piece of equipment at $50$ dollars less 20%. He then wishes to sell the equipment at a gain of 25% of his cost after allowing a 15% discount on his marked price. Determine the price at which the equipment should be marked.


Six points on a circle are numbered 1, 2, 3, 4, 5, and 6 in clockwise order. A bug jumps in a clockwise direction from one point to another around the circle; if it is on an odd-numbered point, it moves two points, and if it is on an even-numbered point, it moves one point. If the bug begins on point 6, after 2023 jumps calculate the point where the bug will be.


A box contains 21 red balls, 17 green balls, 24 yellow balls, 10 blue balls, 14 white balls, and 14 black balls. Find the minimum number of balls that must be drawn without replacement to guarantee that at least 16 balls of a single color will be drawn.


Two cyclists, $C$ and $D$, start at the same time to ride from town X to town Y and town Z respectively, where town Y is $90$ miles away from X and town Z is $120$ miles away. Cyclist $C$ travels $6$ mph slower than cyclist $D$, who reaches town Y and then travels $30$ miles further before meeting cyclist $C$, who is still en route to town Y. Determine the speed of cyclist $C$.


Given that all three vertices of $\bigtriangleup ABC$ lie on the parabola defined by $y = 2x^2$, with $A$ at the origin and $\overline{BC}$ parallel to the $x$-axis, find the length of $BC$ given that the area of the triangle is $128$.


Six different awards are to be given to four students, with each student receiving at least one award, find the total number of different ways the awards can be distributed.


What is the value of $\dfrac{13! - 12!}{10!}$?


Given a wooden cube $n$ units on a side is painted blue on all six faces and then cut into $n^3$ unit cubes, find the value of $n$ if exactly one-third of the total number of faces of the unit cubes are blue.


Two pictures, each 2 feet across, are hung in the center of a wall that is 25 feet wide with 1 foot of space between them. Calculate the distance from the end of the wall to the nearest edge of the first picture.


Three wheels are spun, and each wheel selects one number. Calculate the probability that the sum of the three selected numbers is even, given the first wheel has a probability $\frac{3}{4}$ of landing on an even number, the second wheel has a probability $\frac{1}{2}$, and the third wheel has a probability $\frac{1}{4}$ of landing on an even number.


Given the quadratic equation $y = x^2 + px + r$, where $r = q - \frac{p^2}{4}$, determine the value of $q$ for which the minimum value of $y$ is zero.


Given that Queen Middle School has 1500 students, each student takes 5 classes a day, each teacher teaches 5 classes, and each class has 25 students and 1 teacher, determine the total number of teachers at Queen Middle School.


Find the minimum value of $\sqrt{x^2+y^2}$ if $8x+15y=120$ and $x \geq 0$.


Given the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ written in a $3\times3$ array of squares, with each consecutive pair sharing an edge, and the numbers in the four corners adding up to $20$ and the numbers in the middle row adding up to $15$, calculate the number in the center.


If a box of marbles contains $\frac{2}{3}$ green marbles, while the rest are yellow, and the number of yellow marbles is tripled with the number of green marbles staying the same, what fraction of the marbles will be yellow?


Given $x = 1+3^p$ and $y = 1+3^{-p}$, determine the expression for $y$ in terms of $x$.


Solve the equation where \(\log(3x) - 4 \log 9 = 3\).


Given that Moe needs to mow his larger rectangular lawn measuring $120$ feet by $200$ feet, the swath of the mower remains $28$ inches, but this time he overlaps each pass by $6$ inches to ensure no grass is missed, and Moe walks at the rate of $4000$ feet per hour with the mower, calculate the number of hours it will take Moe to mow the entire lawn.


If five fish are traded for three loaves of bread and two loaves of bread are traded for seven bags of rice, determine the value of one fish in terms of bags of rice.


Anthony's family ordered a 16-slice pizza for dinner. Anthony ate one slice, shared another slice equally with his brother Ben, and shared a third slice equally with his cousin Chris. What fraction of the pizza did Anthony eat?


