Given that a quadratic equation of the form $x^2 + bx + c = 0$ has real roots, where $b$ is chosen from the set $\{-6, -5, -4, -3, -2, -1, 1, 2, 3, 4, 5, 6\}$ and $c$ from the set $\{1, 2, 3, 4, 5, 6\}$, determine the number of such equations.


Let Martha the moose take 50 equal steps between consecutive lamp posts on a city street. Let Percy the pronghorn cover the same distance in 15 equal bounds. The lamp posts are evenly spaced, and the 51st post is exactly 2 miles or 10560 feet from the first post. Calculate the difference in length between Percy's bound and Martha's step.


How many whole numbers between $200$ and $500$ contain the digit $2$?


Given the sequence defined by $g(n) = \dfrac{2 + \sqrt{2}}{4}\left(1 + \sqrt{2}\right)^n + \dfrac{2 - \sqrt{2}}{4}\left(1 - \sqrt{2}\right)^n$, determine $g(n+1) - g(n-1)$ in terms of $g(n)$.


In the republic of Midland, statisticians estimate there is a baby born every $6$ hours and a death every $2$ days. Additionally, an average of $1$ person migrates into Midland every day. Calculate the net change in population of Midland per year, rounded to the nearest ten.


Given the function \( f \) such that \( f\left(\frac{x}{4}\right) = x^2 - x + 2 \), find the sum of all values of \( z \) for which \( f(4z) = 8 \).


A ferry boat transports visitors to a resort starting at 9 AM until the final ride at 5 PM, on an hourly schedule. On the initial 9 AM trip, there are 120 visitors and with each subsequent trip, the number of visitors decreases by 2. Determine the total number of visitors carried to the resort that particular day.


Given that eight students from Dover High worked for 4 days, five students from Edison High worked for 7 days, and six students from Franklin High worked for 6 days, and a total of 884 dollars was paid to the students, determine the total amount earned by the students from Edison High.

Solution: \boxed{300.39\ \text{dollars}}


The pan containing 24-inch by 15-inch brownies is cut into pieces that measure 3 inches by 2 inches. Calculate the total number of pieces of brownies the pan contains.


Given the equation $\frac{a+b^{-1}}{a^{-1}+b} = 9$, find the number of pairs of positive integers $(a, b)$ satisfying $a+b \leq 50$.


Seven positive consecutive integers start with $a$. Find the average of $7$ consecutive integers that start with the average of the original seven integers.

Solution: \boxed{a + 6}


Given that \(a\) is a positive real number and \(b\) is an integer between \(3\) and \(300\), inclusive, find the number of ordered pairs \((a, b)\) that satisfy the equation \((\log_b a)^{101} = \log_b(a^{101})\).


How many whole numbers are between $\sqrt{50}$ and $\sqrt{200}$


Find the number of dimes you must have given that you have eleven coins in total, a collection of pennies, nickels, dimes, and quarters totaling $1.18, and at least one coin of each type is present.


Given two positive numbers $x$ and $y$ in the ratio $a: b^2$, where $0 < a < b$. If $x+y = 2c$, determine the smaller of $x$ and $y$.

Solution: \boxed{\frac{2ac}{a + b^2}}


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

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


A half-sector of a circle with a radius of $6$ inches is rolled up to form the lateral surface area of a right circular cone by joining along the two radii. Determine the volume of the cone in cubic inches.

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


Given the equation $x^2 + px + 7 = 0$, determine the relationship between the absolute value of the sum of its distinct real roots $r_1$ and $r_2$.

Solution: \boxed{|r_1 + r_2| > 2\sqrt{7}}


Calculate the area of the polygon with vertices at $(2,1)$, $(4,3)$, $(6,1)$, $(4,-2)$, and $(3,4)$.


John drove 150 miles in 120 minutes. His average speed during the first 40 minutes was 70 mph, and for the next 40 minutes, he traveled at 75 mph. Determine the average speed during the last 40 minutes, in mph.


Given $A$ can finish a task in $12$ days, $B$ is $20\%$ more efficient than $A$, and $C$ works twice as efficiently as $A$, calculate how many days it will take for $B$ and $C$ together to complete the task.


Given that $x, y$ and $3x - \frac{y}{3}$ are not zero, evaluate the expression
\[
\left( 3x - \frac{y}{3} \right)^{-1} \left[ (3x)^{-1} + \left( \frac{y}{3} \right)^{-1} \right]^2
\]

Solution: \boxed{\frac{(y + 9x)^2}{3x^2y^2(9x - y)}}


Given that point $P$ moves perpendicularly to side $AB$ in $\triangle PAB$, $M$ and $N$ are the midpoints of sides $PA$ and $PB$ respectively, determine how many of the four quantities listed below change as $P$ is moved: the length of segment $MN$, the perimeter of the triangle $PAB$, the area of $\triangle PAB$, the area of trapezoid $ABNM$.


In Mr. Calculation's class, a majority of the 36 students purchased notebooks from the school store. Each of these students bought the same number of notebooks, and each notebook cost more in cents than the number of notebooks bought by each student. The total expenditure for all notebooks was 2275 cents. Determine the cost per notebook in cents.


A circle is inscribed in a triangle with side lengths $10, 15, 19$. Let the segments of the side of length $10$, made by a point of tangency, be $r'$ and $s'$, with $r'<s'$. Find the ratio $r':s'$.


Given Alex had 40 hits with 2 home runs, 3 triples, and 6 doubles, determine the percentage of his hits that were singles.


Given a student travels from her university to her hometown, a distance of 150 miles, in a sedan that averages 25 miles per gallon, and for the return trip she uses a hybrid car averaging 50 miles per gallon due to a different route of 180 miles, calculate the average gas mileage for the entire trip.


Given a box initially containing 3 red, 3 green, and 3 yellow marbles, determine the probability that Cheryl takes 3 marbles of the same color after Carol takes 3 marbles at random from the box and then Cheryl takes the last 3 marbles.


Given that $\frac{2}{3}$ of the marbles are blue and the rest are red, determine the fraction of marbles that will be red after the number of blue marbles is tripled and the number of red marbles stays the same.


A bug starts crawling on a number line at point $3$. It first moves left to $-4$, then turns around and crawls right to $8$. Calculate the total distance the bug crawls.


Given that $(n+1)! + (n+3)! = n! \cdot 1320$, calculate the sum of the positive integer $n$.


A right triangle has one angle measuring $30^\circ$. This triangle shares its hypotenuse with a second triangle that is also right-angled. The two triangles together form a quadrilateral. If the other acute angle in the second triangle is $45^\circ$, find the area of the quadrilateral given that the hypotenuse common to both triangles is 10 units long.

Solution: \boxed{\frac{25\sqrt{3} + 50}{2}}


Given a box contains $4$ shiny pennies and $5$ dull pennies, determine the probability that the third shiny penny appears on the sixth draw.


Given the circle described by $x^2 + y^2 = m$ and the line $x - y = \sqrt{m}$, determine the value of $m$ such that the circle is tangent to the line.


Let $\clubsuit(x)$ denote the sum of the digits of a positive integer $x$. Find the number of two-digit values of $x$ for which $\clubsuit(\clubsuit(x))=4$.


The total price of the microwave from the in-store price is $\textdollar 149.95$, and the microwave from the television commercial is paid in five easy payments of $\textdollar 27.99$ and includes a shipping and handling charge of $\textdollar 14.95$ and an extended warranty fee of $\textdollar 5.95$. Calculate the total amount saved in cents.


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


Given the expression $3000(3000^{1500} + 3000^{1500})$, evaluate this expression.

Solution: \boxed{2 \cdot 3000^{1501}}


Joe has a rectangular lawn measuring 120 feet by 180 feet. His lawn mower has a cutting swath of 30 inches, and he overlaps each cut by 6 inches to ensure no grass is missed. Joe mows at a rate of 4000 feet per hour. Calculate the time it will take Joe to mow his entire lawn.


In a right triangle, the sides other than the hypotenuse are $a$ and $b$, with $a:b = 2:5$. If the hypotenuse is $c$, and a perpendicular from the right angle vertex to $c$ is drawn, dividing it into segments $r$ and $s$, adjacent respectively to $a$ and $b$, find the ratio of $r$ to $s$.


A box contains 4 red marbles, 3 green marbles, and 3 yellow marbles. Carol takes 3 marbles from the box at random; then Claudia takes 3 marbles from the remaining 7 marbles at random; and then Cheryl takes the last 4 marbles. What is the probability that Cheryl gets at least 2 marbles of the same color?
A) $\frac{1}{10}$
B) $\frac{1}{6}$
C) $\frac{3}{4}$
D) $\frac{9}{10}$
E) $1$


Each of two boxes contains three chips numbered 1, 2, 3. A chip is drawn randomly from each box and the numbers on the two chips are multiplied. Calculate the probability that their product is a multiple of 3.


What is the sum of the digits of the square of 1111111?


Given that a line extends from $(c, 0)$ to $(2, 2)$ and divides a region into two regions of equal area in a coordinate plane, where four unit squares form a larger square with a side length of 2 units with the lower left corner at the origin, determine the value of $c$.


Given that the ratio of $8^\text{th}$-graders to $7^\text{th}$-graders is $7:4$ and the ratio of $7^\text{th}$-graders to $6^\text{th}$-graders is $10:9$, find the smallest number of students that could be participating if all grades are involved.


Simplify $\left(\sqrt[4]{81} - \sqrt{8\frac{1}{2}}\right)^2$.

Solution: \boxed{\frac{35}{2} - 3\sqrt{34}}


Given that Nadia makes a fruit punch using 150 grams of orange juice with 45 calories in 100 grams, 200 grams of apple juice with 52 calories in 100 grams, 50 grams of sugar with 385 calories in 100 grams, and 600 grams of water, calculate the total calories in 300 grams of her fruit punch.


Two circles, each of radius 2, are constructed as follows: The center of circle A is chosen uniformly at random from the line segment joining (0,0) and (2,0). The center of circle B is chosen uniformly at random from the line segment joining (0,2) to (2,2), with the additional condition that the center of B must lie above or on the line y = 1. Calculate the probability that circles A and B intersect.


Given the expression $\left(x\sqrt[4]{3} + y\sqrt[3]{5}\right)^{400}$, calculate the number of terms with rational coefficients in its expansion.


Consider a regular tetrahedron with vertices \(A\), \(B\), \(C\), and \(D\), where each edge of the tetrahedron is of length 1. Let \(P\) be a point on edge \(AB\) such that \(AP = t \times AB\) and \(Q\) be a point on edge \(CD\) such that \(CQ = s \times CD\), where \(0 \leq t, s \leq 1\). Find the least possible distance between points \(P\) and \(Q\).


In an isosceles trapezoid, if the smaller base equals the length of a diagonal and the larger base equals twice the altitude, find the ratio of the altitude to the larger base.


The point is chosen at random within the rectangle in the coordinate plane whose vertices are $(0, 0), (3030, 0), (3030, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{3}$. Find $d$ to the nearest tenth.


How many 4-digit numbers greater than 3000 are there that use the four digits of 2033?


Determine the 1234th digit to the right of the decimal point in the number \( x = .123456789101112...498499500 \).


Given a square EFGH with side length 2, an inscribed circle Ω intersects GH at point N. Line EN intersects Ω at a point Q different from N. Determine the length of EQ.


