Two circles are placed outside a square $ABCD$. The first circle is tangent to side $\overline{AB}$, and the second is tangent to side $\overline{CD}$. Both circles are tangent to the lines extended from $BC$ and $AD$. Calculate the ratio of the area of the first circle to that of the second circle.


Mr. A owns a home worth $15,000. He sells it to Mr. B at a $20\%$ profit. Mr. B then sells the house back to Mr. A at a $15\%$ loss. Calculate the amount Mr. A gains or loses in the transaction.


Calculate the value of $\dfrac{13! - 12! + 11!}{10!}$.


At a math contest, $75$ students are wearing blue shirts, and another $105$ students are wearing yellow shirts. The $180$ students are assigned into $90$ pairs. In exactly $30$ of these pairs, both students are wearing blue shirts. Calculate the number of pairs in which both students are wearing yellow shirts.


Given that Nayla has an index card measuring $5 \times 7$ inches, and she shortens the length of one side by $2$ inches, resulting in a card with an area of $21$ square inches, determine the area of the card if instead, she shortens the length of the other side by the same amount.


Consider all triangles ABC such that AB = AC, and point D is on AC such that BD is perpendicular to AC. Given that AC and CD are integers and BD^2 = 85, find the smallest possible value of AC.


Given the function $g(x) = 3x^2 + x - 4$, calculate the expression $[g(x+h) - g(x)] - [g(x) - g(x-h)]$.


How many pairs of positive integers (a, b) satisfy the equation \(\frac{a+b^{-1}}{a^{-1}+b} = 9,\) given that \(a+b \leq 200\)?


The number of points common to the graphs of $(x+2y-3)(2x-y+1)=0$ and $(x-2y+4)(3x+4y-12)=0$ is what?


Three cyclists start from the same point on a 600-meter circular track and travel clockwise with constant speeds of 3.6, 3.9, and 4.2 meters per second. Determine the time in seconds until they meet again at the same starting point.


An urn initially contains two red balls and one blue ball. A supply of extra red and blue balls is available nearby. George performs the following operation five times: he draws a ball from the urn at random and then adds two balls of the same color from the supply before returning all balls to the urn. After the five operations, the urn contains thirteen balls. What is the probability that the urn contains nine red balls and four blue balls?
**A) $\frac{1920}{10395}$**
**B) $\frac{1024}{10395}**
**C) $\frac{192}{10395}**
**D) $\frac{960}{10395}**


Given the graphs of $y = -2|x-a| + b$ and $y = 2|x-c| + d$, and the points of intersection at $(1, 6)$ and $(5, 2)$, find the value of $a+c$.


Casper can buy $16$ pieces of red candy, $18$ pieces of green candy, $20$ pieces of blue candy, or $n$ pieces of yellow candy. A piece of yellow candy costs $\30$ cents. Determine the smallest possible value of $n$.


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


If the side of one square is the diagonal of a second square, and the side of the second square is the diagonal of a third square, calculate the ratio of the area of the first square to the area of the third square.


Given the equations $x^3 + bx + c = 0$ and $x^3 + cx + b = 0$, determine the number of ordered pairs $(b,c)$ of positive integers for which neither equation has any real solutions.


Given the word $COMMITTEE$, calculate the number of distinguishable rearrangements of its letters with both the vowels first.


A computer generates a 5-digit confirmation code where each digit ranges from $0$ to $9$ with repeated digits allowed. However, no code may end with the sequence $0,0,5$. Calculate the total number of valid confirmation codes possible.


Determine the area of a rectangular garden plot which measures 1.5 meters on one side and 0.75 meters on the other side.

Solution: \boxed{1.125 \, \text{m}^2}


Two integers have a sum of 28. When two more integers are added to these, the sum is 45. Finally, when two more integers are added to the sum of the previous 4 integers, the sum is 63. Determine the minimum number of odd integers among the 6 integers.


Triangle ABC has a right angle at C and ∠A = 30°. If BD (D in AC) is the bisector of ∠ABC, determine ∠BDC.


Find the smallest nonprime integer greater than $25$ with no prime factor less than $15$ and the sum of its digits greater than $10$.


Given a classroom of $k > 10$ students with an average quiz score of 8. If the average score of 10 students is 15, calculate the mean score of the remaining $k - 10$ students.


A skateboard rolls down a ramp, traveling $8$ inches in the first second and accelerating such that during each successive $1$-second time interval, it travels $9$ inches more than during the previous $1$-second interval. The skateboard takes $20$ seconds to reach the bottom of the ramp. Calculate the total distance it travels, in inches.


Two pitchers, each with a capacity of 800 mL, contain juices. The first pitcher is 1/4 filled with orange juice and 1/8 with apple juice. The second pitcher is 1/5 filled with orange juice and 1/10 with apple juice. The rest of each pitcher is filled with water. Both pitchers are then poured into one large container. What fraction of the mixture in the large container is orange juice?


How many positive even multiples of $5$ less than $1000$ are perfect squares?


Given a thin sheet of uniform density aluminum in the shape of a square measures 4 inches per side and weighs 8 ounces, find the weight of a similar aluminum sheet in the shape of an equilateral triangle with a side length of 6 inches.


Given a deck consisting of four red cards labeled $A$, $B$, $C$, $D$, four green cards labeled $A$, $B$, $C$, $D$, and four blue cards labeled $A$, $B$, $C$, $D$, calculate the probability of drawing a winning pair, where a winning pair is defined as either two cards of the same color or two cards of the same letter.


Zoey took four tests with varying numbers of problems. She scored 85% on a 30-problem test, 75% on a 50-problem test, 65% on a 20-problem test, and 95% on a 40-problem test. What is the percentage of all problems that Zoey answered correctly?


How many sequences of $0$s and $1$s of length $21$ begin with a $0$, end with a $0$, contain no two consecutive $0$s, and contain no three consecutive $1$s?


Several students are competing in a series of four races. A student earns $6$ points for winning a race, $4$ points for finishing second, and $2$ points for finishing third. There are no ties. What is the smallest number of points that a student must earn in the four races to be guaranteed of earning more points than any other student?


A box contains 35 red balls, 30 green balls, 25 yellow balls, 15 blue balls, 12 white balls, and 10 black balls. What is the minimum number of balls that must be drawn from the box without replacement to guarantee that at least 20 balls of a single color will be drawn?


Calculate the sum $E(1)+E(2)+E(3)+\cdots+E(200)$ where $E(n)$ denotes the sum of the even digits of $n$, and $5$ is added to the sum if $n$ is a multiple of $10$.


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


A circle with area \(A_1\) is contained in the interior of a larger circle with area \(A_1 + A_2\). If the radius of the larger circle is \(4\), and if \(A_1, A_1 + A_2, A_2\) is an arithmetic progression, then find the radius of the smaller circle.


Calculate the sum of all numbers of the form $2k + 3$, where $k$ takes integer values from $0$ to $n-1$.

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


Alice and Bob each draw a number from a hat containing the numbers $1$ to $50$, both without replacement. If Alice cannot determine who has the larger number, and Bob can still determine who has the larger number after hearing Alice's statement, and Bob's number is prime, andAlice multiplies Bob's number by $150$ and adds her own number to get a perfect square, find the sum of the two numbers drawn from the hat.


The hands of a clock will form an angle of $120^{\circ}$ at what times between $7$ and $8$ o'clock, correct to the nearest minute?

Solution: \boxed{7:16 \text{ and 8:00}}


If \( y \) varies inversely as \( x \), and if \( y = 6 \) when \( x = 3 \), find the value of \( y \) when \( x = 12 \).


Queen Secondary School has 1500 students. Each student takes 4 classes a day. Each class has 25 students and is taught by 1 teacher. Each teacher teaches 5 classes a day. Determine the number of teachers at Queen Secondary School.


What is the largest quotient that can be formed using two numbers chosen from the set $\{-12, -4, -3, 1, 3, 9\}$?


Aman, Bella, and Charlie went on a holiday trip and agreed to share the costs equally. During the holiday, they took turns paying for shared expenses like meals and accommodations. At the end of the trip, Aman had paid $X$ dollars, Bella had paid $Y$ dollars, and Charlie had paid $Z$ dollars, where $X < Y < Z$. How many dollars must Aman give to Charlie so that they all share the costs equally?

Solution: \boxed{\frac{Y + Z - 2X}{3}}


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


A square pattern is composed of 5 black and 20 white square tiles. Around the pattern, attach a border consisting of white tiles on all sides. What is the ratio of black tiles to white tiles in the extended pattern?


Let $S$ be the set of the $1700$ smallest positive multiples of $5$, and let $T$ be the set of the $1700$ smallest positive multiples of $9$. Determine the number of elements that are common to both $S$ and $T$.


Given that $5y + 3$ varies inversely as the cube of $x$, and when $y = 8$, $x = 2$, determine the value of $y$ when $x = 4$.


A number $m$ is randomly selected from the set $\{21, 23, 25, 27, 29\}$, and a number $n$ is randomly selected from $\{2010, 2011, 2012, \ldots, 2030\}$. Calculate the probability that $m^n$ ends in the digit $1$.


Determine the number of solutions to the equation $\tan(3x) = \cos(x - \frac{\pi}{4})$ on the interval $[0, \frac{3\pi}{2}]$.


Two poles, one 30 inches high and another 50 inches high, are 120 inches apart. Determine the height of the intersection of the lines joining the top of each pole to the foot of the opposite pole.


Twenty-five percent less than 80 is one-quarter more than what number?


Given Alicia earns $25 per hour, of which $2\%$ is deducted to pay local taxes and an additional $0.5\%$ is deducted for state taxes, calculate the total amount of cents per hour of Alicia's wages that are used to pay these taxes combined.


Calculate the value of the expression $\frac{8 \times 10^{10}}{2 \times 10^5 \times 4}$.


A store owner bought 2000 pencils at $0.15 each. If he sells them for $0.30 each, calculate the number of pencils he must sell to make a profit of exactly $180.00$.


Connie multiplies a number by 4 and gets 200 as her result. She realizes she should have divided the number by 4 and then added 10 to get the correct answer. Find the correct value of this number.


Four friends make cookies from the same amount of dough with the same thickness. Art's cookies are circles with a radius of 2 inches, and Trisha's cookies are squares with a side length of 4 inches. If Art can make 18 cookies in his batch, determine the number of cookies Trisha will make in one batch.


Given the binomial expansion of $\left(\frac{x^3}{3} - \frac{3}{x^2}\right)^{10} \cdot x^2$, find the coefficient of $x^5$.


Given the dimensions of a rectangular tile are reported as 4 inches by 6 inches, considering the uncertainties of at least $x - 1$ inches and at most $x + 1$ inch, find the minimum possible area of this rectangle in square inches.


Find the total number of diagonals that can be drawn in two polygons, the first with 100 sides and the second with 150 sides.


A rabbit sits at a corner of a square park with side length $12$ meters. It hops $7.2$ meters along a diagonal toward the opposite corner. It then makes a $90^{\circ}$ right turn and hops $3$ more meters. Calculate the average of the shortest distances from the rabbit to each side of the square.