Find $n$ which represents the number of integer values of $x$ such that $Q = x^4 + 4x^3 + 10x^2 + 4x + 29$ is the square of an integer. Determine $n$.


Given that the two children from the youngest generation receive a $60\%$ discount, the two members of the middle generation pay full price, and the two members of the oldest generation receive a $20\%$ discount as senior citizens, and the senior ticket costs $\$7.50$, calculate the total amount Grandfather must pay for the concert and a $5 handling fee.


When simplified, $\log_{16}{32} \div \log_{16}{\frac{1}{2}}$, calculate the result.


A positive integer $N$ with three digits in its base ten representation is chosen at random, with each three-digit number having an equal chance of being chosen. Calculate the probability that $\log_3 N$ is an integer.


Two integers have a sum of $24$. When two more integers are added to the first two, the sum becomes $39$. Finally, when two more integers are added to the sum of the previous $4$ integers, the sum becomes $58$. What is the minimum number of even integers among the $6$ integers?


Given the graphs of $y = -2|x-a| + b$ and $y = 2|x-c| + d$, where they intersect at points $(5,10)$ and $(11,6)$, find the value of $a+c$.


The $y$-intercepts, $R$ and $S$, of two perpendicular lines intersecting at the point $B(4,10)$ have a sum of zero. Find the area of $\triangle BRS$.


Carlos is constructing a rectangular storage unit using one-foot cubical blocks. The storage unit is 15 feet long, 12 feet wide, and 8 feet high. The walls and the floor are 1.5 feet thick. Calculate the total number of blocks contained in the storage unit.


Given that $a$, $b$, and $c$ are real numbers, solve the equation $4x - 7 + a = 2bx + c$ for a unique solution $x$, under the condition that $c \neq 0$.

Solution: \boxed{x = \frac{c + 7 - a}{4 - 2b}}


Find the integer value nearest to $(\sqrt{5}+\sqrt{3})^4$.


Julia drives to a mountain retreat, covering 120 miles on the interstate, 40 miles on a mountain road, and 5 miles on a dirt track. She drives twice as fast on the interstate as on the mountain road and four times as fast on the interstate as on the dirt track. If Julia spent 60 minutes driving on the interstate, calculate the total time of her entire trip.


What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $15!$?


When a certain unfair die is rolled, an odd number is $4$ times as likely to appear as an even number. The die is rolled twice. Calculate the probability that the sum of the numbers rolled is odd.


A rectangular floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers and $b > a$. A person paints a rectangle on the floor with the sides of the rectangle parallel to the floor, but now leaving a border of width $2$ feet all around. The unpainted border part occupies half of the total area of the floor. Determine the number of possibilities for the ordered pair $(a, b)$.


In triangle $ABC$, point $D$ is on side $BC$ such that $BD:DC = 1:2$. A line through $A$ and $D$ intersects $BC$ at $E$. If the area of triangle $ABE$ is $30$, find the total area of triangle $ABC$.


Each disc can now hold a maximum of 75 minutes, and the narration takes 6 hours and 45 minutes to complete. How many minutes of narration will each disc contain?


Given a list of the lengths of 24 students' names from a classroom, with 9 names of 4 letters each, 6 names of 5 letters, 2 names of 6 letters, and 7 names of 7 letters, determine the median length of these names.


Let $x=-1008$. Evaluate the expression $\Bigg\vert\Big\vert |x| - 2x\Big\vert - |x|\Bigg\vert - x$.


Given that Max visits Sylvia every 5 days, Nora every 6 days, and Olivia every 7 days, and all three visited her yesterday, calculate the number of days in the next 365-day period when exactly two friends visit her.


Given the sequence of even counting numbers starting from $0$, find the sum of the first $1500$ terms. Then, given the sequence of odd counting numbers, find the sum of the first $1500$ terms, and calculate their difference.


What is the smallest positive integer $n$ such that $\sqrt{n} - \sqrt{n-1} < 0.02$?