How many distinct triangles can be formed by selecting three vertices from a set of points arranged in a 2×4 grid (2 rows and 4 columns)?


Let \( x = -2023 \). Find the value of \(\left\vert\left\vert |x|^2 - x\right\vert - |x|\right\vert - x\).


Given a square in the coordinate plane with vertices at \((0, 0)\), \((3030, 0)\), \((3030, 3030)\), and \((0, 3030)\), find the value of \(d\) to the nearest tenth, where the probability that a randomly chosen point within the square is within \(d\) units of a lattice point is \(\frac{3}{4}\).


In a class of $n > 15$ students, the overall mean score on a test is $10$. A group of $15$ students from the class has a mean score of $16$. What is the mean of the test scores for the rest of the students in terms of $n$?

Solution: \boxed{\frac{10n - 240}{n-15}}


For how many integers \( x \) is the number \( x^4 - 51x^2 + 100 \) negative?


Given Sharon usually takes 4 hours to drive from her house to her workplace, and on one day she drives at her normal speed for half the distance but then reduces her speed by 30 miles per hour for the remaining half. If the total journey on that day took 330 minutes, calculate the total distance from Sharon's house to her workplace.


Dave, Emily, and Fiona take turns tossing an eight-sided die, starting with Dave, followed by Emily, and then Fiona. Determine the probability that Fiona is the first to toss an eight.


The number $15!$ has a certain number of positive integer divisors. Determine the probability that one of its divisors is odd.


Find the sum of the first fifty positive odd integers subtracted from the sum of the first fifty positive even integers.


The product of two positive numbers is 16. The reciprocal of one of these numbers is 3 times the reciprocal of the other number. What is the sum of the two numbers?


In trapezoid $EFGH$, the sides $EF$ and $GH$ are equal, and the height from $F$ to $GH$ is 6 units. Given that the bottom base $GH$ is 12 units long, and the top base $EF$ is 10 units, find the perimeter of trapezoid $EFGH$.

Solution: \boxed{22 + 2\sqrt{37} \text{ units}}


A sequence of $6$ numbers is formed by starting with two initial numbers. Each subsequent number in the sequence is the product of the two previous numbers. Given that the last two numbers of the sequence are $81$ and $6561$, find the value of the first number in the sequence.


Given that Theresa’s parents have promised to buy her tickets if she spends an average of 9 hours per week helping around the house for 6 weeks, and for the first 5 weeks she has helped 9, 12, 6, 13, and 11 hours, determine how many hours she must work in the final week to earn the tickets.


Let $M$ be the second smallest positive integer that is divisible by every positive integer less than 10 and includes at least one prime number greater than 10. Find the sum of the digits of $M$.


Given that nine unit squares are arranged in a 3x3 grid on the coordinate plane, with the lower left corner at the origin, and a line extending from (d,0) to (0,3) divides the entire configuration into two regions with equal areas, calculate the value of d.


The number $15!$ has several positive integer divisors. Find the probability that a randomly chosen divisor is odd.


Given that Liam bought some pencils costing more than a penny each at the school bookstore and paid $\textdollar 2.10$, and Mia bought some of the same pencils and paid $\textdollar 2.82$, determine the number of pencils that Mia bought compared to Liam.


Mr. Green measures his rectangular garden by taking 18 steps by 25 steps. Each of Mr. Green's steps is 2.5 feet long. He expects three-quarters of a pound of potatoes per square foot from his garden. Calculate the area of the garden in square feet, and then find the total weight of potatoes that Mr. Green expects from his garden.


Liam has two older twin sisters. The product of their three ages is 144. Find the sum of their three ages.


Given the sequence of squares with each side length one tile more than the previous square, calculate how many more tiles the eighth square requires than the seventh square.


Given that bricklayer Brenda would take 8 hours to build a wall alone, and bricklayer Brandon would take 12 hours to build it alone, and working together, they build the wall in 6 hours with a 15 bricks per hour reduction in their combined output, find the total number of bricks in the wall.


Given that Mrs. Blue's rectangular garden measures 10 steps by 30 steps, with each step being 3 feet long, determine the total weight of potatoes she expects from her garden, knowing that only 90% of the garden space is suitable for planting and she expects three-quarters of a pound of potatoes per square foot.


How many real numbers \(x\) satisfy the equation \(3^{2x+2} - 3^{x+4} + 5 \cdot 3^x - 15 = 0\)?


The basic subscription price of the online streaming service is $15 per month. Calculate the maximum percentage decrease in the number of subscribers that the service can tolerate in order to keep their total income at least the same despite a 20% price increase.


Determine the minimum value of $\frac{3x^2 + 6x + 19}{8(1+x)}$ for $x \ge 0$.


Given that $\frac{2}{3}$ of the marbles are blue and the rest are red, calculate the fraction of marbles that will be red if the number of red marbles is doubled and the number of blue marbles remains the same.


Let $x$ be the smallest real number greater than 0 such that $\cos(x) = \cos(x^2)$, where the arguments are in radians. Find the value of $x$.


A car travels the first part of its journey, covering a distance of $b$ miles at a speed of 50 mph, and then completes another part where it travels for $2n+b$ miles at a speed of 75 mph. Calculate the total time taken by the car to complete its journey.

Solution: \boxed{\frac{5b + 4n}{150}}


Given that Mr. A initially owns a home worth $\$15,000$, he sells it to Mr. B at a $20\%$ profit, then Mr. B sells it back to Mr. A at a $15\%$ loss, then Mr. A sells it again to Mr. B at a $10\%$ profit, and finally Mr. B sells it back to Mr. A at a $5\%$ loss, calculate the net effect of these transactions on Mr. A.


Homer started peeling a pile of 50 potatoes at the rate of 4 potatoes per minute. Five minutes later, Christen joined him and peeled at the rate of 6 potatoes per minute. Determine the total number of potatoes peeled by Christen.


A top hat contains 5 red chips and 4 green chips. Chips are drawn randomly, one at a time without replacement, until all 5 of the reds are drawn or until all 4 green chips are drawn. Calculate the probability that all 5 red chips are drawn.


At 3:30 o'clock, find the angle between the hour and minute hands of a clock.


Eight students, including Abby and Bridget, are to be seated randomly in two rows of four for a group photo as shown:
\begin{eqnarray*}
\text{X}&\quad\text{X}\quad&\text{X}\quad&\text{X} \\
\text{X}&\quad\text{X}\quad&\text{X}\quad&\text{X}
\end{eqnarray*}
What is the probability that Abby and Bridget are seated adjacent to each other either in the same row or the same column?
A) $\frac{1}{7}$
B) $\frac{2}{9}$
C) $\frac{3}{14}$
D) $\frac{4}{15}$
E) $\frac{5}{14}$


Given the numbers $2^9+1$ and $2^{17}+1$, inclusive, calculate the number of perfect cubes between them.


Given the total degree measure of the interior angles of a convex polygon is $2083^\circ$, determine the degree measure of the omitted angle.


Given the equation \(\frac{\frac{1}{x} - \frac{1}{y}}{\frac{1}{x} + \frac{1}{y}} = 1001\), find the value of \(\frac{x+y}{x-y}\).


Given that $a @ b = \frac{a \times b}{a + b}$ for $a,b$ positive integers, and further operation defined as $c # d = c + d$, calculate the value of $3 @ 7 \# 4$.


The population of a village is $800$, and the graph indicates that the number of females is divided into four equal parts, with three of these parts representing the females. Determine the number of males in the village.


A piece of wood of uniform density in the shape of a right triangle with base length $3$ inches and hypotenuse $5$ inches weighs $12$ ounces. Another piece of the same type of wood, with the same thickness, also in the shape of a right triangle, has a base length of $5$ inches and a hypotenuse of $7$ inches. Calculate the approximate weight of the second piece.


A point is chosen at random from within a circular region with radius $3$, calculate the probability that the point is closer to the center of the region than it is to the boundary of the region.


Given that Mr. Blue receives a $15\%$ raise every year, calculate the percentage increase in his salary after five such raises.


In a classroom with \( n > 15 \) students, the average score of a quiz is 10. A subset of 15 students have an average score of 17. Determine the average of the scores of the remaining students in terms of \( n \).

Solution: \boxed{\frac{10n - 255}{n - 15}}


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


Given points P(0, -3) and Q(5, 3) in the xy-plane; point R(x, m) is taken so that PR + RQ is a minimum where x is fixed to 3, determine the value of m.


What is the largest quotient that can be formed using two numbers chosen from the set $\{ -30, -5, -3, 1, 3, 10, 15 \}$?


Find the sum of all real numbers $x$ for which the median of the numbers $3, 7, 9, 20,$ and $x$ is equal to the mean of those five numbers and $x$ is less than $9$.


Find the fraction exactly halfway between $\frac{1}{4}$, $\frac{1}{6}$, and $\frac{1}{3}$.


What is the tens digit of $(25! - 20!)$?


Given a grid arrangement of eight regular squares surrounding a central square of side length 2, find the area of triangle DEF, where D, E, and F are the centers of three adjacent outer squares.


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


Given that there are 21 students in Dr. Smith's physics class, the average score before including Simon's project score was 86. After including Simon's project score, the average for the class rose to 88. Calculate Simon's score on the project.


The line joining the midpoints of the diagonals of a trapezoid has length $4$. If the longer base is $100$, determine the length of the shorter base.


A positive integer $n$ has $72$ divisors and $5n$ has $90$ divisors. Determine the greatest integer $k$ such that $5^k$ divides $n$.


Given that the town experiences two consecutive years of population increase by 20%, followed by two consecutive years of population decrease by 30%, calculate the net percentage change in the population over these four years, rounded to the nearest percent.


Calculate the value of $(2^3 - 3 + 5^3 - 0)^{-1} \times 7$.


An integer $N$ is selected at random from the range $1 \leq N \leq 1950$. Calculate the probability that the remainder when $N^{14}$ is divided by $5$ is $1$.


Given a circle $O$, points $C$ and $D$ are on the same side of diameter $\overline{AB}$, $\angle AOC = 40^{\circ}$, and $\angle DOB = 30^{\circ}$. Find the ratio of the area of the smaller sector $COD$ to the area of the circle.


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


Given Tom is currently $T$ years old, which is also equal to the sum of the ages of his four children. Three years ago, Tom's age was three times the sum of his children's ages. Determine the value of $\frac{T}{3}$.


Calculate the value of $3^{\left(1^{\left(0^8\right)}\right)}+\left(\left(3^1\right)^0\right)^8$.


Given the first three terms of a geometric progression are $2^2$, $2^{\frac{3}{2}}$, and $2$, respectively, find the fourth term.


Gary the gazelle takes 55 equal jumps to navigate between consecutive street lamps on a city park path, while Zeke the zebra covers the same distance in 15 equal strides. The distance to the 26th lamp from the start is 2640 feet. Calculate how much longer Zeke's stride is than Gary's jump.


Find the greatest number of consecutive non-negative integers whose sum is $120$.


Given the equation $\sin(3x) = \cos(x)$ on the interval $[0, 2\pi]$, find the number of solutions.


Given the range of $10000$ to $99999$, inclusive, determine the number of $5$-digit positive integers having only even digits that are divisible by $5$.


A $4 \times 4$ square is partitioned into $16$ unit squares. Each unit square is painted either white or black with each color being equally likely, independently and at random. The square is then rotated $180\,^{\circ}$ about its center, and every white square that lands in a position formerly occupied by a black square is painted black, while black squares moved to a position formerly occupied by white squares are turned white. Determine the probability that the entire grid is now black.