Determine how many positive even multiples of 3 less than 1500 are perfect squares.


How many whole numbers are there between $\sqrt{50}$ and $\sqrt{200}+1$?


Driving at a constant speed, Tim usually takes 120 minutes to drive from his home to his workplace. One day Tim begins the drive at his usual speed, but after driving half of the way, he encounters heavy traffic and reduces his speed by 30 miles per hour. Due to this, the trip takes him a total of 165 minutes. Calculate the total distance from Tim's home to his workplace in miles.


Given the equation $y = \sqrt{4+\sqrt{4+\sqrt{4+\sqrt{4+\cdots}}}}$, determine the range within which $y$ falls.

Solution: \boxed{\frac{1 + \sqrt{17}}{2}}


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


Given Gilda has given away $30\%$ of her marbles and $15\%$ of what is left to Ebony, and $30\%$ of what is now left to Jimmy, calculate the percentage of her original bag of marbles that Gilda has left for herself.


Samuel traveled the first quarter of his journey on a muddy road, the next 30 miles on a tiled road, and the remaining one-sixth on a sandy road. Calculate the total length of Samuel's journey in miles.


An upright cone filled with water has a base radius of $10 \mathrm{cm}$ and a height of $15 \mathrm{cm}$. This water is then poured into a cylinder with a base radius of $15 \mathrm{cm}$. If the cylindrical container is only $10 \mathrm{cm}$ high and any excess water overflows into a spherical container, calculate the volume of water in the spherical container.


Given a triangle with vertices at points with integer coordinates (2, 3), (5, 7), and (3, 4), determine the area of this triangle.


In 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, determine the maximum number of teams that could be tied for the most wins at the end of the tournament.


The Lions beat the Eagles 3 out of the 4 times they played, then played N more times, and the Eagles ended up winning at least 98% of all the games played; find the minimum possible value for N.


Calculate the sum of the sequence $3\left(1-\dfrac{1}{3}\right) + 4\left(1-\dfrac{1}{4}\right) + 5\left(1-\dfrac{1}{5}\right) + \cdots + 12\left(1-\dfrac{1}{12}\right)$.


In an annual school competition, the ratio of 9th-graders to 10th-graders is $7:4$, and the ratio of 9th-graders to 11th-graders is $21:10$. Determine the smallest number of students that could be participating in the competition from these grades.


Given the product of Kiana and her two older twin brothers' ages is 256, find the sum of their three ages.


Given that the first, middle, and last initials of the baby's monogram must start with 'O' and the remaining two initials must be distinct lowercase letters from the first half of the alphabet, and the initials must be in alphabetical order, determine the total number of possible monogram combinations.


Given the fraction $\frac{987654321}{2^{30}\cdot 5^6}$, determine the minimum number of digits to the right of the decimal point required to express this fraction as a decimal.


Alice and Bob play a game on a circle divided into 12 equally spaced points, numbered 1 to 12 clockwise. Both start at point 12. Alice moves 7 points clockwise, and Bob moves 4 points counterclockwise each turn. Determine the number of turns after which Alice and Bob will meet on the same point.


Jo and Blair take turns counting, but this time, each one says a number that is two more than the last number said by the other person, starting with $1$ as the first number. What is the $30^{\text{th}}$ number said?


Suppose $x$ cows produce $y$ gallons of milk in $z$ days. Calculate the amount of milk that $2x$ cows will produce in $3z$ days if the rate of milk production increases by 10%.


Given the sequence where the first number is $1$, and, for all $n\ge 2$, the product of the first $n$ numbers in the sequence is $2n^2$, calculate the sum of the fourth and the sixth numbers in this sequence.


Given the pan of brownies measures 24 inches by 15 inches and each brownie is cut into pieces measuring 3 inches by 2 inches, calculate the total number of equal-sized pieces of brownie the pan contains.


Given that the product of $450$ and $x$ is a square and the product of $450$ and $y$ is a cube, calculate the sum of $x$ and $y$.


Given the numbers $7350$ and $165$, find the number obtained when the Greatest Common Divisor is first decreased by $15$, and then multiplied by $3$.


Evaluate the expression $\frac{8^5}{4 \times 2^5 + 16}$.

Solution: \boxed{\frac{2^{15}}{4 \times 2^5 + 16} = \frac{2^{15}}{128 + 16} = \frac{2^{15}}{144} = \frac{2^{15-4}}{9} = \frac{2^{11}}{9} = \frac{2048}{9}}


Given that $\log_{b} 1024$ is a positive integer, how many positive integers $b$ satisfy this condition.


The average age of all members including the pet dog is $22$ years, and the father is $50$ years old. The average age of the mother, children, and the pet dog is $18$ years. If the pet dog is $10$ years old, calculate the number of children in the Kasun family.


Given $\angle \text{CBD}$ as a right angle and the sum of angles around point B, including $\angle \text{ABC}$, $\angle \text{ABD}$, and $\angle \text{CBD}$, totals $200^\circ$. If the measure of $\angle \text{ABD}$ is $70^\circ$, find the measure of $\angle \text{ABC}$.


A semipro baseball league mandates that each team consists of 25 players. The league rule specifies that every player must receive a salary of at least $18,000, while the total salaries for all players on a team cannot exceed $1,000,000. Determine the maximum possible salary, in dollars, that a single player can attain under these rules.


How many ordered pairs of integers (x, y) satisfy the equation x^4 + y^2 = 4y?


Soda is sold in packs of 8, 16, and 32 cans. Find the minimum number of packs needed to buy exactly 120 cans of soda.


Given that Alice mixes $150$ grams of lemon juice, $120$ grams of honey, and $330$ grams of water to make a beverage, and that the calorie content for each ingredient is $30$ calories per $100$ grams of lemon juice, $415$ calories per $100$ grams of honey, and water contains no calories, determine the total calorie content in $300$ grams of Alice's beverage.


Given that the sum of $x$ and $y$ is $12$ and the product of $x$ and $3y$ is $108$, find the quadratic polynomial whose roots are $x$ and $y$.

Solution: \boxed{t^2 - 12t + 36}


LeRoy, Bernardo, and Camila went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses. At the end of the trip, it turned out that LeRoy had paid A dollars, Bernardo had paid B dollars, and Camila had paid C dollars. If the sums paid were A < B < C, calculate the total amount LeRoy must give to Bernardo and Camila so that they share the costs equally.

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


Given the city's water tower stands 50 meters high and holds 150,000 liters of water, and Maya's miniature version holds 0.2 liters of water, determine the height of Maya's model water tower.


Evaluate the expression $2^{-(3k+2)} - 3\cdot2^{-(3k)} + 2^{-3k}$ for any integer $k$.

Solution: \boxed{-\frac{7}{4} \cdot 2^{-3k}}


Evaluate the limit of $\frac{x^3 - 3x + 2}{x - 1}$ as $x$ approaches $1$.


Given that the really non-overlapping minor arcs PQ, QR, and RP in circle O are x+85°, 2x+15°, and 3x-32°, respectively, find one interior angle of triangle PQR.


Given the 9 data values $70, 110, x, 60, 50, 220, 100, x, 90$ are such that the mean, median, and mode are all equal to $x$, determine the value of $x$.


Suppose that $\tfrac{3}{4}$ of $16$ apples are worth as much as $6$ pears. How many pears are worth as much as $\tfrac{1}{3}$ of $9$ apples?


Given the equation $\sqrt{9 - 3x} = x\sqrt{9 - 9x}$, determine the number of roots that satisfy this equation.


Consider two points $(a, b)$ and $(c, d)$ on the parabola whose equation is $y = mx^2 + k$. Determine the distance between these two points in terms of $a, c, m$.

Solution: \boxed{|c - a|\sqrt{1 + m^2(c+a)^2}}


A construction manager calculates that one of his two workers would take 8 hours and the other 12 hours to complete a particular paving job alone. When both workers collaborate, their combined efficiency is reduced by 8 units per hour. The manager decides to assign them both to speed up the job, and it takes them precisely 6 hours to finish. What is the total number of units paved?


Given the function $y = 2(x-a)^2 + 3(x-b)^2$, determine the value of $x$ that minimizes $y$.

Solution: \boxed{\frac{2a + 3b}{5}}


Tickets to a musical cost $x$ dollars each, where $x$ is a whole number. A group of 8th graders buys tickets costing a total of $120$, a group of 9th graders buys tickets for $180$, and a group of 10th graders buys tickets for $240$. Determine the number of possible values for $x$.


Given that Alice paid $120, Bob paid $150, and Charlie paid $180, calculate how much Bob should give to Alice so that all three of them share the costs equally.


Given a $60$-question multiple choice test, students earn $5$ points for a correct answer, $0$ points for an answer left blank, and $-2$ points for an incorrect answer. Owen's total score was $150$. Determine the maximum number of questions that Owen could have answered correctly.


Five fish can be traded for three loaves of bread, and one loaf of bread can be traded for six bags of rice, and three bags of rice can be traded for two apples. Express the number of apples equivalent to one fish in terms of a fraction.


The cost for the first $\frac{3}{4}$ mile is $3.50, and the remaining cost for additional mileage is $0.25 for each $0.1 mile. If you plan to give the driver a $3 tip, calculate the number of miles you can ride for a total cost of $15.


A person drives z miles due east at a speed of 3 minutes per mile, and then returns to the starting point driving due west at 3 miles per minute. Determine the average speed for the entire round trip in miles per hour.


Determine the value of $k$ such that the polynomial $2x^3 - 8x^2 + kx - 10$ is divisible by $x-2$.

Solution: \boxed{k = 13}


Given the teacher has 30 cookies to distribute among five students so that each student receives at least three cookies, find the number of ways the cookies can be distributed.


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

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


On circle $O$, points $E$ and $F$ are on the same side of diameter $\overline{AB}$, $\angle AOE = 40^{\circ}$, and $\angle FOB = 60^{\circ}$. Find the ratio of the area of the smaller sector $EOF$ to the area of the circle.


If $x$ cows give $x+1$ cans of milk in $x+2$ days, determine the number of days it will take $x+4$ cows to give $x+7$ cans of milk.


Determine the largest number by which the expression $n^4 - n^2$ is divisible for all possible integral values of $n$.


Determine the distance in feet between the 5th red light and the 23rd red light, where the lights are hung on a string 8 inches apart in the pattern of 3 red lights followed by 4 green lights. Recall that 1 foot is equal to 12 inches.


Elena earns 25 dollars per hour, of which $2\%$ is deducted to pay local taxes and an additional $1.5\%$ for health benefits. Calculate the total amount of cents per hour of Elena's wages that are used to pay these deductions.


Let $x = .123456789101112....497498499$, where the digits are obtained by writing the integers $1$ through $499$ in order. Calculate the $1000$th digit to the right of the decimal point.


Given $(-\frac{1}{343})^{-5/3} + 1$, simplify the expression.


If Lena has a $3 \times 7$ index card, and shortening the length of one side of this card by $2$ inches results in an area of $15$ square inches, find the area of the card in square inches if she shortens the length of the other side by $2$ inches.