On a digital watch that displays time in a 24-hour format, with the time shown as hours and minutes, calculate the largest possible sum of the digits in the display.


Given Marissa has 2035 coins, consisting of pennies and nickels, with at least three pennies and at least two nickels, calculate the difference in cents between the greatest and least amounts of money she can have.


Given that Isabella's fort has dimensions $15$ feet in length, $12$ feet in width, and $6$ feet in height, with one-foot thick floor and walls, determine the number of one-foot cubical blocks required to construct this fort.


Doug can paint a room in 4 hours, and Dave can paint the same room in 6 hours. They start painting together but must stop for a 30-minute meeting after which they continue to finish the room. Let $t$ be the total time, in hours, required for them to complete the job working together, including the meeting. Which equation correctly represents this situation?

Solution: \boxed{\left(\frac{5}{12}\right)(t-0.5) = 1}


Given right triangle $\triangle ABC$ has a hypotenuse $AC$ of length 10, where $A$ is at the right angle. Point $M$ is the midpoint of hypotenuse $\overline{AC}$, and $D$ is a point on leg $\overline{AB}$ such that $AD:DB = 1:2$. Determine the area of $\triangle ADM$.


A 20-quart radiator contains 18 quarts of water and 2 quarts of oil. Five quarts of this mixture are removed and replaced with pure antifreeze. This replacement process is done two more times. Calculate the fractional part of the final mixture that is water.


Determine the value of $m$ such that the polynomial $6x^3 - 12x^2 + mx - 24$ is divisible by $x - 4$.


Given that in a bag of marbles, $\frac{1}{3}$ of the marbles are blue, $\frac{1}{3}$ are red, and the rest are yellow, if the number of red marbles is doubled and the number of blue marbles stays the same, determine the fraction of the marbles that will be red.


Given that points $A$ and $B$ are $8$ units apart, find the number of lines in a given plane containing $A$ and $B$ that are $3$ units from $A$ and $4$ units from $B$, and also intersect the line $y = x$ at a $45^\circ$ angle.


December 31, 2020, was a leap day. What time will it be 2023 minutes after midnight?

Solution: \boxed{9:43 \text{ AM}}


Calculate the sum of the numbers $2143 + 3412 + 4213 + 1324$.


Given the distance light travels in one year is approximately $9,460,800,000,000$ kilometers, calculate the distance light travels in $70$ years.

Solution: \boxed{6.62256 \times 10^{14}}


Given a boat with a speed of 20 mph in still water and a stream current of 4 mph, calculate the ratio of the average speed for a round trip downstream and then upstream to the speed in still water.


Evaluate the expression $3 + \sqrt{3} + \frac{1}{3 + \sqrt{3}} + \frac{1}{\sqrt{3} - 3}$.

Solution: \boxed{3 + \frac{2\sqrt{3}}{3}}


An artist begins filling in a large mosaic at 9:00 AM. By 12:45 PM she has completed one-fourth of the mosaic. Find the time when she will finish the entire mosaic.

Solution: \boxed{12:00 \text{ AM next day}}


How many whole numbers are between $\sqrt[3]{50}$ and $\sqrt{200}$?


What is the value of $\dfrac{12!-10!}{8!}$?


A list of $2023$ positive integers has a unique mode, which occurs exactly $11$ times. Determine the least number of distinct values that can occur in the list.


In Marco's first $6$ soccer games, he scored $2, 5, 1, 4, 6,$ and $3$ goals. In his seventh game, he scored fewer than $10$ goals, and his goals-per-game average for these $7$ games was an integer. In his eighth game, he also scored fewer than $10$ goals and the goals-per-game average for the $8$ games became an integer too. Find the product of the number of goals he scored in the seventh and eighth games.


Twelve points are spaced around a $3 \times 3$ square at intervals of one unit. Two of the 12 points are chosen at random. Find the probability that the two points are one unit apart.