Let $x$ be a real number selected uniformly at random between 0 and 1000. If $\lfloor \sqrt{x} \rfloor = 14$, find the probability that $\lfloor \sqrt{10x} \rfloor = 44$.


Given that a four-digit integer is between 1000 and 9999, has only even digits, is divisible by both 5 and 10, and the hundreds digit is 0, determine the number of such integers.


Four years ago, Liam was twice as old as his sister Mia. Six years before that, Liam was three times as old as Mia. Determine the number of years until the ratio of their ages will be 3:2.


Paul originally had enough paint for 50 identically sized rooms, but lost 5 cans on the way. Determine the number of cans of paint he used for the remaining 40 rooms.


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


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


The numbers $\log(a^2b^4)$, $\log(a^6b^9)$, and $\log(a^{10}b^{14})$ form the first three terms of an arithmetic sequence, and the $10^\text{th}$ term of this sequence is $\log(a^n)$. Determine the value of $n$.


Given that Jackie has $40$ thin rods, one of each integer length from $1 \text{ cm}$ through $40 \text{ cm}$, with rods of lengths $5 \text{ cm}$, $12 \text{ cm}$, and $20 \text{ cm}$ already placed on a table, find the number of the remaining rods that she can choose as the fourth rod to form a quadrilateral with positive area.


Given two numbers chosen from the set $\{-25, -4, -1, 1, 3, 9\}$, determine the largest quotient that can be formed.


Given the equation $16^{-3} = \frac{4^{60/y}}{4^{32/y} \cdot 16^{24/y}}$, find the value of y that satisfies this equation.


The smallest positive integer that is neither prime nor square and that has no prime factor less than 70.


Given that in a class test, $15\%$ of the students scored $60$ points, $50\%$ scored $75$ points, $20\%$ 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.


The sum of three numbers is $125$. The ratio of the first to the second is $\frac{3}{4}$, and the ratio of the second to the third is $\frac{7}{6}$. Find the second number.


Calculate the result of $\frac{3}{10} + \frac{5}{100} - \frac{2}{1000}$.


Given Alex, Jamie, and Casey play a game over 8 rounds, and for each round, the probability Alex wins is $\frac{1}{3}$, and Jamie is three times as likely to win as Casey, calculate the probability that Alex wins four rounds, Jamie wins three rounds, and Casey wins one round.


Determine the value of $x$ that gives the minimum value of the quadratic function $x^2 + px + qx$.


Solution: \boxed{\textbf{(E)}\ m = -\frac{3}{2}}


Given that the plane is tiled by alternative patterns of congruent squares and congruent pentagons, where each large square is divided into 16 smaller squares, out of which 5 smaller squares are turned into congruent pentagons, calculate the percent of the plane that is enclosed by the pentagons.


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


The average age of the members of the family is 25, the mother is 45 years old, and the average age of the father and children is 20. If $x$ is the number of children in the family, determine the value of $x$.


Given a cyclic quadrilateral $ABCD$ with side $AB$ extended beyond $B$ to point $E$, if $\measuredangle BAD=80^\circ$ and $\measuredangle ADC=110^\circ$, find the measure of $\measuredangle EBC$.


Given that a tiled surface consists of $n^2$ identical blue square tiles, each measuring $s$ inches per side, and each surrounded by a yellow border $d$ inches wide, forming an overall large square pattern, and $n=30$ being observed to cover $81\%$ of the area of the entire square region, determine the ratio $\frac{d}{s}$.


Given that a circle is divided into 15 sectors and the central angles of these sectors form an arithmetic sequence, find the degree measure of the smallest possible sector angle.


How many whole numbers between 200 and 500 contain the digit 3?


A 3x3x3 cube is assembled from $27$ standard dice. Each die has the property that opposite faces sum to $7$. Calculate the largest possible sum of all of the values visible on the $6$ faces of the cube.


Alice has two pennies, three nickels, four dimes, and two quarters in her purse. Calculate the percentage of one dollar that is in her purse.


The real number \(x\) satisfies the equation \(x + \frac{1}{x} = \sqrt{3}\). Evaluate the expression \(x^{7} - 5x^{5} + x^{2}\).


A digital watch now displays time in a 24-hour format, showing hours and minutes. Find the largest possible sum of the digits when it displays time in this format, where the hour ranges from 00 to 23 and the minutes range from 00 to 59.


Given a deck with five red cards labeled $1, 2, 3, 4, 5$ and four green cards labeled $1, 2, 3, 4$, find the probability of drawing a winning pair, where a winning pair is defined as two cards of the same color or two cards with the same number.


Given that $\angle XYZ = 36^\circ$ and $\angle XYW = 15^\circ$, calculate the smallest possible degree measure for $\angle WYZ$.


Evaluate the expression $(3(3(3(3(3 - 2 \cdot 1) - 2 \cdot 1) - 2 \cdot 1) - 2 \cdot 1) - 2 \cdot 1)$.


Calculate the sum of the first one hundred positive odd integers subtracted from the sum of the first one hundred positive even integers.


Given that ten friends dined at a restaurant and Sara's nine friends each paid an extra $3 to cover her part of the bill, determine the total value of the bill.


Given the integer $2083$, express it as the sum of two positive integers whose difference is as small as possible and calculate this difference.


What is the maximum possible product of three different numbers from the set $\{-9, -5, -3, 1, 4, 6, 8\}$?


Given the sets of consecutive integers $\{1\}$, $\{2, 3\}$, $\{4,5,6\}$, $\{7,8,9,10\}$, $\ldots$, where each set contains one more element than the preceding one, and where the first element of each set is one more than the last element of the preceding set, let $S_n$ be the sum of the elements in the nth set. Find $S_{30}$.


Given $2025$ is written as a product of two positive integers whose difference is as small as possible, calculate the difference.


An item, when sold for $x$ dollars, results in a loss of 20%. However, when sold for $y$ dollars, it leads to a gain of 25%. Find the ratio of $y:x$.


Determine the expression for $g(-x)$ in terms of $g(x)$, given that $g(x) = \frac{x^2 - x - 1}{x^2 + x - 1}$.


George observed a train crossing a bridge and began timing it as it passed. He counted 8 cars in the first 12 seconds of the train's passage. The entire train took 3 minutes and 30 seconds to completely pass the bridge at a constant speed. Estimate the number of cars in the train.


LeRoy paid $L$ dollars, Bernardo paid $B$ dollars, and Cecilia paid $C$ dollars, where $L < B < C$. Calculate the amount in dollars that LeRoy must give to Cecilia so that they share the costs equally.

Solution: \boxed{\frac{B + C - 2L}{3}}


Big Bea, the bear, ate 196 bananas from April 1 through April 7. Each day she ate eight more bananas than on the previous day. Calculate the number of bananas Big Bea ate on April 7.


Given a rhombus $EFGH$ with a perimeter of $64$ meters and the length of diagonal $\overline{EH}$ of $30$ meters, calculate the area of the rhombus $EFGH$.


Determine the area of a quadrilateral with vertices at the coordinates $(2,1)$, $(1,6)$, $(4,5)$, and $(9,9)$.


Given that Sarah had the capacity to paint 45 rooms and 4 cans of paint were dropped, determine the number of cans of paint used to paint 35 rooms.


How many positive integers less than $500$ are $8$ times the sum of their digits?


Given the data set $[8, 15, 21, 29, 29, 35, 39, 42, 50, 68]$, where the median $Q_2 = 32$, the first quartile $Q_1 = 25$, and the third quartile $Q_3 = 45$, find the number of outliers present if an outlier is defined as a value that is more than $2.0$ times the interquartile range below the first quartile or above the third quartile.


The length of track required to rise 800 feet at a 4% grade is approximately 20,000 feet. Calculate the additional length of track required to reduce this grade to 3%.


Given two distinct dice: Die A with numbers from $1$ to $6$, and Die B with numbers from $1$ to $7$, if Die B shows a $7$, its face value counts as zero in the sum, calculate the probability that the sum of the numbers on the two top faces is exactly $7$.


Given a $5\times 5$ block of calendar dates from 1 to 25 arranged consecutively, reverse the order of the numbers in the second, third, and fifth rows, and then calculate the positive difference between the sums of the numbers on the two diagonals.


Jo and Blair take turns counting, starting with Jo who says "3". Each subsequent number said by either Jo or Blair is two more than the last number said by the other person. What is the $100^{\text{th}}$ number said?


Given the data set $[2, 11, 23, 23, 25, 35, 41, 41, 55, 67, 85]$, median $Q_2 = 35$, first quartile $Q_1 = 23$, and third quartile $Q_3 = 55$, determine how many outliers this data set has.


Determine the number of positive integers $x$ that satisfy the inequality $(10x)^4 > x^8 > 2^{16}$.


A circle is inscribed within a right angle triangle with sides $3$, $4$, and $5$. Simultaneously, a square is concentric with the circle and has side length $s$. Determine the value of $s$ and compute the area inside the circle but outside the square.

Solution: \boxed{\pi - 2}


Determine the area of a quadrilateral with vertices at $(0, 0)$, $(4, 0)$, $(4, 3)$, and $(2, 5)$.


Every high school in the city of Newton sent a team of $4$ students to a math contest. Andrea scored the highest among all contestants, and her teammates Beth, Carla, and David placed $50$th, $75$th, and $100$th, respectively. Determine how many schools are in the city of Newton.


If Alice sells an item at $15 less than the list price and receives 15% of her selling price as her commission, and Charles sells the item at $18 less than the list price and receives 18% of his selling price as his commission, and if Alice and Charles both receive the same commission amount, determine the list price of the item.


In the complex plane, let $A$ be the set of solutions to $z^3 - 27 = 0$ and let $B$ be the set of solutions to $z^3 - 9z^2 - 27z + 243 = 0,$ find the distance between the point in $A$ closest to the origin and the point in $B$ closest to the origin.

Solution: \boxed{3(\sqrt{3} - 1)}


Two angles of an isosceles triangle measure $60^\circ$ and $x^\circ$. What is the sum of the three possible values of $x$?


Three numbers have a mean that is $20$ more than the smallest number and $30$ less than the largest number. The median of the three numbers is $10$. Find their sum.


Given $x \spadesuit y = (x+y+1)(x-y)$, calculate $2 \spadesuit (3 \spadesuit 6)$.


Given $x$ is real and positive and grows beyond all bounds, and $y$ is a positive constant, determine the limit of $\log_4{(8x + 3y - 7)} - \log_4{(4x + y + 2)}$.


Given the set of all fractions $\frac{x}{y}$, where $x$ and $y$ are relatively prime positive integers, how many of these fractions have the property that if both numerator and denominator are increased by $1$, the value of the fraction is increased by $20\%$?


Shop owner purchases 2000 pencils at $0.15 each and sells them at $0.30 each, determine how many pencils he must sell to make a profit of exactly $150.


Consider a set of 16 integers from -8 to 7, inclusive, arranged to form a 4-by-4 square. Find the value of the common sum of the numbers in each row, column, and main diagonal.


When George places his marbles into bags with 8 marbles per bag, he has 5 marbles left over. When John does the same with his marbles, he has 6 marbles left over. George then acquires 5 more marbles. Announce the number of marbles leftover when their total number of marbles is placed into as many bags as possible, carrying 8 marbles per bag.