Given the taxi fare of $3.00 for the first $\frac{1}{2}$ mile and then $0.30 for each additional 0.1 mile, and a $3 tip, calculate the number of miles you can ride for a total of $15.


For how many $n$ in $\{1, 2, 3, ..., 50 \}$ is the tens digit of $n^2$ odd when $n$ ends in 3 or 7?


If Haruto has a $6 \times 8$ index card, and he shortens the length of one side by $2$ inches and the area would be $36$ square inches, determine the area of the card in square inches if he instead shortens the length of the other side by $2$ inches.


Given a list of $2057$ positive integers with a unique mode occurring exactly $15$ times, find the least number of distinct values that can occur in the list.


Given Ben's current test scores are $95, 85, 75, 65,$ and $90$, find the minimum score he needs for the next test to increase his average score by at least $4$ points.


How many 4-digit numbers greater than 1000 can be formed using all the digits from the number 2013?


Let $f(x) = |x-3| + |x-5| - |2x-8|$ for $3 \leq x \leq 10$. Find the sum of the largest and smallest values of $f(x)$.


Determine the number of distinct terms in the simplified form of the expansion \( [(2a+4b)^2(2a-4b)^2]^3 \).


How many $3$-digit positive integers have digits whose product equals $30$ and at least one of the digits is an even number?


The sum of the numerical coefficients in the expansion of the binomial $(a-b)^7$ is what value?


Let \( S \) be a set of \( 7 \) integers taken from \( \{1, 2, \dots, 15\} \) with the property that if \( a \) and \( b \) are elements of \( S \) with \( a < b \), then \( b \) is not a multiple of \( a \). Find the median value of an element in \( S \).


Determine the least positive period $q$ of the functions $g$ such that $g(x+2) + g(x-2) = g(x)$ for all real $x$.


Chloe selects a real number uniformly at random from the interval $[0, 2020]$. Independently, Laurent selects a real number uniformly at random from the interval $[0, 4040]$. Calculate the probability that Laurent's number is greater than Chloe's number.


Given the equation $x^4 + y^2 = 4y$, find the number of ordered pairs of integers $(x, y)$ that satisfy this equation.


Given positive integers $a$ and $b$ are members of a set where $a \in \{2, 3, 5, 7\}$ and $b \in \{2, 4, 6, 8\}$, and the sum of $a$ and $b$ must be even, determine the smallest possible value for the expression $2 \cdot a - a \cdot b$.


Simplify the expression $\sqrt{1+ \left (\frac{x^6-1}{3x^3} \right )^2}$.

Solution: \boxed{\frac{\sqrt{x^{12} + 7x^6 + 1}}{3x^3}}


What is the smallest prime number dividing the sum $2^{14} + 7^{12}$?


Consider a sequence defined by $y_{k+1} = y_k + k$ for $k=1, 2, \dots, n-1$ and $y_1=2$, find the value of $y_1 + y_2 + \dots + y_n$.

Solution: \boxed{2n + \frac{(n-1)n(n+1)}{6}}


Find the area bounded by the x-axis, the line x = 10, and the curve defined by y = x^2 when 0 ≤ x ≤ 3, and y = 3x - 6 when 3 ≤ x ≤ 10.


The two digits in Jack's age are the same as the digits in Bill's age, but in reverse order. In three years, Jack will be three times as old as Bill will be then. Calculate the difference in their current ages.


Given two integers have a sum of 34, and when two more integers are added to the first two integers, the sum is 51, and finally, when two more integers are added to the sum of the previous four integers, the sum is 72, determine the minimum number of odd integers among the 6 integers.


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


The point $P$ moves circumferentially around the square $ABCD$. Points $M$ and $N$ are the midpoints of sides $PA$ and $PB$ respectively. Considering this motion, how many of the four quantities listed change: the length of the diagonal $AC$, the perimeter of square $ABCD$, the area of square $ABCD$, the area of quadrilateral $MBCN$?


Two concentric circles have radii $2$ and $4$. Find the probability that the chord joining two points chosen independently and uniformly at random on the outer circle does not intersect the inner circle.


Square $ABCD$ has side length $10$ and $\angle B = 90$°. Region $R$ consists of all points inside the square that are closer to vertex $B$ than any of the other three vertices. Find the area of $R$.


At the end of $1994$, Walter was one-third as old as his mother. The sum of the years in which they were born was $3900$. Calculate Walter's age at the end of $2004$.


How many different four-digit numbers can be formed by rearranging the four digits in $2023$?


Consider a positive real number to be very special if it has a decimal representation that consists entirely of digits $0$ and $5$. For instance, $\frac{500}{99}= 5.\overline{05}= 5.050505\cdots$ and $55.005$ are very special numbers. Find the smallest $n$ such that $1$ can be written as a sum of $n$ very special numbers using only up to three decimal places in each number.


A rectangular prism has dimensions $A$, $B$, and $C$ with face areas $30$, $30$, $40$, $40$, $60$, and $60$ square units, respectively. Find $A + B + C$.


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


Simplify the expression $(-\frac{1}{343})^{-2/3}$.


In a sequence where integer $n$ appears $n$ times for $1 \leq n \leq 300$, find the median of this list.


A right circular cone and a sphere share the same radius, $r$. The volume of the cone is one-third that of the sphere. Determine the ratio of the cone's altitude, $h$, to the sphere's radius.


The smallest positive integer n such that $\sqrt{n} - \sqrt{n-1} < 0.005$


Three distinct numbers are randomly selected from the set $\{-3, -2, 0, 1, 2, 5, 6\}$. Calculate the probability that the product of these three numbers is positive.


Ace runs with constant speed and Flash runs \( x^2 \) times as fast, where \( x>1 \). Flash gives Ace a head start of \( y^2 \) yards, and, at a given signal, they start off in the same direction. Determine the number of yards Flash must run to catch Ace.

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


Given that Anne, Cindy, and Ben repeatedly take turns tossing a die in the order Anne, Cindy, Ben, find the probability that Cindy will be the first one to toss a five.


Given that a CNC machine starts its daily operations at $9:00 \text{ AM}$ and completes one-fourth of the day's planned tasks by $12:30 \text{ PM}$, calculate the time at which the CNC machine will finish all tasks.

Solution: \boxed{11:00 \text{ PM}}


Let $n$ be the number of ways $15$ dollars can be changed into nickels and half-dollars, with at least one of each coin being used. Determine the value of $n$.


Given the numbers 144 and 756, determine the ratio of the least common multiple to the greatest common factor.


How many pairs $(m,n)$ of integers satisfy the equation $m + n = mn - 3$?


In a city of 400 adults, every adult owns a car, a motorcycle, or both, and some also own bicycles. If 350 adults own cars, 60 adults own motorcycles, and 30 adults own bicycles, calculate the number of car owners who do not own a motorcycle but may or may not own a bicycle.


Let $\clubsuit(x)$ denote the sum of the digits of the positive integer $x$. For how many two-digit values of $x$ is $\clubsuit(\clubsuit(x))=4$?


Sarah's flight took off from Denver at 11:07 AM and landed in Chicago at 2:45 PM. Both cities are in the same time zone. If her flight took $h$ hours and $m$ minutes, with $0 < m < 60$, find the sum $h + m$.


Given the product of 0.125 and 3.84, calculate the correct result if the decimal points were included.


Given a circle is inscribed in a square, with the circle tangent to two adjacent sides of the square at points $B$ and $C$, determine the fraction of the area of the square that lies outside the circle.

Solution: \boxed{1 - \frac{\pi}{4}}


In a particular year, the price of a commodity increased by 15% in January, decreased by 25% in February, increased by 30% in March, and fell by \(x\%\) in April. At the end of April, the price of the commodity was the same as it was at the beginning of January. Determine \(x\) to the nearest integer.


A right triangle $ABC$ with sides $AB = 12$ cm and $BC = 12$ cm is repeatedly subdivided by joining the midpoints of each triangle’s sides, and the bottom-left triangle is shaded in each iteration. Calculate the total shaded area after 100 iterations.

Solution: \boxed{24 \text{ cm}^2}


Given that $\log_b 256$ is a positive integer, calculate the number of positive integers $b$.


Given a positive integer $n$ that has $72$ divisors and $5n$ has $96$ divisors, find the greatest integer $k$ such that $5^k$ divides $n$.


Given a baseball league consisting of two five-team divisions, where each team plays every other team in its division N games and each team plays every team in the other division M games with N > 2M and M > 4, and each team plays an 82 game schedule, determine the number of games a team plays within its own division.


What is the greatest number of consecutive integers whose sum is $36$?


Evaluate the expression $\sqrt{\log_4{8} + \log_8{16}} + \sqrt{\log_2{8}}$.

Solution: \boxed{\sqrt{\frac{17}{6}} + \sqrt{3}}


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


If Amara has a $5 \times 7$ index card, and if she shortens one side of this card by $2$ inches, the card would have an area of $21$ square inches, determine the area of the card if instead she reduces the length of the other side by $2$ inches.


Two cyclists are $2k$ miles apart. When traveling in the same direction, they meet in $3r$ hours, and when traveling in opposite directions, they pass each other in $2t$ hours. Given that the speed of the faster cyclist is twice the speed of the slower cyclist when they travel towards each other, find the ratio of the speed of the faster cyclist to that of the slower cyclist when they travel in the same direction.


From the set $\{ -3, 0, 0, 4, 7, 8\}$, find the probability that the product of two randomly selected numbers is $0$.


Given that Ahn chooses a two-digit integer, subtracts twice the integer from 300, and triples the result, find the largest number Ahn can get.


Determine the smallest number of students that could be participating from these grades, given that the ratio of $10^\text{th}$-graders to $8^\text{th}$-graders is $3:2$, and the ratio of $10^\text{th}$-graders to $9^\text{th}$-graders is $5:3$.


Solve the equation $x - \frac{8}{x-2} = 5 - \frac{8}{x-2}$, where $x$ is an integer.


Given Alice has $30$ apples and each person must receive at least $3$ apples, calculate the number of ways she can distribute them equally among herself, Becky, and Chris.


Seven points on a circle are given. Four of the chords joining pairs of the seven points are selected at random. Calculate the probability that the four chords form a convex quadrilateral.


Construct a square and attach an equilateral triangle to one of its sides. On a non-adjacent side of the triangle, construct a pentagon. On a non-adjacent side of the pentagon, construct a hexagon. Continue this pattern until you construct a heptagon. Calculate the total number of visible sides.


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


Given that a password is composed of five digits (0 to 9) and no password may begin with the sequence $9,1,1,1$, calculate the number of valid passwords that are possible.


Determine the number of different total scores the basketball player could have achieved by making 8 baskets, each worth either 2, 3, or 4 points.


Three pitchers with capacities of $500$ mL, $700$ mL, and $800$ mL have orange juice. The first pitcher is $1/5$ full, the second is $3/7$ full, and the third is $1/4$ full. After topping each pitcher off with water, all the contents are poured into a large container. What fraction of the mixture in the large container is orange juice?


Given $145^9$, determine the total number of positive integer divisors that are perfect squares or perfect cubes, or both.