A rectangle with a width of 5 meters and a length of 15 meters is cut diagonally from one corner to the opposite corner. Determine the dimensions of one of the resulting right-angled triangles.

Solution: \boxed{5\ \text{m}, 15\ \text{m}, 5\sqrt{10}\ \text{m}}


If the original price of a jacket is $\textdollar 250$, find the final price after consecutive discounts of 40% and 15% and a sales tax of 5%.


A product originally priced at \$120 receives a discount of 8%. Calculate the percentage increase needed to return the reduced price to its original amount.


Given that $\log_{3x}729 = x$, where $x$ is a real number, find the value of $x$.


Given that $x$ varies directly as the square of $y$, and $y$ varies directly as the cube root of $z$, determine the power $n$ such that $x$ varies as $x^n$.


Given eleven books consisting of three Arabic, two English, four Spanish, and two French, calculate the number of ways to arrange the books on the shelf keeping the Arabic books together, the Spanish books together, and the English books together.


A school has $120$ students and $6$ teachers. The class sizes are $40, 35, 25, 10, 5, 5$. Calculate the difference $t - s$ where $t$ is the average number of students a randomly selected teacher sees, and $s$ is the average class size a randomly selected student experiences.


Given the binomial $\left(\frac{x^3}{3} - \frac{1}{x^2}\right)^9$, find the coefficient of $x^3$ in its expansion.


A 4x4 grid is to be filled with four A's, four B's, four C's, and four D's such that each row and column contains one of each letter. If 'A' is placed in the upper left corner, calculate the number of possible arrangements.


If \(\frac{x}{x-1} = \frac{y^3 + 2y - 1}{y^3 + 2y - 3}\), calculate the value of \(x\).

Solution: \boxed{\frac{y^3 + 2y - 1}{2}}


Using the digits 1, 2, 3, 7, 8, and 9, find the smallest sum of two 3-digit numbers that can be obtained by placing each of these digits in the addition problem shown:

____  ____  + ____ = _______


Given that an item is sold for $x$ dollars with a $20\%$ loss based on the cost and $y$ dollars with a $25\%$ profit based on the cost, find the ratio of $y:x$.


Given that a group of students sit around a large circular table and a bag containing $120$ pieces of candy is passed around, with each person taking two candies and the bag starting and ending with Chris, determine the possible number of students at the table.


Given that there are 30 students in the school club with 15 participating in chess, 18 in soccer, and 12 in music, and 14 participating in at least two activities, determine the number of students participating in all three activities.


Given y = x^2 - px + q, and the least possible value of y is 1, determine q.

Solution: \boxed{1 + \frac{p^2}{4}}


Given that the mean weight of the rocks in A is 45 pounds, the mean weight of the rocks in B is 55 pounds, the mean weight of the rocks in the combined piles A and B is 48 pounds, and the mean weight of the rocks in the combined piles A and C is 50 pounds, calculate the greatest possible integer value for the mean in pounds of the rocks in the combined piles B and C.


Given that Route X is 8 miles long with an average speed of 40 miles per hour, and Route Y is 7 miles long with 6.5 miles traveled at 50 miles per hour and a 0.5-mile stretch traveled at 10 miles per hour, determine the difference in time taken between the two routes.


What is the smallest whole number larger than the perimeter of any triangle with sides of length $5$ and $12$?


What is the value of $\dfrac{13! - 12!}{10!}$?


When simplified and expressed with negative exponents, the expression $(x^2 + y^2)^{-1}(x^{-2} + y^{-2})$ equals what value in terms of $x$ and $y$.


Given rocks of three different types are available: 6-pound rocks worth $18 each, 3-pound rocks worth $9 each, and 2-pound rocks worth $3 each, and there are at least 15 rocks available for each type. Carl can carry at most 20 pounds. Determine the maximum value, in dollars, of the rocks Carl can carry out of the cave.


If the first jar has a volume $V$ and an alcohol to water ratio of $2p:3$, and the second jar has a volume $2V$ and an alcohol to water ratio of $q:2$, determine the ratio of the volume of alcohol to the volume of water in the mixture.