Given the polynomial expansion of $(1+3x-x^2)^5$, find the coefficient of $x^9$.


Al and Carol start their new jobs on the same day. Al's schedule is 4 work-days followed by 2 rest-days, and Carol's schedule is 5 work-days followed by 1 rest-day. Determine the number of their first 1000 days in which both have rest-days on the same day.


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


A parabolic arch has a height of $20$ inches and a span of $30$ inches. Determine the height of the arch at $3$ inches from the center.


Given that Store P offers a 20% discount on the sticker price plus a $120 rebate, and Store Q offers a 30% discount on the same sticker price with no rebate, and Clara saves $30 more by purchasing at Store P, determine the sticker price of the laptop.


What is $\sum_{n = 1}^{1023}{f(n)}$, where $f(n) = \left\{\begin{matrix}\log_{4}{n}, &\text{if }\log_{4}{n}\text{ is rational,}\\ 0, &\text{otherwise.}\end{matrix}\right.$


Consider a sequence of consecutive integer sets where each set starts one more than the last element of the preceding set and each set has one more element than the one before it. For a specific n where n > 0, denote T_n as the sum of the elements in the nth set. Find T_{30}.


Given the expression $\left(\frac{x^2+2}{x}\right)\left(\frac{y^2+2}{y}\right)+\left(\frac{x^2-2}{y}\right)\left(\frac{y^2-2}{x}\right)$, where $xy \neq 0$, simplify the expression.

Solution: \boxed{2xy + \frac{8}{xy}}


If $1764$ is written as a product of two positive integers whose difference is as small as possible, calculate the difference of these two positive integers.


Given $A$ and $B$ together can do a job in $3$ days; $B$ and $C$ can do it in $6$ days; and $A$ and $C$ in $3.6$ days, calculate the number of days $C$ needs to do the job alone.


Given \( n = x - y^{x - (y+1)} \), find the value of \( n \) when \( x = 3 \) and \( y = -3 \).


Queen High School has $1500$ students, and each student takes $6$ classes per day. Each teacher teaches $5$ classes, with each class having $25$ students and $1$ teacher. How many teachers are there at Queen High School?


Given that the digits of the integers must be from the set $\{1,3,4,5,6,8\}$ and the integers must be between $300$ and $800$, determine how many even integers have all distinct digits.


Given a pentagon is inscribed in a circle, calculate the sum of the angles that are inscribed into each of the five segments outside the pentagon, each segment being the region outside the pentagon but inside the circle.


Let \( g(x) \) be a polynomial with leading coefficient 1, whose four roots are the reciprocals of the four roots of \( f(x) = x^4 + ax^3 + bx^2 + cx + d \), where \( 1 < a < b < c < d \). Find \( g(1) \) in terms of \( a, b, c, \) and \( d \).

Solution: \boxed{\frac{1 + a + b + c + d}{d}}


Calculate the value of
\[
\frac{(.5)^4}{(.05)^3}
\]


How many 4-digit numbers greater than 1000 can be formed using each of the digits 3, 0, 3, 1 exactly once?


A positive integer $n$ has $72$ divisors and $5n$ has $90$ divisors. What is the greatest integer $j$ such that $5^j$ divides $n$?


Solution: \boxed{\frac{a + 2b}{2a}}


Each of 3000 boxes in a line contains a single red marble, and for $1 \le k \le 3000$, the box in the $k\text{th}$ position also contains $k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $P(n)$ be the probability that Isabella stops after drawing exactly $n$ marbles. Determine the smallest value of $n$ for which $P(n) < \frac{1}{3000}$.


A box contains 30 red balls, 24 green balls, 22 yellow balls, 15 blue balls, 12 white balls, 10 black balls, and 5 purple balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least 16 balls of a single color will be drawn?


Given the listed price of an item is greater than $\textdollar 150$, and the shopper can use one of three coupons: Coupon A offers 15% off the listed price, Coupon B provides a $\textdollar 30$ flat discount, and Coupon C offers 20% off the amount by which the listed price exceeds $\textdollar 150$, determine the difference between the smallest and largest prices for which Coupon A saves more dollars than both Coupon B and Coupon C.


A frog starts at point $(2, 3)$ on a grid and makes jumps of length $1$ parallel to the coordinate axes. Each jump direction (up, down, right, or left) is chosen randomly. The jumping sequence stops when the frog reaches any point on the boundary of the rectangle defined by vertices $(0,0), (0,5), (5,5),$ and $(5,0)$. Calculate the probability that the sequence of jumps ends at either the top or bottom horizontal side of the rectangle.


Two-thirds of a pitcher is filled with orange juice. The pitcher is then emptied by pouring an equal amount of juice into each of 6 cups. Calculate the percent of the total capacity of the pitcher that each cup received.


Two boxes contain colored balls. Box 1 has $40$ red balls, $30$ green balls, $25$ yellow balls, and $15$ blue balls. Box 2 has $35$ red balls, $25$ green balls, and $20$ yellow balls. Find the minimum number of balls that must be drawn from these boxes (without replacement, drawing from any box at will) to guarantee that at least $20$ balls of a single color are drawn.


John has two identical cups. Initially, he puts 6 ounces of tea into the first cup and 6 ounces of milk into the second cup. He then pours one-third of the tea from the first cup into the second cup and mixes thoroughly. After stirring, John then pours half of the mixture from the second cup back into the first cup. Finally, he pours one-quarter of the mixture from the first cup back into the second cup. What fraction of the liquid in the first cup is now milk?


For how many integers x does a triangle with side lengths 15, 20 and x have all its angles acute?


Jo and Blair take turns counting numbers starting from 1, with each person adding 2 to the last number said by the other person. Given that Jo starts by saying "1", find the $30^{\text{th}}$ number said.


Sharon drives from her house to her mother's house in a known time and distance relationship with a variable speed. Given the original travel time is 180 minutes, and if she decreases her speed by 30 miles per hour after driving $\frac{1}{2}$ of the distance, the total travel time extends to 300 minutes. Find the distance from Sharon's house to her mother's house.


Given two altered dice where one die has faces labeled with the numbers $1, 1, 3, 3, 5, 5$ and the other has faces labeled $2, 2, 4, 4, 6, 6$, find the probability that the sum of the numbers on the top faces will be $7$ after the two dice are rolled.


Given that the cost of a school play ticket is $x$ dollars, where $x$ is a whole number, and a group of 9th graders spends $36$, a group of 10th graders spends $54$, and a group of 11th graders spends $72$ dollars, determine the number of possible values for $x$.


For all real numbers $x$, evaluate the expression $x[x\{x(3-x)-5\}+12]+2$.

Solution: \boxed{-x^4 + 3x^3 - 5x^2 + 12x + 2}


Jeremy's father drives him to school at a normal speed of 15 minutes, but on a certain day, he drove at a speed 25 miles per hour faster and arrived at school in 9 minutes. Find the distance to school.


A total of 10 men can paint a house in 6 days. If 4 additional skilled men join them, and the skilled men work 25% more efficiently than the original group, calculate the number of days it will take for all the men to paint the same house.


Evaluate $\sqrt{\frac{x}{1 - \frac{3x - 2}{2x}}}.$

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


Using only pennies, nickels, dimes, quarters, and half-dollars, find the smallest number of coins Freddie would need to pay any amount of money less than a dollar.


Alex sent 150 text messages and talked for 28 hours, given a cell phone plan that costs $25 each month, $0.10 per text message, $0.15 per minute used over 25 hours, and $0.05 per minute within the first 25 hours. Calculate the total amount Alex had to pay in February.


Given two circular pulleys with radii of $10$ inches and $6$ inches, and the distance between the points of contact of the belt with the pulleys is $30$ inches, calculate the distance between the centers of the pulleys in inches.


Given a basketball player made 7 baskets during a game, with each basket being worth either 2 or 3 points, determine the total number of different numbers that could represent the total points scored by the player.


Solution: \boxed{-\frac{1}{2} + \frac{i\sqrt{3}}{6}}


Each of two boxes contains three chips numbered $1$, $4$, $3$. A chip is drawn randomly from each box. Find the probability that the product of the numbers on the two chips is even.


When $n$ standard 8-sided dice are rolled, the probability of obtaining a sum of 3000 is greater than zero and is the same as the probability of obtaining a sum of S. Find the smallest possible value of S.


What is the radius of a circle inscribed in a rhombus with diagonals of length 16 and 30?


Consider a 4x4x4 cube made out of 64 normal dice, where each die's opposite sides still sum to 7. Find the smallest possible sum of all of the values visible on the 6 faces of the larger cube.


Given that quadrilateral EFGH is a rhombus with a perimeter of 80 meters and the length of diagonal EG is 30 meters, calculate the area of rhombus EFGH in square meters.


Given the digits 2, 0, 2, and 1, and the restriction that no zero can be in the first or last position, determine the number of different four-digit numbers which can be formed by rearranging these digits.


Suppose that $\tfrac{3}{4}$ of $16$ apples are worth as much as $10$ grapes. Calculate the number of grapes worth as much as $\tfrac{1}{3}$ of $9$ apples.


Jane buys either a $60$-cent muffin or an $80$-cent bagel every day for a seven-day week. Given that her total weekly expenditure is a whole number of dollars, determine the number of bagels she bought this week.


Given $(xy+1)^2 + (x-y)^2$, find the smallest possible value for this expression for real numbers $x$ and $y$.


A laser is placed at the point $(2,3)$. The laser beam travels in a straight line. Larry wants the beam to hit and bounce off the $y$-axis, then hit and bounce off the $x$-axis, finally hitting the point $(6,3)$. Calculate the total distance the beam will travel along this path.


Consider the arithmetic sequence defined by the set $\{2, 5, 8, 11, 14, 17, 20\}$. Determine the total number of different integers that can be expressed as the sum of three distinct members of this set.


If $y$ cows produce $y+2$ cans of milk in $y+3$ days, determine the number of days it will take for $y+4$ cows to give $y+7$ cans of milk.


Given that 8 balls are randomly and independently painted either red or blue with equal probability, find the probability that exactly 4 balls are red and exactly 4 balls are blue, and all red balls come before any blue balls in the order they were painted.


Given that a collector offers to buy state quarters for 2500% of their face value, and for every set of five quarters, the collector adds a $2 bonus, calculate the amount Bryden will get for his seven state quarters.


Calculate the area of $\triangle AMC$ in rectangle $ABCD$ where $AB=10$, $AD=12$, and $AM=9$ meters.


How many 3-digit positive integers have digits whose product equals 36?


Given an arithmetic sequence with a first term of $7$, a last term of $88$, and the sum of all terms equal to $570$, determine the common difference.


Suppose the estimated cost to establish a colony on Mars is $50$ billion dollars. This amount is to be equally shared by $300$ million tax-paying citizens of a country. Calculate each person’s share.


A square-shaped floor is covered with congruent square tiles. Along with the two diagonals of the square and a horizontal line through its center that includes additional tiles, the total number of tiles along the two diagonals and this central line is 55. Determine the number of tiles that cover the entire floor.


Given that Three-digit powers of 3 and 7 are used in this "cross-number" puzzle, find the only possible digit for the outlined square.


Determine the number of distinct positive integral pairs (x, y) satisfying x^6y^6 - 13x^3y^3 + 36 = 0.


Given every 7-digit whole number is a possible telephone number except those that begin with 0, 1, or 2, determine the fraction of telephone numbers that begin with a digit from 3 to 9 and end with an even digit.