In rectangle ABCD, AB = 3 ⋅ AD. Points P and Q are on AB such that AP = PQ = QB. Let E be the midpoint of CD. If lines PE and QE are drawn to meet CD at points R and S respectively, find the ratio of the area of ∆PQS to the area of rectangle ABCD.


A dart board is a regular hexagon divided into regions as shown. The center of the board is a smaller regular hexagon, and each of the six surrounding regions is a trapezoid. The side length of the larger hexagon is twice that of the smaller hexagon. If a dart thrown at the board is equally likely to land anywhere on the board, determine the probability that the dart lands within the center hexagon.


Given a $3\times3$ block of calendar dates shown, first, the order of the numbers in the second row is reversed, and then the numbers on each diagonal are added. What will be the positive difference between the two diagonal sums?


A particle moves such that its speed starting from the third mile varies inversely with the square of the integral number of miles already traveled plus 1. Given that the third mile is traversed in 3 hours, determine the time, in hours, required to traverse the nth mile, where n ≥ 3.


A triangle has an area of $24$, one side of length $8$, and the median to that side of length $7.2$. Let $\theta$ be the acute angle formed by that side and the median. Calculate $\sin{\theta}$.


Dianna's teacher gave her the task to substitute numbers for $a$, $b$, $c$, $d$, $e$, and $f$ in the expression $a-(b-(c-(d+(e-f))))$ and compute the result. Dianna misinterpreted the parentheses but performed the addition and subtraction accurately, coincidentally arriving at the correct result. Dianna used the numbers $1$, $2$, $3$, $4$, and $5$ respectively for $a$, $b$, $c$, $d$, and $e$. Find the number that Dianna substituted for $f$.


Given a 24-inch by 15-inch pan of brownies cut into pieces that measure 3 inches by 2 inches, calculate the number of pieces of brownies the pan contains.


At 3:30 o'clock, calculate the angle formed by the hour and minute hands of a clock.


Given that a new kitchen mixer is listed in a store for $\textdollar 129.99$ and an online advertisement offers the same mixer for four easy payments of $\textdollar 29.99$ and a one-time shipping and handling fee of $\textdollar 19.99$, calculate how many cents are saved by purchasing the mixer through the online advertisement instead of in-store.


A 6 × 6 grid of blocks is used. In how many different ways can 4 blocks be selected such that no two blocks are in the same row or column?


The number $b = \frac{r}{s}$, where $r$ and $s$ are relatively prime positive integers, has the property that the sum of all real numbers $y$ satisfying
\[
\lfloor y \rfloor \cdot \{y\} = b \cdot y^2
\]
is $360$, where $\lfloor y \rfloor$ denotes the greatest integer less than or equal to $y$ and $\{y\} = y - \lfloor y \rfloor$ denotes the fractional part of $y$. What is $r + s$?
**A)** 861
**B)** 900
**C)** 960
**D)** 1021


A laser is placed at the point $(4,6)$. The laser beam travels in a straight line. The requirement is for the beam to hit and bounce off the $y$-axis, then the $x$-axis, again off the $y$-axis, and finally hit the point $(8,6)$. Calculate the total distance the beam will travel along this path.


For how many positive integer values of \(N\) is the expression \(\dfrac{48}{N+3}\) an integer?


Let $u = 3 + 3^q$ and $v = 3 + 3^{-q}$. Express $v$ in terms of $u$.


Let $\angle ABC = 40^{\circ}$ and $\angle ABD = 28^{\circ}$. If there is a point $E$ on line segment $BD$ such that $\angle DBE = 10^{\circ}$, determine the smallest possible degree measure for $\angle CBE$.


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


Jessica has $3p + 2$ nickels, and Samantha has $2p + 6$ nickels. Compute the difference in their money in pennies.

Solution: \boxed{5p - 20}


A right circular cone and a sphere have bases with the same radius, denoted as \( r \). If the volume of the cone is one-third that of the sphere, find the ratio of the altitude of the cone to the radius of its base.


How many ordered pairs (a, b) of positive integers satisfy the equation $a \cdot b + 82 = 25 \cdot \text{lcm}(a, b) + 15 \cdot \text{gcd}(a, b)$?


Suppose $n$ standard 6-sided dice are rolled, and the probability of obtaining a sum of 2027 equals the probability of obtaining another sum $S$. Find the smallest possible value of $S$.


Expand and simplify the expression $(1-x^2)(1+x^4+x^6)$.

Solution: \boxed{1 - x^2 + x^4 - x^8}


If $x^4$, $x^2+\frac{1}{x^2}$, and $1+\frac{1}{x}+\frac{1}{x^3}$ are multiplied, determine the degree of the resulting polynomial.


Consider the set of numbers $\{1, 10, 10^2, 10^3, \ldots, 10^{12}\}$. Find the ratio of the largest element of the set to the sum of the other twelve elements of the set rounded to the nearest tenth.


Given that the ratio of AD to AB is 4:3, AB is 36 inches, determine the ratio of the area of the rectangle to the combined area of the semicircles.


Given the area of a rectangle is $18$, plot all the ordered pairs $(w, l)$ of positive integers for which $w$ is the width and $l$ is the length of the rectangle.

Solution: \boxed{(1, 18), (2, 9), (3, 6), (6, 3), (9, 2), (18, 1)}


The number of revolutions of a wheel, with a fixed center and with an outside diameter of 10 feet, required to cause a point on the rim to go two miles, can be calculated.


Given the set $\{-9, -7, -1, 2, 4, 6, 8\}$, find the minimum possible product of three different numbers.


Given that Mona's age is a perfect square and at least half of the guesses $16, 25, 27, 32, 36, 40, 42, 49, 64, 81$ are too low, calculate Mona's age.


Given a rectangular box with dimensions A, B, and C, and surface areas of its faces given as 40, 40, 90, 90, 100, and 100 square units, determine the sum A + B + C.


In a circle with center $O$ and radius $r$, a chord $AB$ is drawn with length equal to $2r$. From $O$, a perpendicular to $AB$ meets $AB$ at $M$. From $M$ a perpendicular to $OA$ meets $OA$ at $D$. Determine the area of triangle $MDA$ in square units.


For how many integer values of $n$ does the value of $4800 \cdot \left(\frac{2}{3}\right)^n$ result in an integer?


Rectangle $ABCD$ has a length of 8 inches and a width of 6 inches. The diagonal $AC$ is divided into four equal segments by points $E$, $F$, and $G$. Determine the area of triangle $BFG$.


Given a rectangular pan of brownies that measures $24$ inches by $15$ inches is cut into pieces that measure $3$ inches by $2$ inches, calculate the number of brownies the pan contains.


Evaluating the expression $\left(\left((3+1)^{-1} \cdot 2\right)^{-1} \cdot 2\right)^{-1} \cdot 2$.

Solution: \boxed{\left(\left((3+1)^{-1} \cdot 2\right)^{-1} \cdot 2\right)^{-1} \cdot 2 = \frac{1}{2}}


A triangle has sides of length $7$ and $23$. What is the smallest whole number greater than the perimeter of any triangle with these side lengths?


When a certain unfair die is rolled, an even number is $5$ times as likely to appear as an odd number. The die is rolled twice. Calculate the probability that the sum of the numbers rolled is odd.


Given a regular dodecagon (12-sided polygon), determine the number of diagonals it has and double that number to account for possible symmetrical line segments inside the polygon that don't necessarily connect vertices.


Calculate the percentage discount d and the total amount paid if the original price of an item is P dollars and it is sold for Q dollars, where P > Q > 0, and 10 items are purchased.


If it costs $206.91 to label all the lockers consecutively, where each digit in a locker number costs three cents, determine the number of lockers at Wellington Middle School.


Three distinct vertices of a regular tetrahedron are chosen at random. Determine the probability that the plane determined by these three vertices contains points inside the tetrahedron.


Given an n-digit positive integer, determine how many charming n-digit integers exist, where the digits are an arrangement of the set {1,2,...,n} and its first k digits form an integer that is divisible by k for k = 1,2,...,n, and the entire number ends in digit 7.


Maria starts working at 7:25 A.M. and works for 9 hours, not including her 1-hour lunch break, and goes for lunch at noon. At what time will Maria's working day end?

Solution: \boxed{5:25 \text{ P.M.}}


How many whole numbers between 1 and 2000 do not contain the digits 1 or 2?


Consider a region bounded by the lines \((x-a)^2 = 25\) and \(y^2 = 9x\). Calculate the area of this region for \(a = 5\).


Given a tesseract (4-dimensional hypercube), calculate the sum of the number of edges, vertices, and faces.


Calculate the average monthly sales from Booster's Club sales data, given that the sales from January to June are $150, 90, X, 180, 210, 240$ respectively, where $X$ represents the sales in March, which were 50% greater than those in February.


Given Aniyah has a $5 \times 7$ index card and if she shortens the length of one side of this card by $2$ inches, the card would have an area of $21$ square inches, determine the area of the card in square inches if she shortens the length of the other side by $2$ inches.


Twelve points are evenly spaced, with one unit intervals, around a 3x3 square. If two of these twelve points are selected at random, calculate the probability that the two points are one unit apart.


Find the smallest interval necessarily containing the probability $p$ that both events $C$ and $D$ occur if $P(C) = \frac{5}{6}$ and $P(D) = \frac{7}{9}$.

Solution: \boxed{\left[\frac{11}{18},\frac{7}{9}\right]}


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


Given the container $3$ centimeters high, $5$ centimeters wide, and $7$ centimeters long can hold $105$ grams of clay, and a second container with triple the height, double the width, and the same length as the first container can hold $n$ grams of clay, determine the value of $n$.


The AMC 8 was first given in 1985 and has been administered annually. Jessica turned 15 years old the year she took the tenth AMC 8, in what year was Jessica born?


Given that circle O's diameter AB, angle AOE is 60 degrees, and angle FOB is 90 degrees, find the ratio of the area of the smaller sector EOF to the area of the circle.


What is the minimum number of digits to the right of the decimal point needed to express the fraction $\frac{987654321}{2^{15} \cdot 5^7}$ as a decimal?


Given that triangle ABC has a right angle at B, point D is the foot of the altitude from B, with AD = 5 and DC = 7, find the area of triangle ABC.


The taxi fare in Rivertown is $3.00 for the first $\frac{3}{4}$ mile and additional mileage charged at the rate of $0.25 for each additional 0.1 mile. You plan to give the driver a $3 tip. Determine the number of miles that can be ridden for a total of $15.


The "High School Elite" basketball league consists of $8$ teams. Each team plays every other team three times and also plays $5$ games against non-conference opponents. Calculate the total number of games in a season involving the "High School Elite" league teams.


Given a square in the coordinate plane with vertices at $(0,0)$, $(100,0)$, $(100,100)$, and $(0,100)$, determine the value of $d$ (to the nearest tenth) for which the probability that a randomly chosen point within the square is within $d$ units of a lattice point is $\tfrac{1}{4}$.


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

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


Find the number of terms with rational coefficients in the expansion of $\left(x\sqrt[4]{2}+y\sqrt[5]{3}\right)^{1250}$.