Solution: \boxed{\frac{\frac{2p}{2p+3} + \frac{2q}{q+2}}{\frac{3}{2p+3} + \frac{4}{q+2}}}


Jasmine is a contractor who had paint for 50 identically sized rooms. On her route to the project site, she loses 4 cans which reduces her capacity to now cover only 36 rooms. Each can of another type she has adds a capacity of 2 rooms. If she finally manages to cover all 50 rooms using additional cans of another type, determine the number of the second type of cans she used.


In a classroom of 32 students, a majority bought the same number of pens, with each student purchasing more than 1 pen, and the cost of each pen in cents exceeded the number of pens bought by each student. If the total sum spent on the pens was $21.16, determine the cost of one pen in cents.


If a purchase is made at the gadget shop, calculate the final price of an electronic item whose original price is $\textdollar 240$, given a $30\%$ discount before noon, an additional $15\%$ discount on the already reduced price, and a $8\%$ tax applied to the final price.


For how many integers $n$ is $\dfrac{n}{24-n}$ the square of an integer?


A triangle has sides measuring $40$, $90$, and $100$ units. If an altitude is dropped to the side measuring $100$ units, find the length of the longer segment created on this side.


Let $N = 17^3 + 3 \cdot 17^2 + 3 \cdot 17 + 1$. Determine the number of positive integers that are factors of $N$.


Suppose that $f(x+5) = 4x^2 + 9x + 2$ and $f(x) = ax^2 + bx + c$. Calculate the value of $a+b+c$.


Find the Greatest Common Divisor (GCD) of 7254 and 156, then reduce the result by 10.


Let a sequence $\{v_n\}$ be defined by $v_1 = 7$ and the relationship $v_{n+1} - v_n = 2 + 5(n-1), n = 1, 2, 3, \dots$. If $v_n$ is expressed as a polynomial in $n$, find the algebraic sum of its coefficients.


Given the expression $n^3 - n + 2$, determine the largest number by which this expression is divisible for all possible integral values of $n$.


One dimension of a cube is tripled, another is decreased by `a/2`, and the third dimension remains unchanged. The volume gap between the new solid and the original cube is equal to `2a^2`. Calculate the volume of the original cube.


If the sum of all the angles except two of a convex polygon is $2340^{\circ}$, calculate the number of sides of the polygon.


In a tournament, there are eight teams that play each other twice. A team earns 3 points for a win, 1 point for a draw, and 0 points for a loss. Furthermore, the top four teams earned the same number of total points. Calculate the greatest possible number of total points for each of the top four teams.


Calculate Mr. $X$'s net gain or loss from the transactions, given that he sells his home valued at $12,000$ to Mr. $Y$ for a $20\%$ profit and then buys it back from Mr. $Y$ at a $15\%$ loss.


Given that Josanna has six test scores: $95, 85, 75, 65, 90, 70$, and she wants to achieve an exact average of $5$ points higher than her current average with her next test, determine the score she needs on her seventh test.


Let $x$ and $y$ be two-digit integers such that $y$ is obtained by reversing the digits of $x$. The integers $x$ and $y$ satisfy the equation $x^2 - y^2 = 9m^2$ for some positive integer $m$. Determine $x + y + 2m$.


Given that the speed of sound is 1100 feet per second and a wind is slowing down the sound by 60 feet per second, and considering the time elapsed between Charlie Brown seeing the lightning and hearing the thunder is 15 seconds, determine the distance at which Charlie Brown was from the lightning, rounded off to the nearest half-mile.


Calculate $\log 216$ and determine its value in terms of simpler logarithmic expressions.

Solution: \boxed{3(\log 2 + \log 3)}


A cylindrical water tank, placed horizontally, has an interior length of 15 feet and an interior diameter of 8 feet. If the surface area of the water exposed is 60 square feet, find the depth of the water in the tank.

Solution: \boxed{4 - 2\sqrt{3}}