Given that each of two boxes contains three chips numbered $1$, $4$, and $5$, calculate the probability that the product of the numbers on the two chips is a multiple of $3$.


What is the greatest possible product of the digits in the base-seven representation of a positive integer less than $2300$?


Given Ricardo has $3000$ coins comprised of pennies ($1$-cent coins), nickels ($5$-cent coins), and dimes ($10$-cent coins), with at least one of each type of coin, calculate the difference in cents between the highest possible and lowest total value that Ricardo can have.


Given three congruent circles with centers $P$, $Q$, and $R$ tangent to the sides of rectangle $ABCD$, the circle centered at $Q$ has a diameter of $6$ and passes through points $P$ and $R$. Compute the area of rectangle $ABCD$.


Given that the probability of a biased coin landing heads is $\frac{1}{3}$ and tails is $\frac{2}{3}$, and Tom, Dick, and Harry flip the coin repeatedly until they get their first head, calculate the probability that all three flip their coins the same number of times.


Given that $3x - 2y - 2z = 0$ and $x - 4y + 8z = 0$, where $z \neq 0$, calculate the value of $\frac{3x^2 - 2xy}{y^2 + 4z^2}$.


Find the ratio of AD:DC in triangle ABC, where AB=6, BC=8, AC=10, and D is a point on AC such that BD=6.


Students from Arlington school worked for $4$ days, students from Bradford school worked for $7$ days, and students from Clinton school worked for $8$ days. If a total of eight students, six students, and seven students from Arlington, Bradford, and Clinton schools respectively earned a total of $1,456, calculate the total earnings from Clinton school.


Given each piece of candy in the store costs a whole number of cents, and a piece of yellow candy costs $15$ cents, determine the smallest possible value of $n$ if Joe has exactly enough money to buy $10$ pieces of red candy, $16$ pieces of green candy, $18$ pieces of blue candy, or $n$ pieces of yellow candy.


Given $d$ is a digit, determine for how many values of $d$ the number $3.014d$ is greater than $3.015$.


An ellipse with a horizontal major axis and center at $O$ has a pair of parallel tangents that are vertical and equidistant from $O$. Find the number of points equidistant from the ellipse and these two parallel tangents.


Given the repeating decimal $5.1717171717\ldots$, express the number as a fraction and calculate the sum of the numerator and denominator of this fraction when reduced to lowest terms.


Let $n$ be the largest integer that is the product of exactly three distinct prime numbers $d$, $e$, and $d^2 + e^2$, where $d$ and $e$ are distinct single-digit primes such that $d < e$. Find the sum of the digits of $n$.


Given that Ms. Demeanor's class consists of 50 students, more than half of her students bought crayons from the school bookstore, each buying the same number of crayons, with each crayon costing more than the number of crayons bought by each student, and the total cost for all crayons was $19.98, determine the cost of each crayon in cents.


A group of students is organizing a fundraiser. Initially, $60\%$ of the group are girls. Later on, three girls leave and three boys join, after which $50\%$ of the group are girls. Determine the number of girls initially in the group.


A triangle $\bigtriangleup ABC$ has vertices lying on the parabola defined by $y = x^2 + 4$. Vertices $B$ and $C$ are symmetric about the $y$-axis and the line $\overline{BC}$ is parallel to the $x$-axis. The area of $\bigtriangleup ABC$ is $100$. $A$ is the point $(2,8)$. Determine the length of $\overline{BC}$.


Find the ratio of $AE:EC$ in $\triangle ABC$ given that $AB=6$, $BC=8$, $AC=10$, and $E$ is on $\overline{AC}$ with $BE=6$.


Min-jun's video had a score of $120$, and $75\%$ of the votes were likes. Determine how many total votes were cast on his video at that point.


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


Two numbers have a difference, sum, and product that are in the ratio \(1:8:15\). Find the product of the two numbers.


The product of the three ages of Kiana and her twin brothers is $72$. Express their ages as $x$, $x$, and $y$ and determine their sum.


The ratio of $a$ to $b$ is $5:2$, the ratio of $c$ to $d$ is $2:1$, and the ratio of $d$ to $b$ is $2:5$. Determine the ratio of $a$ to $c$.


Given that a circle is divided into 18 sectors with central angles forming an arithmetic sequence, find the degree measure of the smallest possible sector angle.


How many positive factors of 48 are also multiples of 8?


A cell phone plan costs $30$ each month, plus $4$ cents per text message, and $15$ cents for each minute used over $40$ hours. Given that Tim sent 200 text messages and talked for 42 hours in March, calculate the total cost of his plan for the month.


Determine how many different prime numbers are factors of $N$ if $\log_2 ( \log_3 ( \log_5 (\log_ 7 (\log_{11} N)))) = 16$.


Given nine squares are arranged in a 3x3 grid and numbered from 1 to 9 from left to right, top to bottom. The paper is folded three times in sequence: fold the right third over to the middle third, fold the left third over the previous fold covering the right and middle thirds, and fold the bottom third up to the top third. Determine the numbered square that ends up on top after these folds.


If $5(3x + 7\pi) = Q$, solve for the value of $10(6x + 14\pi)$.


A triangle with integral sides has a perimeter of 12. Determine the area of this triangle.


Cameron has $90$ red tokens and $60$ blue tokens. He can exchange three red tokens for one gold token and two blue tokens, or two blue tokens for one gold token and one red token. What is the total number of gold tokens Cameron ends up with after exchanging his tokens until no more exchanges can be made?


There are several sets of three different numbers whose sum is 21, which can be chosen from {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}. How many of these sets contain a 7?


Given $(1296^{\log_6 1728})^{\frac{1}{4}}$, evaluate the resulting expression.


Lucy and her five friends are ages $4$, $6$, $8$, $10$, $12$, and $14$. Given that two of her friends whose ages sum to $18$ went to the skatepark, two friends less than $12$ went to the swimming pool, and Lucy and the $6$-year-old went to the library, determine Lucy's age.


Calculate the average of the shortest distances from the lemming to each side of a square field with a side length of $12$ meters.


Let $x$ be the smallest real number greater than $5$ such that $\sin(x) = \cos(2x)$, where the arguments are in degrees. Find the value of $x$.


Evaluate the expression $\frac{(0.5)^4}{(0.05)^3} + 3$.


The sum of the numerical coefficients in the expansion of $(2x^2 - 3xy + y^2)^6$ is 64.


Given a round-robin tournament with 7 teams, each team plays one game against each other team, and each game results in one team winning and one team losing, what is the maximum number of teams that could be tied for the most wins at the end of the tournament?


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


Given the four-digit number $5056$, find the number of different numbers that can be formed by rearranging its digits.


Given that Alice sells an item at $15 less than the list price and receives $15\%$ of her selling price as her commission, while Bob sells the same item at $25 less than the list price and receives $25\%$ of his selling price as his commission, determine the list price of the item.


Evaluate the double summation $\sum^{50}_{i=1} \sum^{50}_{j=1} 2(i+j)$.


Determine the number of distinct terms in the expansion of $\left[(a+2b)^2 (a-2b)^2\right]^3$.


Given a rectangular track consisting of two straight sections of length $80$ meters each and two semicircular ends, the track has a width of $10$ meters, and it takes $60$ seconds longer to jog around the outer edge than around the inner edge, determine Mira's speed in meters per second.


Given the set $\{2,3,5,7,11,13\}$, add one of the numbers twice to another number, and then multiply the result by the third number. What is the smallest possible result?


Given four spheres of radius $2$, each tangent to three orthogonal planes at the corners of a room, find the radius of the smallest sphere centered at the room's geometric center that contains these four spheres.

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


Given the sequence where each integer \( n \) appears \( n \) times, for \( 1 \leq n \leq 150 \), calculate the median of the numbers in this list.


Given the data set $[8, 22, 36, 36, 44, 45, 45, 48, 56, 62]$, with a median $Q_2 = 44.5$, first quartile $Q_1 = 36$, and third quartile $Q_3 = 48$, determine the number of outliers in the data set, where an outlier is defined as a value that is more than $2$ times the interquartile range below the first quartile ($Q_1$) or more than $2$ times the interquartile range above the third quartile ($Q_3$).


In an island kingdom, five fish can be traded for three jars of honey, and a jar of honey can be traded for six cobs of corn. Find the number of cobs of corn one fish is worth.


Given John the painter had enough paint initially to paint 50 similarly sized rooms, and after dropping five cans of paint he was left with enough to paint 40 rooms, determine how many cans of paint he used for these 40 rooms.


Jack had a bag of $150$ apples. He sold $20\%$ of them to Jill. Next, he sold $30\%$ of the remaining apples to June. Afterwards, he decides to sell $10\%$ of what's left to Jeff. Finally, Jack donates $5\%$ of the remaining apples to a local school. What is the number of apples that Jack has left after all these transactions?


Expand $\left ( 1 - \frac{1}{a} \right )^8$ and find the sum of the last three coefficients.


What is the number of terms in the simplified expansion of $[(2a+5b)^3(2a-5b)^3]^3$?


The sum of the angles of the pentagon inscribed in a circle cut off by its sides is 180 degrees.


A square piece of wood with a side length of 4 inches and uniform density weighs 16 ounces. Calculate the weight, in ounces, of a square piece of the same wood with a side length of 6 inches.


A hemisphere with a radius of 3 rests on the base of a triangular pyramid of height 9, and the hemisphere is tangent to all three faces of the pyramid. Calculate the length of each side of the base of the equilateral triangle.


Suppose the product $\dfrac{4}{3}\cdot \dfrac{5}{4}\cdot \dfrac{6}{5}\cdot \ldots\cdot \dfrac{c}{d} = 16$, find the sum of $c$ and $d$.


Let $x$ be the number of students in Mina's high school. Given that Nina's high school has 5 times as many students as Mina's high school, the number of students in Nina's high school is $5x$. Furthermore, if 200 students were to transfer from Nina's high school to Mina's high school, the total number of students in Mina's high school would be $x+200$ and the total number of students in Nina's high school, after the transfer, would be $5x-200$. The problem states that after the transfer, the two schools would have twice as many students as the smaller school. Therefore, the equation $2(x+200) = 5x-200$ can be formed. Solving for $x$, we find that $x=800. Thus, there are $\boxed{4000}$ students in Nina's high school.


Consider a number formed by subtracting one from $2^{15}$, i.e., $32,767$. Find the sum of the digits of the greatest prime number that is a divisor of this number.


A worker's salary is reduced by 30%. What percentage increase is needed on the new salary to restore it to the original salary?


Given the digits in the set $\{1, 3, 4, 5, 6, 9\}$, count the number of even integers between 300 and 800 whose digits are all different.


Suppose the estimated $50$ billion dollar cost to construct a space station is shared equally by the $500$ million people in a coalition of countries, calculate the amount each person's share.

Solution: \boxed{\ 100}


Find the number of ordered pairs $(b,c)$ of positive integers such that both $x^2+bx+c=0$ and $x^2+cx+b=0$ have exactly one real solution.


Calculate the expression $200(200-5)-(200\cdot200-5)$.


Given $i$ be the imaginary unit such that $i^2 = -1$ and $a$ is a non-zero real number, calculate $(ai-i^{-1})^{-1}$.


Four A's, four B's, four C's, and four D's are placed in the sixteen spaces of a 4x4 grid so that each row and column contains one of each letter. If A is placed in the upper left corner, determine the number of possible arrangements.