Let $M$ be the greatest five-digit number whose digits have a product of $180$. Find the sum of the digits of $M$.


Two 8-sided dice are rolled once. Let the sum of the numbers rolled be the diameter of a circle. Determine the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference.


Given that a magician's hat contains 4 blue marbles and 3 yellow marbles, marbles are drawn randomly, one at a time without replacement, until all 4 blue marbles are drawn or until all 3 yellow marbles are drawn, calculate the probability that all 4 blue marbles are drawn.


An 18-inch pizza can perfectly accommodate eight equal circles of pepperoni across its diameter, and thirty-six circles of pepperoni are placed on this pizza without overlapping. Calculate the fraction of the pizza's surface that is covered by pepperoni.


Choose three different numbers from the set $\{2,3,5,7,11,13\}$. Add any two of these numbers. Multiply their sum by the third number. What is the smallest result that can be obtained from this process?


Given the expression \[
\left(\frac{x^3+1}{x}\right)\left(\frac{y^3+1}{y}\right)+\left(\frac{x^3-1}{y}\right)\left(\frac{y^3-1}{x}\right), \quad xy \neq 0
\], simplify the expression.

Solution: \boxed{2x^2y^2 + \frac{2}{xy}}


Given that \( (a_1, a_2, \ldots, a_{15}) \) is a list of the first 15 positive integers such that for each \( 2 \leq i \leq 15 \), either \( a_i + 1 \) or \( a_i-1 \) (or both) appear somewhere before \( a_i \) in the list, and \( a_1 \) must be either 1 or 2, and \( a_{15} \) must be either 14 or 15, determine the number of such lists.

Solution: \boxed{4 \times 4096 = 16384}


Five friends did yard work over the weekend, earning amounts of $10, $30, $50, $40, and $70. Calculate the amount the friend who earned $70 will give to the others when they equally share their total earnings.


The ferry boat begins transporting tourists to an island every hour starting at 9 AM until its last trip, which starts at 4 PM. On the first trip at 9 AM, there were 120 tourists, and on each successive trip, there were 2 fewer tourists than on the previous trip. Determine the total number of tourists the ferry transported to the island that day.


Given the function g(n) = log27n if log27n is rational, and 0 otherwise, find the value of the sum from n=1 to 7290 of g(n).


Given that $40\%$ of all students initially answered "Yes" and $60\%$ answered "No", and at the end of the year $80\%$ answered "Yes" and $20\%$ answered "No", determine the minimum possible percentage of students that changed their mind during the year.


Given that John's flight departed from Chicago at 3:42 PM and arrived in Denver at 6:57 PM, both cities being in the same time zone, compute the sum of the hours and minutes that his flight took.


Determine the number of points C in a plane that exist such that the perimeter of triangle ABC is 36 units and the area of triangle ABC is 48 square units.


Given the arithmetic series starting with the term $k^2 - 1$ and each subsequent term increasing by $1$, calculate the sum of the first $2k$ terms of this series.

Solution: \boxed{2k^3 + 2k^2 - 3k}


Given that Liam has three older siblings who are triplets and the product of their four ages is 216, calculate the sum of their four ages.


Given the set $\{-10, -7, -3, 0, 2, 4, 8, 9\}$, calculate the minimum possible product of three different numbers.


Given $a_1,a_2,\dots,a_{2023}$ be a strictly increasing sequence of positive integers such that $a_1+a_2+\cdots+a_{2023}=2023^{2023}$. Determine the remainder when $a_1^3 - a_2^3 + a_3^3 - a_4^3 + \cdots - a_{2022}^3 + a_{2023}^3$ is divided by $6$.


Given two cubic polynomials $P(x)$ and $Q(x)$ where $P(x) = x^3 + 3x^2 - 3x - 9$, construct $Q(x)$ by replacing each nonzero coefficient of $P(x)$ with their arithmetic mean. Determine the value of $Q(1)$.


A school organizes a talent show with 120 students participating, each capable of singing, dancing, or acting. No student excels in all three. It's known that 50 students cannot sing, 75 students cannot dance, and 35 students cannot act. Determine how many students have exactly two of these talents.


John covers an initial segment of 40 miles in 4 hours, then takes a 1-hour break, and finally covers another 15 miles in 3 hours. Calculate John's average speed in miles per hour for the entire trip.


Simplify $\left(\sqrt[3]{8} + \sqrt{4 \frac{1}{4}}\right)^2$.

Solution: \boxed{\frac{33 + 8\sqrt{17}}{4}}


Given the number $2025$, calculate the number of different four-digit numbers that can be formed by rearranging the four digits.


Given $x^3y = k$ for a positive constant $k$, find the percentage decrease in $y$ when $x$ increases by $20\%$.


Given John owes Jenny $60$ cents and has a pocket full of $5$-cent coins, $20$-cent coins, and $50$-cent coins, calculate the difference between the largest and the smallest number of coins he can use to pay her exactly.


For \(p=1, 2, \ldots, 10\), let \(S_p\) be the sum of the first 50 terms of the arithmetic progression whose first term is \(p\) and whose common difference is \(3p\). Find the value of \(S_1 + S_2 + \cdots + S_{10}\).


The sum of the final two numbers after 4 is added to each number and then each of the resulting numbers is tripled is given by what expression in terms of \( S \)?

Solution: \boxed{3S + 24}


Two subsets of the set S = {a, b, c, d, e, f} are to be chosen so that their union is S and their intersection contains exactly three elements. In how many ways can this be done, assuming that the order in which the subsets are chosen does not matter?


The expression $a^4 - a^{-4} = (a^2 - a^{-2})(a^2 + a^{-2}) = (a^2 - a^{-2})\left( (a^2)^2 + (a^{-2})^2 + 2a^2a^{-2}\right)$. Further simplify this expression to factor it into the product of three simpler expressions.

Solution: \boxed{(a - a^{-1})(a + a^{-1})(a^2 + a^{-2})}


If $x$ and $y$ are distinct nonzero real numbers such that $x + \frac{3}{x} = y + \frac{3}{y}$ and the sum of $x$ and $y$ is 4, find the product $xy$.


Star writes down the whole numbers $1$ through $40$. Emilio copies Star's numbers, but he replaces each occurrence of the digit $3$ by the digit $2$. Calculate the difference between Star's sum of the numbers and Emilio's sum of the numbers.


When the mean, median, and mode of the list
\[4, 9, x, 4, 9, 4, 11, x\]
are arranged in increasing order, they form a non-constant geometric progression. What is the sum of all possible values of $x$?
A) 3
B) 6.83
C) 15.5
D) 17
E) 20


Calculate the minimum number of digits to the right of the decimal point needed to express the fraction $\frac{987654321}{2^{30} \cdot 5^6 \cdot 3}$.


Positive integers $a$ and $b$ are such that $a < 6$ and $b < 9$. Find the smallest possible value for $3a - 2ab$.


When $n$ standard 6-sided dice are rolled, find the smallest possible value of $S$ such that the probability of obtaining a sum of 2000 is greater than zero and is the same as the probability of obtaining a sum of $S$.


Teams $X$ and $Y$ are playing in a basketball league where each game results in a win for one team and a loss for the other team. Team $X$ has won $\tfrac{3}{4}$ of its games and team $Y$ has won $\tfrac{2}{3}$ of its games. Additionally, team $Y$ has won $5$ more games and lost $5$ more games than team $X$. Determine the number of games that team $X$ has played.


Determine the number of points $C$ such that the perimeter of $\triangle ABC$ is $60$ units, and the area of $\triangle ABC$ is $120$ square units.


Given that the merchant purchased $1200$ keychains at $0.15$ each and desired to reach a target profit of $180$, determine the minimum number of keychains the merchant must sell if each is sold for $0.45$.


Jenna bakes a $24$-inch by $15$-inch pan of chocolate cake. The cake pieces are made to measure $3$ inches by $2$ inches each. Calculate the number of pieces of cake the pan contains.


Given the numbers $5-2\sqrt{8}$, $2\sqrt{8}-5$, $12-3\sqrt{9}$, $27-5\sqrt{18}$, and $5\sqrt{18}-27$, find the smallest positive number.


If tree saplings are planted around a square garden, each sapling being 2 feet apart, calculate the number of saplings needed if one side of the garden measures 30 feet.


$R$ varies directly as $S$ and inversely as $T$. When $R = 2$ and $T = \frac {8}{5}$, $S = \frac{1}{2}$. Find $S$ when $R = 16$ and $T = \sqrt {50}$.

Solution: \boxed{\frac{25 \sqrt{2}}{2}}


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


In a chess tournament, 64 players are wearing red hats, and 68 players are wearing green hats. A total of 132 players are paired into 66 teams of 2. If exactly 20 of these teams consist of players both wearing red hats, find the number of teams composed of players both wearing green hats.


Points $A$ and $B$ are on a circle of radius $7$ and $AB = 8$. Point $C$ is the midpoint of the minor arc $AB$. Find the length of the line segment $AC$.

Solution: \boxed{\sqrt{98 - 14\sqrt{33}}}


Triangle OAB has O=(0,0), B=(8,0), and A in the first quadrant. In addition, ∠ABO=90° and ∠AOB=45°. If OA is rotated 90° counterclockwise about O, find the coordinates of the image of A.

Solution: \boxed{(-8, 8)}


Points A and B are 8 units apart. Determine the number of lines in a given plane containing A and B that are 3 units from A and 5 units from B.


If each team consists of 18 players, and each player must be paid at least $20,000, and the total salary for the entire team cannot exceed $900,000, calculate the maximum possible salary that a single player on a team can receive.


Given Shauna's scores on the first four tests are 79, 88, 94, and 91, and the maximum points for each test is 100, find the lowest score she could earn on one of the remaining two tests if she must score at least 75 on each test and maintain an average of 85 for all six tests.


Determine the coefficient of $x^6$ in the expansion of $\left(\frac{x^3}{3} - \frac{3}{x^2}\right)^9$.


A triangle $PQR$ is formed in the first quadrant such that vertex $P$ is at $(0,0)$ and points $Q$ and $R$ are on the lines $y=2x$ and $y=3x$, respectively, with both $Q$ and $R$ as lattice points. The area of triangle $PQR$ is $500,000$. Find the number of such triangles.


$800$ students were surveyed about their pasta preferences. According to the survey results: $320$ students preferred spaghetti and $160$ students preferred manicotti. Calculate the ratio of the number of students who preferred spaghetti to the number of students who preferred manicotti.


Given two poles that are $30''$ and $70''$ high standing $150''$ apart, find the height of the intersection of lines joining the top of each pole to the foot of the opposite pole.


A bag contains five pieces of paper, each labeled with one of the digits $1$, $2$, $3$, $4$, or $5$, with no repeats. Three of these pieces are drawn, one at a time without replacement, to construct a three-digit number. Determine the probability that the three-digit number is a multiple of $3$.


Given that a and b are non-zero constants, evaluate the expression $(2a^{-1} + 3b^{-1})^{-1}$.


Josanna's test scores to date are $88, 92, 75, 81, 68, 70$. What is the minimum test score she would need to reach a new average of her test scores that is $5$ points higher than the current average?