If Alice is cycling east at a speed of 10 miles per hour and Sam is also cycling east but at a faster speed of 16 miles per hour, with Sam being 3 miles west of Alice, calculate the time in minutes it will take for Sam to catch up to Alice.


Let $ABCD$ be a square, and let $E, F, G, H$ be the midpoints of the hypotenuse of right-angled triangles with one leg as $\overline{AB}, \overline{BC}, \overline{CD}, \overline{DA}$ respectively, each exterior to the square. Given that each triangle has legs equal to the side of the square, find the ratio of the area of square $EFGH$ to the area of square $ABCD$.


Let a sequence $\{u_n\}$ be defined by $u_1=4$ and the recurrence relation $u_{n+1}-u_n=2n^2 - 2n + 1, n=1,2,3,\dots$. If $u_n$ is expressed as a polynomial in $n$, find the algebraic sum of its coefficients.


A solid cube of side length $4$ has a solid cube of side length $2$ removed from each corner. How many edges does the remaining solid have?


Jessica has 30 coins consisting of nickels and dimes. If her nickels were dimes and her dimes were nickels, she would have $1.20 more. Calculate the total value of her coins.


A rectangular board of 10 rows is numbered in a snaking pattern, starting in the upper left corner. In the first row, the squares are numbered from 1 to 10, in the second row from 20 to 11 (backwards), and so on alternating directions each row. A student shades square 1, then skips one square and shades square 3, skips two squares and shades square 6, skips three squares and shades square 10, continuing in this manner. Determine the number of the shaded square that is the first to ensure at least one shaded square in each row.


Given two numbers such that their difference, their sum, and their product are to one another as $1:8:40$, calculate the product of these two numbers.


When all four valves are open, the tank fills in 1.2 hours, and with valves A, B, and D open, it takes 2 hours. With valves A, C, and D open it takes 1.5 hours. Given the rate at which valve D releases water into the tank is half the rate of valve C, determine how many hours it will take to fill the tank with only valves A, B, and C open.


The Tigers beat the Sharks 3 out of the 4 times they initially played. They then played N more games, all won by the Sharks. Afterwards, the Sharks ended up winning at least 90% of all the games played. Determine the minimum possible value for N.


In the expansion of $\left(a - \frac{1}{\sqrt{a}}\right)^8$, determine the coefficient of the $a^{-\frac{1}{2}}$ term.


Given the first four terms of an arithmetic sequence $a, y, b, 3y$, determine the ratio of $a$ to $b$.


Given that $63$ students are wearing red shirts and $69$ students are wearing green shirts, and the $132$ students are assigned into $66$ pairs, with $26$ pairs consisting of two red shirts, determine the number of pairs that consist entirely of students wearing green shirts.


Given that $x$ is a real number, evaluate the expression $x[x\{x(3-x)-5\}+12]+2$.

Solution: \boxed{-x^4 + 3x^3 - 5x^2 + 12x + 2}


Given Liam has written one integer three times and another integer four times. The sum of these seven numbers is 131, and one of the numbers is 17, determine the value of the other number.


Given a set of tiles numbered from 1 through 150, find the smallest number of operations required to reduce the number of tiles in the set to less than ten.


If a package weighs $W$ ounces, calculate the total mailing cost in cents, given that the service charges five cents for every ounce or portion thereof and there is a fixed handling fee of three cents per package.

Solution: \boxed{5\lceil W \rceil + 3}


Given a positive real number is called doubly special if its decimal representation consists entirely of the digits $0$ and $5$, find the smallest integer $n$ such that $1$ can be written as a sum of $n$ doubly special numbers.


Five friends — Sarah, Lily, Emma, Nora, and Kate — performed in a theater as quartets, with one friend sitting out each time. Nora performed in 10 performances, which was the most among all, and Sarah performed in 6 performances, which was the fewest among all. Calculate the total number of performances.


Let $x = .123456789101112...495496497498499$, where the digits are obtained by writing the integers from $1$ through $499$ in order. Determine the $1234$th digit to the right of the decimal point.


Determine the sum of all possible x-coordinates where the lines given by the equations $y=2x+4$ and $y=4x+c$ intersect the x-axis.


A sphere with center $O$ has radius $9$. A right triangle with sides of length $12, 35, 37$ is situated in space so that each of its sides is tangent to the sphere. Calculate the distance between $O$ and the plane determined by the triangle.


Four balls numbered $1, 2, 3$, and $4$ are placed in an urn. One ball is drawn, its number noted, and then returned to the urn. This process is repeated three times. If the sum of the numbers noted is $9$, determine the probability that the ball numbered $3$ was drawn all three times.


A positive integer $n$ has 72 divisors and $5n$ has 90 divisors. What is the greatest integer $k$ such that $5^k$ divides $n$?


Given Chloe chooses a real number uniformly at random from the interval $[ 0,3000 ]$ and Laurent chooses a real number uniformly at random from the interval $[ 0,6000 ]$, find the probability that Laurent's number is greater than Chloe's number.


Determine the number of positive factors of 147,456.


There are twenty-four $4$-digit numbers that use each of the four digits $1$, $3$, $6$, and $8$ exactly once. List the numbers in numerical order from smallest to largest and find the number in the $15^{\text{th}}$ position in the list.


In an exam, $15\%$ of the students scored $60$ points, $20\%$ scored $75$ points, $25\%$ scored $85$ points, $25\%$ scored $90$ points, and the rest got $100$ points. Calculate the difference between the mean and the median score on this exam.


Let \( S = \omega^n + \omega^{-n} \), where \( \omega = e^{2\pi i / 5} \) is a complex fifth root of unity, and \( n \) is an integer. Determine the total number of possible distinct values for \( S \).


Numbers between $200$ and $500$ that are divisible by $5$ contain the digit $3$. How many such whole numbers exist?


How many positive factors of 48 are also multiples of 8?


In the coordinate plane, a point $(x, y)$ needs to be the same distance from the x-axis, the y-axis, and the line $x + y = 4$. What is the x-coordinate of such a point?
A) $\sqrt{2}$
B) $2+\sqrt{2}$
C) $2$
D) $1$
E) Not uniquely determined


Given $15!$ has multiple positive integer divisors. If one of these divisors is chosen at random, calculate the probability that it is odd.


Let $U$ be the set of the first $2010$ positive multiples of $5$, and let $V$ be the set of the first $2010$ positive multiples of $7$. Calculate how many elements are common to both sets $U$ and $V$.


The average age of the members of the Jansen family is $25$, the father is $50$ years old, and the grandfather is $70$ years old. Given that the average age of the mother, grandfather, and children is $20$, calculate the number of children in the family.


Given a decorative window that includes a rectangle with semicircles on either end and an equilateral triangle centered on top of one semicircle, where the ratio of the length of the rectangle's length $AD$ to the width $AB$ is $3:2$ and $AB$ is 20 inches, determine the ratio of the area of the rectangle to the combined area of the semicircles and the triangle.

Solution: \boxed{\frac{6}{\pi + \sqrt{3}}}


Given that the speed of sound in the current weather conditions is 1100 feet per second, calculate to the nearest quarter-mile the distance Daisy was from the lightning flash 12 seconds after hearing the thunder.


If two factors of $3x^3 - hx + k$ are $x+2$ and $x-1$, find the value of $|3h-2k|$.


Two pictures, each 4 feet across, are hung in the center of a wall that is 30 feet wide. The pictures are spaced 1 foot apart. Find the distance from the end of the wall to the nearest edge of the first picture.


Given the yearly changes in the population census of a city for three consecutive years are, respectively, a 40% increase, a 15% decrease, and a 15% decrease, calculate the net change over the three years, rounded to the nearest percent.


A circular region has a radius of 4 units. What is the probability that a point chosen at random from within this region is closer to the center than it is to the boundary of the region?


A ticket for a concert costs $x$ dollars, where $x$ is a whole number. A group of middle schoolers buys tickets costing a total of $90$, and a group of high schoolers buys tickets costing a total of $150$. Find the number of values for $x$ that are possible.


The smallest whole number larger than the perimeter of any triangle with sides of lengths $6$ and $21$ and a third side of length $x$ can be found by solving for x in the triangle inequality, $x>21-6$ and $x<6+21$.


Given that the endpoints of a line segment are $(7,31)$ and $(61,405)$, determine how many lattice points are on the line segment, including both endpoints.


Consider the sequence of numbers where each integer \( n \) appears \( n \) times, for \( 1 \leq n \leq 100 \). What is the median of the numbers in this list?


Given that a rectangular piece of metal with dimensions 8 units by 10 units is used to cut out the largest possible circle, which is then used to cut out the largest possible square, calculate the total amount of metal wasted.


Given a rectangular prism with distinct edge lengths, determine the number of unordered pairs of edges that determine a plane.


The taxi fare in New Metro City is $3.00 for the first mile and additional mileage charged at the rate $0.25 for each additional 0.1 mile. You plan to give the driver a $3 tip. Calculate the number of miles that can be ridden for $15.


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 Carlos took $65\%$ of a whole pie, then Maria took half of what remained. Calculate the portion of the whole pie that was left.


In the $xy$-plane, find the number of lines whose $x$-intercept is a positive integer and whose $y$-intercept is a positive prime number that pass through the point $(5,4)$.


Given that the team earned $72$ points in total, where the points from three-point shots were double the points from two-point shots, and the number of successful free throws was twice the number of successful two-point shots, determine the number of free throws they made.


For a triangle with side lengths $12, 30$, and $x$ to have all its angles acute, determine the number of integers $x$ that satisfy the condition.


A line $x=k$ intersects the graph of $y=\log_2 x$ and the graph of $y=\log_2 (x + 2)$. The distance between the points of intersection is $1$. Given that $k = a + \sqrt{b}$, where $a$ and $b$ are integers, find the value of $a+b$.


Let $ABCDEF$ be an equiangular convex hexagon with a perimeter of 2. Determine the perimeter $s$ of the six-pointed star formed by the pairwise intersections of the extended sides of the hexagon.


Paul owes Paula $60$ cents and has a pocket full of $5$-cent coins, $10$-cent coins, $25$-cent coins, and $50$-cent coins that he can use to pay her. Calculate the difference between the largest and smallest number of coins he can use to pay her.


If the sum of the first 15 terms and the sum of the first 85 terms of an arithmetic progression (AP) are 45 and 255, respectively, then find the sum of the first 100 terms.


Calculate the equivalent single discount for successive discounts of $15\%$ and $25\%$ on an item.


A tetrahedron has 4 vertices. You choose three vertices at random to form a triangle.


Given that points $P$ and $Q$ are on a circle of radius $10$ and $PQ=12$, and point $R$ is the midpoint of the minor arc $PQ$, calculate the length of the line segment $PR$.


Given the expression $\frac{x^2-4x+3}{x^2-6x+9} \div \frac{x^2-3x+2}{x^2-4x+4},$ simplify the expression.


How many positive integer divisors of $221^{10}$ are perfect squares or perfect cubes (or both)?


Find the quadratic equation whose roots sum up to $7$ and the absolute value of whose difference is $9$.


Determine the smallest positive integer \(x\) for which \(1800x = M^3\), where \(M\) is an integer.


Determine the number of distinct terms in the expansion of $[(x+4y)^2(x-4y)^2]^3$ when simplified.