Alice and Bob start at point 15 on a circle divided into 15 equally spaced points. Alice moves 7 points clockwise each turn, while Bob moves 11 points counterclockwise. Determine the number of turns it takes for them to meet at the same point.


A rug is designed with three colors, creating three distinct rectangular regions. The areas of these regions form an arithmetic progression. The innermost rectangle has a width of 2 feet, and each of the two shaded regions surrounding it is 2 feet wide on all sides. Find the length in feet of the innermost rectangle.


Given the expression $2-(-3)-4\times(-5)-6-(-7)-8\times(-9)+10$, evaluate this expression.


Given a cell phone plan that costs $20$ dollars each month, plus $10$ cents per text message sent, plus $15$ cents for each minute used over 25 hours, determine Alex's total cost for the month of February given that he sent 150 text messages and talked for 32 hours.


Consider 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 the numerator and the denominator are increased by 1, the value of the fraction is increased by $20\%$?


Given the sum of the degree measures of the interior angles of a convex polygon is $3025^\circ$, calculate the degree measure of the missing angle.


It takes Clea 70 seconds to walk down an escalator when it is not moving, and 30 seconds when it is moving. Her walking speed increases by 50% when the escalator is moving. Determine the time it would take Clea to ride the escalator down when she is not walking.


Given a square with side length \( s \), find the ratio of the combined areas of the inscribed semicircle and quarter-circle to the area of the square.


Marie starts working on four identical tasks at 9:00 AM. She finishes the first two tasks at 11:20 AM. Calculate the time when all four tasks will be completed.

Solution: \boxed{1:40 \text{ PM}}


Given the volume expansion rate of $5$ cubic centimeters for every $4^\circ$ rise in temperature, and the initial volume of $30$ cubic centimeters at $28^\circ$, determine the volume of the gas in cubic centimeters when the temperature was $12^\circ$.


Let $x$ be the total number of marbles, then $\frac{2}{3}x$ of them are green and the rest, $\frac{1}{3}x$, are yellow. If the number of yellow marbles is tripled, find the fraction of marbles that will be yellow.


The average of 4, 6.5, 8, x, and y is 18, so what is the average of x and y?


Given the equation $y^2 + x^4 = 2x^2 y + 1$, find the absolute difference between the y-coordinates of the distinct points $(\sqrt{e}, a)$ and $(\sqrt{e}, b)$.


Given the number $2025$, calculate the number of different four-digit numbers that can be formed by rearranging its digits.


A cube has a square pyramid placed on one of its faces. Determine the sum of the combined number of edges, corners, and faces of this new shape.


Given Jason rolls three fair standard six-sided dice, determine the probability that he needs to reroll either exactly two or all three dice for the best chance of winning.


Calculate the number of positive integers less than $1200$ that are divisible by neither $8$ nor $7$.


At Fibonacci Middle School, the ratio of 10th-graders to 8th-graders is 7:4, and the ratio of 10th-graders to 9th-graders is 9:5. Find the smallest number of students that could be participating in a specific school activity.


Given that Ms. Blue receives a 12% raise each year, calculate the total percent increase in her salary after five consecutive raises.


Alex earned $x$ amount of money from a summer job. He used one-fourth of his money to buy one-half of the video games he wanted, where each video game costs the same amount. After that, he also spends one-sixth of his money on gaming accessories. Determine the fraction of his money he has left after buying all the video games and the accessories.


Given m = \(\frac{c^2ab}{a - kb}\), express b in terms of m.

Solution: \boxed{\frac{ma}{c^2a + mk}}


Two containers each contain the same number of marbles, and every marble is either red or yellow. In Container $A$ the ratio of red to yellow marbles is $7:3$, and the ratio of red to yellow marbles in Container $B$ is $4:1$. There are $120$ yellow marbles in total. Calculate the difference between the number of red marbles in Container $A$ and the number of red marbles in Container $B$.


Ana and Bonita were born on the same date in different years, n years apart. Last year Ana was 7 times as old as Bonita. This year Ana's age is the cube of Bonita's age. What is n?


Given that the calculator initially displays $\frac{1}{2}$, and the special key that replaces the number $x$ with $\frac{1}{1-x}$ is pressed 50 times, calculate the value displayed on the calculator after the 50th application of the key.


How many distinguishable rearrangements of the letters in the word "COMPLEX" have the vowels first?


A list of $2030$ positive integers has a unique mode, which occurs exactly $11$ times, what is the least number of distinct values that can occur in the list?


The vertex of the parabola $y = 2x^2 - 10x + c$ will be a point on the $x$-axis. Determine the value of $c$ for which this condition holds.


Two lines with slopes $-1$ and $3$ intersect at $(1,3)$. Find the area of the triangle enclosed by these two lines and the line $x-y=2$.


The graphs of $y = -2|x-a| + b$ and $y = 2|x-c| + d$ intersect at points $(1,7)$ and $(9,1)$. Find the sum $a+c$.


Given that Lucy bakes a $24$-inch by $20$-inch pan of brownies and she wants to cut the brownies into rectangular pieces that measure $4$ inches by $3$ inches, calculate the number of brownie pieces she can cut from the pan.


Given that 80 students were from Pinecrest Academy, 60 students were from Maple Grove School, 70 students attended from Maple Grove School, 30 of the boys were from Pinecrest Academy, and 90 students were girls, calculate the number of girls who were from Maple Grove School.


Alberto bikes at a constant rate and covers 80 miles in 4 hours. Bjorn bikes at a rate of 20 miles per hour for the first 2 hours and then increases his speed to 25 miles per hour for the next 2 hours. Determine the difference in the distance Alberto has biked compared to Bjorn after four hours.


Given that Liam's test score in base 7 has tens digit x and unit digit y, where 0 ≤ x, y < 7, determine the possible discrepancy between his incorrect sum and the correct one, after reversing the digits of his test score.

Solution: \boxed{0, 6, 12, 18, 24, 30, 36}


Determine the least number of integers among $a, b, c, d$ that can be negative such that the equation $2^a + 2^b = 5^c + 5^d$ holds.


Given the book's cover dimensions are 5 inches by 7 inches, where each reported dimension can be 0.5 inches more or less than stated, find the minimum possible area of the book cover.


Given that each of a group of $60$ girls is either redhead or brunette, and either green-eyed or brown-eyed, 20 are green-eyed redheads, 35 are brunettes, and 25 are brown-eyed, determine the number of brown-eyed brunettes.


Given two numbers whose sum is $10$ and the absolute value of whose difference is $12$, determine the equation that has these two numbers as its roots.


For how many positive integers m is $\dfrac{2310}{m^2-2}$ a positive integer?


The largest product one could obtain by multiplying two numbers in the set $\{ -8, -3, 0, 2, 4 \}$ is.


Given the operation $a\textdollar b = (a - b)^2$, evaluate $(x - y + z)^2\textdollar(y - x - z)^2$.


Estimate the population of the island of Atlantis in the year 2060, assuming the current population in 2000 is 400 and it doubles every 20 years, but reduces by 25% in 2040.


Given two equiangular polygons \(P\) and \(Q\) with different numbers of sides; each angle of \(P\) is \(p\) degrees, and each angle of \(Q\) is \(q\) degrees, determine the number of possibilities for the pair \((p, q)\) where either \(p = 2q\) or \(q = 2p\).


Given $100 \leq a \leq 300$ and $400 \leq b \leq 800$, and $a + b \leq 950$, find the smallest possible value of the quotient $\frac{a}{b}$.


If $x$ cows produce $x+3$ cans of milk in $x+4$ days, calculate the number of days it will take $x+4$ cows to produce $x+6$ cans of milk.


Given that two points $(p, q)$ and $(r, s)$ lie on the line whose equation is $y=nx+m$, determine the distance between these two points in terms of $p, r, m,$ and $n$.

Solution: \boxed{|r - p|\sqrt{1+n^2}}


Find the coefficient of \(x^9\) in the polynomial expansion of \((1+3x-2x^2)^5\).


Let \( p \), \( q \), \( r \), and \( s \) be positive integers with \( p < 3q \), \( q < 4r \), and \( r < 5s \). If \( s < 90 \), find the largest possible value for \( p \).


On a different math exam, 15% of the students got 80 points, 20% got 85 points, 25% got 90 points, 30% got 95 points, and the rest got 100 points. Find the difference between the mean and the median score on this exam.


How many subsets of two elements can be removed from the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$ so that the mean (average) of the remaining numbers is $7$?


For real numbers $a$ and $b$, define $a\textdollar b = (a - b)^2$. Calculate $(2x + y)^2\textdollar(x - 2y)^2$.

Solution: \boxed{\left(3x^2 + 8xy - 3y^2\right)^2}


Determine the number of distinct ordered pairs $(x,y)$ where both $x$ and $y$ have positive integral values, and they satisfy the equation $x^6y^6 - 19x^3y^3 + 18 = 0$.


Given that Emma wrote down one integer four times and another integer three times, and the sum of these seven numbers is 140, and one of the numbers is 20, find the other number.


Let the base and exponent of $c^d$ be quadrupled, resulting in $s=(4c)^{4d}$. Also, let $s$ equal the square of the product of $c^d$ and $y^d$, resulting in $s=(c^d)(y^d)^2$. Express y in terms of c.


Given the expression \[4^{2+4+6} - (4^2 + 4^4 + 4^6)\], evaluate this expression.


Given that runner A can beat runner B by 40 yards, and runner B can beat runner C by 30 yards, and runner A can beat runner C by 65 yards, determine the length of the race.


Given that Emily places 6 ounces of tea into a twelve-ounce cup and 6 ounces of honey into a second cup of the same size, and then adds 3 ounces of lemon juice to the second cup, then she pours half the tea from the first cup into the second, mixes thoroughly, and finally pours one third of the mixture in the second cup back into the first, calculate the fraction of the mixture in the first cup that is now lemon juice.


Given points $E$ and $F$ lie on line segment $\overline{GH}$. The length of $\overline{GE}$ is $3$ times the length of $\overline{EH}$, and the length of $\overline{GF}$ is $7$ times the length of $\overline{FH}$. What fraction of the length of $\overline{GH}$ is the length of $\overline{EF}$?


Let N be the second smallest positive integer that is divisible by every positive integer less than 8. Calculate the sum of the digits of N.


Given that a 3-digit number has a digit-sum of $27$, how many of these numbers are even?


Given the average age of 8 people in Room A is 30, the average age of 5 people in Room B is 35, and the average age of 7 people in Room C is 40, find the average age of the people in the combined group of 20 individuals.


A grocer decides to create a larger display for a special event, stacking oranges in a pyramid-like structure whose new rectangular base measures $7$ oranges by $10$ oranges. Each layer above the first still follows the rule that each orange rests in a pocket formed by four below, reducing by one orange in each dimension per layer above. Moreover, this time the display features not one but three single oranges aligned vertically at the very top. Calculate the total number of oranges used in the entire display.


Given that Alice has 30 apples, calculate the number of ways she can share them with Becky and Chris so that each of the three people has at least three apples.