Positive integers $a$ and $b$ are such that $a$ is less than $6$ and $b$ is less than $10$. Calculate the smallest possible value for $2 \cdot a - a \cdot b$.


Determine the largest number by which the expression $2n^3 - 2n$ is divisible for all integral values of $n$.


Determine the maximum value of the function $10x - 4x^2$ for any real value of $x$.


Find the minimum value of $\sqrt{x^2+y^2}$ given the constraints $3x+4y=24$ and $x \geq 0$.


On a redesigned dartboard, the outer circle radius is increased to $8$ units and the inner circle has a radius of $4$ units. Additionally, two radii divide the board into four congruent sections, each labeled inconsistently with point values as follows: inner sections have values of $3$ and $4$, and outer sections have $2$ and $5$. The probability of a dart hitting a particular region is still proportional to its area. Calculate the probability that when three darts hit this board, the total score is exactly $12$.
A) $\frac{9}{2048}$
B) $\frac{9}{1024}$
C) $\frac{18}{2048}$
D) $\frac{15}{1024}$


The mean of three numbers is $20$ more than the least of the numbers and $30$ less than the greatest. The median of the three numbers is $10$. Calculate the sum of these three numbers.


Given four integers $p, q, r,$ and $s$, such that their sums taken three at a time are $210, 230, 250,$ and $270$, find the largest of these four integers.


A can complete a piece of work in 12 days. B is 33% less efficient than A. Determine the number of days it takes B to do the same piece of work.


In parallelogram $ABCD$, $BE$ is the height from vertex $B$ to side $AD$, and segment $ED$ is extended from $D$ such that $ED = 8$. The base $BC$ of the parallelogram is $14$. The entire parallelogram has an area of $126$. Determine the area of the shaded region $BEDC$.


Calculate the harmonic mean of 2, 3, and 6.5.


How many unordered pairs of edges in a regular tetrahedron determine a plane?


A positive integer n has 72 divisors, and 3n has 96 divisors. What is the greatest integer k such that 3^k divides n?


Given the set $\{41, 57, 82, 113, 190, 201\}$, find the number of subsets containing three different numbers such that the sum of the three numbers is odd.


A telephone number is represented as $\text{ABC-DEF-GHIJ}$, with each letter representing a unique digit. The digits in each segment are in decreasing order: $A > B > C$, $D > E > F$, and $G > H > I > J$. Furthermore, $D, E,$ and $F$ are consecutive digits, not limited to even or odd. $G, H, I,$ and $J$ are also consecutive digits that include both odd and even numbers. Given that $A + B + C = 17$, determine the value of $A$.


What is the smallest prime number dividing the sum $4^{15} + 6^{17}$?


A parking lot has 20 spaces in a row. Fifteen cars arrive, each of which requires one parking space, and their drivers choose spaces at random from among the available spaces. Auntie Em then arrives in her SUV, which requires 2 adjacent spaces. Calculate the probability that she is able to park.


Given that one root of $3x^2 + rx + s = 0$, with $r$ and $s$ real numbers, is such that its square is $4-3i$, determine the value of $s$.


The average of the numbers $4, 6, 9, a, b, c, d$ is $20$. Find the average of $a$, $b$, $c$, and $d$.


What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 55?


Given that twelve points are equally spaced around a $3 \times 3$ square, calculate the probability that two randomly chosen points are one unit apart.


David drives from his home to the airport to catch a flight. He drives 40 miles in the first hour, but realizes that he will be 45 minutes late if he continues at this speed. He increases his speed by 20 miles per hour for the rest of the way to the airport and arrives 15 minutes early. Determine the distance from his home to the airport.


Given the speed of sound is approximated as 1100 feet per second, the time between the lightning flash and the thunder is 15 seconds, and one mile is 5280 feet, estimate, to the nearest quarter-mile, how far Charlie Brown was from the flash of lightning.


Given that $\frac{1}{4}$ of all eighth graders are paired with $\frac{1}{3}$ of all fifth graders for a buddy system, find the fraction of the total number of fifth and eighth graders that have a buddy.


Evaluate the expression $2 + \frac{1}{2 + \frac{1}{2 + \frac{1}{3}}}$.


Given three wheels with Wheel One having three even and one odd number, Wheel Two having two even and two odd numbers, and Wheel Three having one even and three odd numbers, calculate the probability that the sum of the results from spinning all three wheels is even.


The region consisting of all points in three-dimensional space within $5$ units of line segment $\overline{CD}$ has a volume of $900 \pi$. What is the length of $\overline{CD}$?
A) 26.67
B) 27.33
C) 28.67
D) 29.33

Solution: \boxed{D) = 29.33}


A chord, which is the perpendicular bisector of a radius of length 10 in a circle, intersects the radius at its midpoint. Calculate the length of the chord.


If 1 quart of paint is required to paint a statue 8 ft. high, determine the total amount of paint required to paint 360 statues, each 2 ft. high, with each statue needing a double layer of paint for full coverage.


Given two different third degree polynomial functions y=p(x) and y=q(x), each with leading coefficient 1, calculate the maximum number of points of intersection of their graphs.


Six test scores have a mean of $92$, a median of $93$, and a mode of $94$. Find the sum of the two lowest test scores.


Given a number $2310$, find how many of its positive integer factors have more than four factors.


Given that triangle PQR is an isosceles triangle with an area of 100 and contains 20 smallest identical triangles, each with an area of 1, and PQS is composed of 6 of these smallest triangles and is similar to PQR, find the area of trapezoid RQS.


Given that Elmer's new car provides $80\%$ better fuel efficiency than his old car, and the diesel used by the new car is $30\%$ more expensive per liter than the gasoline used by the old car, calculate the percentage Elmer will save on fuel if he uses his new car for a long trip.


Find the smallest integer value of $m$ such that $3x(mx-5)-2x^2+7=0$ has no real roots.


A rectangle in the coordinate plane has vertices at $(0, 0), (1000, 0), (1000, 1000),$ and $(0, 1000)$. Compute the radius $d$ to the nearest tenth such that the probability the point is within $d$ units from any lattice point is $\tfrac{1}{4}$.


A pair of standard $8$-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. Calculate the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference.


Given two circles of radii $4$ and $5$, and distance $d$ between their centers, determine how many different values of $k$ are possible when drawing all possible lines simultaneously tangent to both circles.


A rope is cut at a random point along its length of 1 meter. What is the probability that the longer piece is at least three times as large as the shorter piece?


Given Jasmine has two types of bottles, one that can hold 45 milliliters and another that can hold 675 milliliters, and a vase that can hold 95 milliliters, determine the total number of small bottles she must buy to fill the large bottle as much as possible and the vase.


Given a two-digit positive integer $N$, find how many such integers have the property that the sum of $N$ and the number obtained by reversing the order of the digits of $N$ equals twice a perfect square.


Determine how many ordered pairs of integers \( (x, y) \) satisfy the equation \( x^{4} + y^2 = 4y. \)


Given the progression $2^{\dfrac{1}{7}}, 2^{\dfrac{2}{7}}, 2^{\dfrac{3}{7}}, 2^{\dfrac{4}{7}}, \dots , 2^{\dfrac{n}{7}}$, find the least positive integer $n$ such that the product of the first $n$ terms of the progression exceeds $1,024$.


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


Given triangle $\triangle DEF$ has vertices $D = (4,0)$, $E = (0,4)$, and $F$ is on the line $x + y = 9$, determine the maximum area of $\triangle DEF$.


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


Find the coefficient of \(x^8\) in the expansion of \(\left(\frac{x^3}{3} - \frac{3}{x^2}\right)^9\).


Find $f(2x)$ for $x \neq 1$ and $x \neq 2$, where $f(x) = \frac{2x+3}{x-2}$.


Four years ago, Sam was four times as old as his brother Tim. Six years before that, Sam was five times as old as Tim. Find the number of years after which the ratio of their ages will be 3 : 1.


Given that $A$, $B$, and $C$ are sets such that $n(A) + n(B) + n(C) = n(A \cup B \cup C)$, with $|A| = 50$, $|B| = 60$, and $|A \cap B| = 25$, determine the minimum possible value of $|A \cap B \cap C|$.


How many distinguishable rearrangements of the letters in "BALANCE" have all the vowels at the end.


Given points $A(-3, -4)$ and $B(6, 3)$ in the xy-plane; point $C(1, m)$ is taken so that $AC + CB$ is a minimum. Find the value of $m$.


A regular hexagon $ABCDEF$ with side length 2 has two circles positioned outside it. The first circle is tangent to $\overline{AB}$ and the second circle is tangent to $\overline{CD}$. Both circles are also tangent to lines $BC$ and $FA$. Find the ratio of the area of the second circle to that of the first circle.


The total in-store price for a blender is $\$75.99$. A television commercial advertises the same blender for four easy payments of $\$17.99$ and a one-time shipping fee of $\$6.50$ coupled with a handling charge of $\$2.50$. Calculate the number of cents saved by purchasing the blender from the television advertiser.


A box 2 centimeters high, 4 centimeters wide, and 6 centimeters long can hold 48 grams of clay. A second box with three times the height, twice the width, and one and a half times the length as the first box can hold n grams of clay. Determine the value of n.


Given that the product of 900 and $x$ is a square and the product of 900 and $y$ is a fifth power, determine the sum of $x$ and $y$.


A school has $120$ students and $6$ teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are $40, 40, 20, 10, 5, 5$. Let $t$ be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let $s$ be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. Find the value of $t-s$.


A beam of light originates from point $A$ on a plane and is reflected several times between lines $AD$ and $CD$ before finally striking a point $B$ on line $CD$ at a $15^\circ$ angle and then reflecting back to point $A$. If the angle $\measuredangle CDA=10^\circ$, determine the maximum number of times the light can be reflected before striking $B$.


The line segment whose endpoints are $(5, 23)$ and $(60, 353)$ contains how many lattice points?


John drove 180 miles in 180 minutes. His average speed during the first 45 minutes was 40 mph, during the second 45 minutes was 50 mph, and during the third 45 minutes was 60 mph. Calculate John's average speed, in mph, during the last 45 minutes.

Solution: \boxed{90 \text{ mph}}


Given the number $825$, convert it into a factorial base of numeration, that is, $825 = a_1 + a_2 \times 2! + a_3 \times 3! + \dots + a_n \times n!$ where $0 \le a_k \le k$. Find the value of $a_5$.


Given the equations $2002C - 3003A = 6006$ and $2002B + 4004A = 8008$, as well as $B - C = A + 1$, calculate the average of A, B, and C.


Given Maya takes 45 minutes to ride her bike to a beach 15 miles away, and Naomi takes 15 minutes to cover half the distance on a bus, stops for 15 minutes, and then takes another 15 minutes to cover the remaining distance. Find the difference, in miles per hour, between Naomi's and Maya's average speeds.


A rectangle has length $AC = 40$ and width $AE = 25$. Point $B$ is the midpoint of $\overline{AC}$, making $AB = BC = 20$, but point $F$ is now three-fifths of the way along $\overline{AE}$ from $A$ to $E$, making $AF = 15$ and $FE = 10$. Find the area of quadrilateral $ABDF$.


A rectangular floor that is 12 feet wide and 18 feet long is tiled with 216 one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and the last tile, calculate the number of tiles the bug visits.


The taxi fare in Metropolis City is $3.00 for the first $\frac{1}{2}$ mile and additional mileage charged at the rate $0.30 for each additional 0.1 mile. You plan to give the driver a $3 tip. Calculate how many miles can you ride for $15.