Given a group of students, consisting of boys and girls, such that $50\%$ are girls initially, and after two girls leave and two boys join, $40\%$ of the group are girls, determine the initial number of girls.


Given that 63 students are wearing green shirts, 69 students are wearing red shirts, and there are 132 students in total, and the students are grouped into 66 pairs with exactly 27 pairs consisting of students wearing green shirts, determine how many pairs are composed of students both wearing red shirts.


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


Calculate the value of angle C in a triangle given that angle A = 45 degrees, angle B = 3x, and angle C = (1/2) angle B.


Determine the number of different selections Jamie can make from 7 cookies, where he has the option to choose from 4 types of cookies.


Each of two boxes contains three chips numbered $1$, $2$, $4$. Calculate the probability that the sum of the numbers on the two chips drawn from the two boxes is even.


Evaluate $(x^x)^{\sqrt{x^x}}$ at $x = 3$.

Solution: \boxed{27^{3 \sqrt{3}}}


Given that the car ran exclusively on its battery for the first $60$ miles, then ran exclusively on gasoline for the rest of the trip, using gasoline at a rate of $0.03$ gallons per mile, and averaged $50$ miles per gallon on the whole trip, determine the total distance of the trip in miles.


When simplified, $(-\frac{1}{216})^{-2/3}$, evaluate the expression.


Let $S_n = 1 - 2 + 3 - 4 + \cdots + (-1)^{n-1}n$, where $n$ is a positive integer. Calculate $S_{21} + S_{34} + S_{45}$.


If the product of two numbers is $4$ and their difference is $2$, calculate the sum of their fourth powers.


A machine is scheduled to process jobs starting at $\text{9:45}\ {\small\text{AM}}$. It finished half of the jobs by $\text{1:00}\ {\small\text{PM}}$. At what time will all the jobs be completely processed?

Solution: \boxed{4:15\text{ PM}}


Given that the video score was $50$, and $75\%$ of the votes were likes, calculate how many votes had been cast on Sangho's video at that point.


Given the sum of the interior angles of a convex polygon is $3239^\circ$, and one angle was incorrectly omitted, determine the measure of the missing angle.


Given that Suzanna increases her distance by 1.5 miles every 7 minutes and takes a 5-minute break after 21 minutes, then continues at the same speed for another 14 minutes, calculate the total distance Suzanna rides.


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


Given that segment $AB$ is both a diameter of a circle with radius $\sqrt{3}$ and a side of an equilateral triangle $ABC$, and the circle intersects side $AC$ at point $D$, find the length of $AE$, where $E$ is the midpoint of $AC$.


Given the factorial $15!$, calculate the probability that a randomly chosen positive integer divisor is an odd number.


Given the functions $f$ that satisfy $f(x+6) + f(x-6) = f(x)$ for all real $x$, determine the least common positive period $p$ for all such functions.


Calculate the value of the product: $\left(\frac{1\cdot5}{3\cdot3}\right)\left(\frac{3\cdot7}{5\cdot5}\right)\left(\frac{5\cdot9}{7\cdot7}\right)\cdots\left(\frac{95\cdot99}{97\cdot97}\right)\left(\frac{97\cdot101}{99\cdot99}\right)$.


Given $a$ and $b$ are two distinct non-zero numbers, determine the relationship between $\frac{a^2 + b^2}{2}$, $\frac{a+b}{2}$, and $\sqrt{ab}$.

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


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

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


Given that a discount sign indicating “$\frac{1}{4}$ off” was displayed, and the actual cost of the laptops is $\frac{5}{6}$ of the price at which they were sold, calculate the ratio of the cost to the posted price.


Given that Paula has exactly enough money to buy 50 popcorn bags at full price, and there is a promotional deal where one bag is full price and the next two bags are $\frac{1}{4}$ off the regular price, calculate the maximum number of popcorn bags Paula could buy.


Given the function \( g_1 \) defined on the positive integers by \( g_1(1) = 1 \) and \( g_1(n) = (p_1-1)^{e_1-1} (p_2-1)^{e_2-1} \cdots (p_k-1)^{e_k-1} \), where \( n = p_1^{e_1} p_2^{e_2} \cdots p_k^{e_k} \) is the prime factorization of \( n > 1 \), and for every \( m \geq 2 \), let \( g_m(n) = g_1(g_{m-1}(n)) \), determine the number of integers \( N \) in the range \( 1 \leq N \leq 100 \) for which the sequence \( (g_1(N), g_2(N), g_3(N), \dots) \) is unbounded.


Triangle PQR has P=(0,0), R=(8,0), and Q in the first quadrant. Additionally, ∠QRP=90° and ∠QPR=45°. Determine the coordinates of the image of Q after P is rotated 120° counterclockwise about P.

Solution: \boxed{(-4 - 4\sqrt{3}, 4\sqrt{3} - 4)}


Given two polygons $P_1$ and $P_2$, where $P_1$ is an equiangular decagon (10 sides) and each angle of $P_2$ is $kx$ degrees, with $k$ being an integer greater than $1$. If $\frac{kx}{x}$ determines the ratio between the number of sides of $P_2$ and $P_1$, find the smallest integer value of $k$ such that $P_2$ can exist as a valid polygon whose interior angles are also expressed in terms of $x$.


Given $8^y - 8^{y-1} = 56$, calculate $\left(3y\right)^y$.


Let $M$ be the greatest four-digit number whose digits have a product of 36. Find the sum of the digits of $M$.


Given a rectangular park split into a trapezoidal lawn area and two congruent isosceles right triangle flower beds, the parallel sides of the trapezoidal lawn are 15 meters and 30 meters long. Calculate the fraction of the park's total area that is occupied by the flower beds.


Let M be the greatest five-digit number whose digits have a product of 210. Find the sum of the digits of M.


Given that a regular octagon and an equilateral triangle have equal areas, determine the ratio of the length of a side of the triangle to the length of a side of the octagon.

Solution: \boxed{\sqrt{ \frac{8\sqrt{3}(1+\sqrt{2})}{3} }}


Given a regular decagon, a triangle is formed by connecting three randomly chosen vertices of the decagon. Calculate the probability that at least one of the sides of the triangle is also a side of the decagon.


Nine points are chosen on a circle, and chords are drawn connecting every pair of points. No three chords intersect at a single point inside the circle. Find the number of triangles with all three vertices in the interior of the circle.


In a three-horse race with horses A, B, and C, where no ties can occur, the odds against A winning are $4:1$, and the odds against B winning are $3:4$. Determine the odds against C winning.


A farmer buys 600 cows. He sells 500 of them for the price he paid for all 600 cows. The remaining 100 cows are sold for 10% more per cow than the price of the 500 cows. Calculate the percentage gain on the entire transaction.


Each vertex of a convex hexagon $ABCDEF$ is to be assigned a color. There are $7$ colors to choose from, and no two adjacent vertices can have the same color, nor can the vertices at the ends of each diagonal. Calculate the total number of different colorings possible.


The smallest whole number larger than the perimeter of any triangle with a side of length $6$ and a side of length $19$.


The angle between the hands of a clock at 3:30 can be calculated.


In a set of $36$ square blocks arranged into a $6 \times 6$ square, how many different combinations of $4$ blocks can be selected from that set so that no two are in the same row or column?


A point P is situated 15 inches from the center of a circle. A secant from P intersects the circle at points Q and R such that the external segment PQ measures 10 inches and segment QR measures 8 inches. Calculate the radius of the circle.


A cinema has 150 seats arranged in a row. What is the minimum number of seats that must be occupied such that the next person to come must sit adjacent to someone already seated?


Two different numbers are randomly selected from the set $\{1, 2, 5, 7, 11, 13\}$ and multiplied together. Calculate the probability that the product is an even number.


The number $25!$ has a significantly large number of positive integer divisors. Determine the probability that an odd divisor is chosen at random from these.


A square and a regular octagon have equal perimeters. If the area of the square is 16, calculate the area of the octagon.

Solution: \boxed{8 + 8\sqrt{2}}


A skateboard rolls down a slope, traveling 8 inches in the first second and accelerating so that during each successive 1-second time interval, it travels 9 inches more than during the previous 1-second interval. The skateboard takes 20 seconds to reach the end of the slope. Find the total distance, in inches, that it travels.


Nine rectangles each have widths $1, 3, 5, 7, 9, 11, 13, 15, 17$ and lengths $1^2, 2^2, 3^2, 4^2, 5^2, 6^2, 7^2, 8^2, 9^2$. Calculate the sum of the areas of all nine rectangles.


Mina writes down one integer three times and another integer four times. The total sum of these seven numbers is $140$, and one of the numbers is $20$. Determine the value of the other number.


Given that soda is sold in packs of 8, 16, and 32 cans, find the minimum number of packs needed to buy exactly 120 cans of soda.


A triangular array of $3003$ coins has $1$ coin in the first row, $2$ coins in the second row, $3$ coins in the third row, and so on up to $N$ coins in the $N$th row. Calculate the sum of the digits of $N$.


Given the original price of a backpack is $120.00 and the sales tax rate is 7%, if Alex calculates the final price by adding 7% sales tax on the original price first and then applying a 15% discount to this total, and if Sam calculates the final price by subtracting 15% of the price first and then adding 7% sales tax on the reduced price and finally adding a fixed packaging fee of $2.50, calculate Alex's total minus Sam's total.


Using three-digit powers of $3$ and $7$ in a cross-number puzzle, determine the unique possible digit for the intersecting square.


For how many values of $d$ is $2.d05 > 2.005$?


Let the roots of the quadratic equation $ax^2 + bx + c = 0$ be $\alpha$ and $\beta$. Given that one root is triple the other, determine the relationship between $a$, $b$, and $c$.

Solution: \boxed{3b^2 = 16ac}


Given the equation \( 4 + 2\cos\theta - 3\sin2\theta = 0 \), determine the number of values of \( \theta \) in the interval \( 0 < \theta \leq 2\pi \) that satisfy the equation.


Given a sequence of 0s and 1s of length 21 that begins with a 0, ends with a 0, contains no two consecutive 0s, and contains no three consecutive 1s, determine the number of such sequences.


A square has a computed area of $1.4456$ square feet rounded to the nearest ten-thousandth of a square foot after its original area has been increased by $0.0001$ square feet. Find the number of significant digits in the measurement of the side length of the original square.


Given that $20\%$ of the students are juniors and $80\%$ are seniors with an overall average score of $86$, and the average score of the seniors is $85$, calculate the score received by each of the juniors.


Given that John scored 85% on 28 tests out of 40 tests, in order to achieve an 85% score on 60 tests, determine the minimum number of remaining tests he can afford to score below 85%.


If Menkara has a $5 \times 7$ index card, and if she shortens the length of one side of this card by 2 inches, the card would have an area of 21 square inches, determine the area of the card in square inches if instead, she doubles the length of the other side.


A child has $3$ red, $3$ blue, and $4$ green cubes and wants to build a tower $9$ cubes high. How many different towers can the child build if exactly one cube is left out?


Points M and N are the midpoints of sides PC and PD of triangle PCD. As P moves along a line that is parallel to side CD, determine the number of the four quantities listed that change: the length of the segment MN, the perimeter of triangle PCD, the area of triangle PCD, the area of trapezoid CDNM.