Consider a sequence \(v_n = n^4 + 2n^2 + 2\). Define the sequence of differences \(\Delta^k(v_n) = \Delta^1(\Delta^{k-1}(v_n))\). Identify for which \(k\) the sequence \(\Delta^k(v_n)\) becomes zero for all \(n\).


Given Mia works 3 hours a day and is paid $0.40 per hour for each full year of her age, and during a nine-month period, Mia worked 80 days and earned $960, determine Mia's age at the end of the nine-month period.


Given a cyclic quadrilateral $ABCD$ with point $E$ such that $AB$ is extended to $E$. Assume $\measuredangle BAD = 85^\circ$ and $\measuredangle ADC = 70^\circ$, calculate the angle $\measuredangle EBC$.


Consider those functions $f$ that satisfy $f(x+6) + f(x-6) = f(x)$ for all real $x$. Find the least common positive period $p$ for all such functions.


In a circle with a radius of 8 units, points A and B are such that the length of chord AB is 10 units. Point C is located on the major arc AB such that the arc length AC is a third of the circumference of the circle. Calculate the length of the line segment AC.


Mr. C owns a home worth $15,000$. He sells it to Mr. D at a 15% profit based on the worth of the house. Mr. D then sells the house back to Mr. C at a 5% loss. Calculate Mr. C's net gain or loss from these transactions.

Solution: \boxed{862.50 \text{ dollars}}


Let x be the total number of seventh graders and y be the total number of tenth graders. If $\frac{1}{4}$ of all tenth graders are paired with $\frac{3}{7}$ of all seventh graders, what fraction of the total number of seventh and tenth graders have a partner?


Let A, B, and C be three piles of rocks. The mean weight of the rocks in A is 30 pounds, and in B it is 55 pounds. The mean weight of the rocks in the combined piles A and B is 35 pounds, and in the combined piles A and C it is 32 pounds. Find the greatest possible integer value for the mean in pounds of the rocks in the combined piles B and C.


What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 60?


Susie buys 5 muffins and 4 bananas. Calvin spends three times as much as Susie for 3 muffins and 20 bananas. Express the cost ratio of a muffin to a banana as a fraction.


A cylindrical oil tank, standing vertically, has a height of 12 feet and an interior diameter of 4 feet. The oil covers the entire circular base and reaches a height of $h$ feet inside the tank. If the volume of the oil is 48 cubic feet, determine the height of the oil in the tank.

Solution: \boxed{\frac{12}{\pi}\text{ feet}}


In a round-robin tournament with 8 teams, where each team plays one game against each other team, and each game results in one team winning and one team losing, what is the maximum number of teams that could be tied for the most wins at the end of the tournament?


Given that $x$ and $y$ are distinct nonzero real numbers such that $x^2 + \frac{2}{x} = y + \frac{2}{y}$, find the value of $xy$.


Given the number of blue chips is at least one third of the white chips and at most one quarter of the red chips, and the number of chips that are either white or blue amounts to at least 72, determine the minimum number of red chips.


Find the remainder when the polynomial $x^4 + 3x^3 - 4$ is divided by the polynomial $x^2 - 1$.

Solution: \boxed{3x - 3}


Given that the sum of two numbers is 15 and their product is 36, find the sum of their reciprocals and the sum of their squares.


Triangle PQR has vertices P = (1,2), Q = (-2,5), and R, where R is on the line x + y = 9. Calculate the area of triangle PQR.


Peter and his family ordered a 16-slice pizza. Peter ate two slices and shared a third of another slice equally with his sister, Lily. What fraction of the pizza did Peter eat?


Given a $4\times 4$ block of dates starting from 5 and increasing by 1 sequentially, first reverse the order in the columns, then calculate the positive difference between the sums of the two diagonals.


A number y is given. x is p percent less than y and z is q percent less than y. Express p and q in terms of y given that x is 10 units less than y and z is 20 units less than y.

Solution: \boxed{p = \frac{1000}{y}, q = \frac{2000}{y}}


Given that 20% of the students scored 60 points, 15% scored 75 points, 40% scored 85 points, and the remaining students scored 95 points, determine the difference between the mean and the median score of the students' scores on this test.


Given letters $A, B, C,$ and $D$ represent four different digits selected from $1, 2, \ldots, 9$, calculate the value of $A+B+1$, where $\frac{A+B+1}{C+D}$ is an integer that is as large as possible.


A circle passes through the vertices of a right triangle with side lengths $8, 15, 17$. What is the radius of this circle?


Two distinct numbers are selected from the set $\{1,2,3,4,\dots,38\}$ so that the sum of the remaining $36$ numbers equals the product of these two selected numbers plus one. Find the difference of these two numbers.


In Mr. Lee's classroom, there are six more boys than girls among a total of 36 students. What is the ratio of the number of boys to the number of girls?


Given trapezoid ABCD with side $\overline{AD}$ perpendicular to side $\overline{DC}$, lengths AD = AB = 3, and DC = 7. Point E is on side $\overline{DC}$ such that $\overline{BE}$ is parallel to $\overline{AD}$, $\overline{BE}$ equally divides DC into sections with segment DE = 3, calculate the area of $\triangle BEC$.


Given the two-digit number $10a+b$ and its reverse $10b+a$, where $a$ and $b$ are the tens and units digits respectively, the difference between the two numbers is $10b+a - (10a+b) = 9b-9a = 7(a+b)$. Find the sum of the two-digit number and its reverse.


A $3 \times 3$ grid is populated randomly with the numbers $1,2,...,9$, where each number is used exactly once. Calculate the probability that the product of numbers in each row is odd.


Determine the probability that a 4 × 4 square grid becomes a single uniform color (all white or all black) after rotation.


A tetrahedron has a triangular base with sides all equal to 2, and each of its three lateral faces are squares. A smaller tetrahedron is placed within the larger one so that its base is parallel to the base of the larger tetrahedron and its vertices touch the midpoints of the lateral faces of the larger tetrahedron. Calculate the volume of this smaller tetrahedron.


Solve for the number of distinct real solutions to the equation $|x - |3x - 2|| = 5$.


Given parallelogram ABCD has AB = 24, BC = 30, and the height from A to DC is 12. Find the area of the parallelogram.


A rectangular yard has two flower beds in the form of congruent isosceles right triangles. The rest of the yard is a trapezoid. The parallel sides of the trapezoid measure $18$ and $30$ meters. Determine the fraction of the yard occupied by the flower beds.


Given $g(x) = x^2 + 4x + 4$, count the number of integers $t$ in the set $\{0, 1, 2, \dots, 30\}$ such that $g(t)$ has a remainder of zero when divided by $10$.


Given a circle is inscribed in a triangle with side lengths $9, 12,$ and $15$. Let the segments of the side of length $9$, made by a point of tangency, be $u$ and $v$, with $u<v$. Find the ratio $u:v$.


For any positive integer $N$, the notation $N!$ denotes the product of the integers $1$ through $N$. Determine the largest integer $n$ for which $5^n$ is a factor of the sum $120! + 121! + 122!$.


Given the equation $x^4 + y^2 = 6y - 3$, determine how many ordered pairs of integers $(x, y)$ satisfy the equation.


How many positive factors of 72 are also multiples of 9?


Four friends — Alex, Betty, Clara, and Dave — participated in a relay race by running in pairs, with one pair sitting out each race. Dave ran in 8 races, which was more than any other friend, and Betty ran in 3 races, which was fewer than any other friend. Determine the total number of races those pairs completed.


In triangle $ABC$, $AC=2 \cdot AB$. Points $D$ and $E$ lie on $\overline{AB}$ and $\overline{BC}$, respectively, such that $\angle BAE = \angle ACD$. Let $F$ be the intersection of segments $AE$ and $CD$, and suppose $\triangle AFE$ is isosceles with $AF=FE$, and that $\triangle CFE$ is equilateral. Find the measure of $\angle ACB$.
A) $30^\circ$
B) $45^\circ$
C) $60^\circ$
D) $75^\circ$
E) $90^\circ$


Determine the value of x that minimizes the expression \( y = (x - a)^2 + (x - b)^2 + c(x - d)^2 \).


Determine the value of $\sqrt{x^2 + y^2}$, given that $8x + 15y = 120$ and both $x$ and $y$ must be non-negative.


Find a value of \( y \) that satisfies the equation \( y^2 + (2b)^2 = (3a - y)^2 \).

Solution: \boxed{\frac{9a^2 - 4b^2}{6a}}


Tom ate $60\%$ of a chocolate cake. Jenny took one fourth of what was left. What portion of the cake was still not eaten?


Casper started with a certain number of candies. On the first day, he ate $\frac{1}{2}$ of his candies and gave $3$ candies to his brother. On the second day, he ate another $\frac{1}{2}$ of his remaining candies and then gave $5$ candies to his sister. On the third day, he again ate half of the candies left and gave $2$ candies to a friend. On the fourth day, Casper ate the last $10$ candies. Determine the initial number of candies Casper had.


Given the product of $x^5$, $x^2 + \frac{1}{x^2}$, and $1 + \frac{2}{x} + \frac{3}{x^2}$, determine the degree of the resulting polynomial.


Given a positive number $y$, determine the condition that satisfies the inequality $\sqrt{2y} < 3y$.

Solution: \boxed{y > \frac{2}{9}}


Given Jo and Blair take turns counting numbers where each player adds 2 to the last number said by the other person, find the 30th number said, given that Jo starts by saying 3.


Anthony earns $25 per hour, and there are two types of taxes deducted from his wage: 2% for federal taxes and 0.5% for state taxes. Calculate the total amount of cents per hour of Anthony's wages used to pay both taxes combined.


The perimeter of an equilateral triangle exceeds the perimeter of a regular hexagon by $2001 \ \text{cm}$, and the length of each side of the triangle exceeds the length of each side of the hexagon by $d \ \text{cm}$. Given that the hexagon has a perimeter greater than 0, determine the number of positive integers that are NOT a possible value for $d$.


Determine the values of $k$ such that the points $(1, -2)$, $(3, k)$, and $(6, 2k - 2)$ are collinear.


Given Jane lists the whole numbers $1$ through $50$ once and Tom copies Jane's numbers, replacing each occurrence of the digit $3$ by the digit $2$, calculate how much larger Jane's sum is than Tom's sum.


Given that Ryan took three tests, the first with 30 questions and a score of 85%, the second with 35 questions (20 math and 15 science) with scores of 95% and 80% respectively, and the third with 15 questions and a score of 65%, calculate the percentage of all problems Ryan answered correctly.


Given the graphs of $y = -|x-a| + b$ and $y = |x-c| + d$, which intersect at points $(3,6)$ and $(9,2)$, find the value of $a+c$.


Express $y$ in terms of $x$, given that $x = 1 + 3^p$ and $y = 1 + 3^{-p}$.


Given the expression $(3n + 2i)^6$, find the number of integers $n$ for which this expression is a pure integer.


A circle is inscribed in a triangle with side lengths $9, 14$, and $20$. Let the segments of the side of length $14$, made by a point of tangency, be $r$ and $s$, with $r<s$. What is the ratio $\frac{r}{s}$?


A point is chosen at random within a rectangle in the coordinate plane whose vertices are $(0, 0), (3030, 0), (3030, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{3}{4}$. Find the value of $d$ to the nearest tenth.