What is the remainder when $1^2 + 2^2 + 3^2 + \cdots + 25^2$ is divided by 6?


Find the smallest whole number that is larger than the sum $3\dfrac{1}{3}+5\dfrac{1}{4}+7\dfrac{1}{6}+9\dfrac{1}{8}$.


Given Jo and Blair take turns counting, with each number said by each person being two more than the last number said by the other person, determine the 30th number said, starting from 2 and including 20, and then from 23 incrementing by 3 instead of 2.


Given a circle of radius $3$ units, find the area of the region consisting of all line segments of length $6$ units that are tangent to the circle at their midpoints.


Given $200(200 + 5) - (200 \cdot 200 + 5)$, calculate the expression.


Lila has a rectangular notecard measuring $5 \times 7$ inches. She decides to shorten one side of the notecard by $2$ inches, resulting in a new area of $21$ square inches. Determine the new area of the notecard if Lila instead shortens the length of the other side by $2$ inches.


What number should be removed from the set $\{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}$ so that the average of the remaining numbers is $7.1$?


The population doubles every 20 years starting from an initial population of 500 in the year 2023. Determine the year when the population is closest to 8,000.


Given the graph of $y = mx + 3$ passes through no lattice point with $0 < x \le 50$ for all $m$ such that $\frac{1}{3} < m < b$, find the maximum possible value of $b$.


Ten points are spaced evenly along the perimeter of a rectangle measuring $3 \times 2$ units. What is the probability that the two points are one unit apart?


A circle of radius 3 is centered at point $A$. An equilateral triangle with side length 4 has one vertex at $A$. Calculate the difference between the area of the region that lies inside the circle but outside the triangle and the area of the region that lies inside the triangle but outside the circle.

Solution: \boxed{9\pi - 4\sqrt{3}}


Determine the number of pairs of regular polygons with sides of unit length that have a ratio of their interior angles of $5:3$.


Given that 4500 Euros equals 3900 pounds, calculate how many Euros the tourist will receive for exchanging 3000 pounds.


In a rhombus ABCD, the longer diagonal is three times the length of the shorter diagonal. If the area of the rhombus is $L$ square units, express the side length of the rhombus in terms of $L$.


Points $B$, $C$, and $E$ lie on $\overline{AD}$. The length of $\overline{AB}$ is $3$ times the length of $\overline{BD}$, and the length of $\overline{AC}$ is $7$ times the length of $\overline{CD}$. Point $E$ is situated such that $\overline{DE}$ is twice the length of $\overline{CE}$. Express the length of $\overline{BC}$ as a fraction of $\overline{AD}$.


Given the square defined by the vertices (0,0), (0,5), (5,5), and (5,0), an initially positioned frog at point (2, 2) makes random jumps of length 1 parallel to the coordinate axes. Calculate the probability that the sequence of jumps ends on a vertical side of the square.


In square $ABCD$, point $E$ is on $AB$ such that $AE = 3EB$, and point $F$ is on $CD$ such that $DF = 3FC$. Determine the ratio of the area of triangle $BFC$ to the area of square $ABCD$.


What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 70?


A rectangular floor that is 15 feet wide and 20 feet long is tiled with one-foot square tiles. A bug starts at the midpoint of one of the shorter sides and walks in a straight line to the opposite side's midpoint. Calculate the number of tiles the bug visits, including the first and the last tile.


Given Ella rode her bicycle for 4 days, with distances of 3 miles each day, at speeds of 6 miles per hour on Monday, 4 miles per hour on Tuesday, 5 miles per hour on Thursday, and 3 miles per hour on Friday. If Ella had always cycled at 5 miles per hour, determine the difference in time spent bicycling.


Given Angie, Bridget, Carlos, Diego, and Eliza are seated at random around a circular table with each person occupying one side, find the probability that Angie and Carlos are seated opposite each other, and Bridget is not sitting next to Carlos.


What is the value of the expression $\sqrt{25\sqrt{16\sqrt{9}}}$?


Given that Chloe's estimate of 40,000 is within $5\%$ of the actual attendance in Chicago, Derek's estimate of 55,000 is within $15\%$ of the actual attendance in Denver, and Emma's prediction of 75,000 is within $10\%$ of the actual count in Miami, calculate the largest possible difference between the numbers attending any two of these three events to the nearest 1,000.


Given Suzanna rides her bike at a constant rate as shown by a graph, if she rides for 40 minutes at the same speed, how many miles would she have ridden?


A supermarket has 150 crates of oranges. Each crate contains at least 125 oranges and at most 149 oranges. What is the largest integer n such that there must be at least n crates containing the same number of oranges?


When Cheenu was a boy, he could run 20 miles in 4 hours. As a middle-aged man, he could bicycle 30 miles in 2 hours. Now, as an old man, he can walk 8 miles in 4 hours. Determine the difference in time it takes for him to walk a mile now compared to when he bicycled a mile as a middle-aged man.


The three numbers taken in pairs have sums of 18, 23, and 27. Find the middle number.


Given a wheel with a fixed center and an outside diameter of $8$ feet, determine the number of revolutions required for a point on the rim to travel half a mile.


A point is chosen at random within a rectangle in the coordinate plane whose vertices are (0, 0), (4040, 0), (4040, 2020), and (0, 2020). The probability that the point is within $d$ units of a lattice point is $\frac{1}{4}$. What is $d$ to the nearest tenth?


The length AB of the line segment where the region in three-dimensional space consisting of all points within 4 units from AB has a total volume of 320π.


Given the integer $2020$, find the sum of its prime factors.


Let $m = 2011^2 + 2^{2011}$. Determine the units digit of $m^3 + 3^m$.


Three fair coins are to be tossed once. For each head that results, one fair die is to be rolled. Determine the probability that the sum of the die rolls is odd.


Given that $15\%$ of the students scored $60$ points, $25\%$ scored $75$ points, $35\%$ scored $85$ points, $20\%$ scored $95$ points, and the rest scored $110$ points, calculate the difference between the mean and the median score of the students' scores on this test.


Given the numbers $4, 5, 3, 3, 6, 7, 3, 5$, calculate the sum of the mean, median, and mode.


Given positive real numbers $x \neq 1$ and $y \neq 1$ satisfy $\log_3{x} = \log_y{81}$ and $xy = 243$, calculate $(\log_3{\tfrac{x}{y}})^2$.


Given a cube with side length $n$ painted red on all six faces, cut into $n^3$ smaller cubes of unit side length, and having exactly one-third of the total number of faces of these small cubes be red, determine the value of $n$.


Calculate the geometric mean of the expressions $\frac{x + a}{x}$ and $\frac{x - a}{x}$.

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


The runners run at speeds of 3.2, 4.0, 4.8, 5.6, and 6.4 meters per second. They stop once they are all together again somewhere on the circular 400-meter track. Determine the time in seconds the runners run.


Let $a$, $b$, $c$, and $d$ be positive integers with $a < 3b$, $b < 2c$, and $c < 5d$. If $d < 150$, calculate the largest possible value for $a$.


The product of two 999-digit numbers $400400400...\text{(300 times)}...400$ and $606606606...\text{(300 times)}...606$ has tens digit \( A \) and units digit \( B \). What is the sum of \( A \) and \( B \)?


A regular tetrahedron has four vertices. Three vertices are chosen at random. Determine the probability that the plane determined by these three vertices contains points inside the tetrahedron.


Two congruent squares, $ABCD$ and $EFGH$, each have a side length of $20$. They overlap to form the $20$ by $35$ rectangle $AEHD$. Calculate the percentage of the area of rectangle $AEHD$ that is shaded and also find the unshaded area.


Given that Angie, Bridget, Carlos, and Diego are seated at random around a square table, one person to a side, determine the probability that Bridget and Carlos are seated next to each other.


Two numbers are such that their difference, their sum, and twice their product are to one another as $2:8:30$. Find the product of the two numbers.


Karl's rectangular vegetable garden is 22 feet by 50 feet, while Makenna's garden, initially 30 feet by 46 feet, has a 1-foot wide walking path taking up space around the entire inside perimeter, effectively reducing the planting area. Calculate the difference in area between the two gardens.


Given the expression $(16^{\log_2 2023})^{\frac{1}{4}}$, evaluate the value of the given expression.


Jill's grandmother takes three quarters of a pill every three days to manage her condition, and a medication supply contains 60 pills. Determine the approximate time in months that this supply will last.


Find the number halfway between $\frac{1}{6}$, $\frac{1}{7}$, and $\frac{1}{8}$.


Six test scores have a mean of 85, a median of 88, and a mode of 90. The highest score exceeds the second highest by 5 points. Find the sum of the three highest scores.


Given the length of the straight sections of the track is 200 meters, the width between the inside and outside of the track is 8 meters, and it takes Keiko 48 seconds longer to jog around the outside edge of the track than around the inside edge, determine Keiko's jogging speed in meters per second.


Given the number $1225$, find the number of different four-digit numbers that can be formed by rearranging its digits.


A triangle and a trapezoid share the same altitude and have equal areas. The base of the triangle is 24 inches. If one base of the trapezoid is 10 inches longer than the other base, find the median of the trapezoid.


Given that any even integer greater than 7 can be expressed as the sum of two different prime numbers, determine the largest possible difference between the two primes for the even number 156.


Given that a 7-digit palindrome must consist of the digits 1, 1, 4, 4, 4, 6, 6, calculate the total number of such palindromes.


A point is chosen randomly from within a circular region with radius $r$. A related concentric circle with radius $\sqrt{r}$ contains points that are closer to the center than to the boundary. Calculate the probability that a randomly chosen point lies closer to the center than to the boundary.


Given the expression $(3(3(3(3(3(3+2)+2)+2)+2)+2)+2)$, calculate its value.


How many different integers can be expressed as the sum of three distinct members of the set $\{2, 5, 8, 11, 14, 17, 20, 23\}$?


What number should be removed from the list $1,2,3,4,5,6,7,8,9,10,12$ so that the average of the remaining numbers is $6.5$?


Given that the monogram of a person's first, middle, and last initials is in alphabetical order with no letter repeated, and the last initial is 'M', calculate the number of possible monograms.


Given that the euro is worth 1.1 dollars, and Charles has 600 dollars and Fiona has 450 euros, calculate the percentage by which the value of Fiona's money is greater than or less than the value of Charles' money.


Given the graphs of $y = -|x-(a+1)| + b$ and $y = |x-(c-1)| + (d-1)$ intersect at points $(3,4)$ and $(7,2)$. Find $a+c$.


Given that M/8 = 8/N, where M ≤ 64, calculate the number of ordered pairs of positive integers (M,N).


Given that the base $b$ is a positive integer that satisfies the equation $\log_{b} 1024$ is a positive integer, calculate the number of values of $b$.


What is the hundreds digit of $(30! - 20! +10!)$?


Given that John drove at an average speed of 60 km/h, then stopped 30 minutes for lunch, and then drove at an average speed of 90 km/h, covering a total distance of 300 km in 4 hours including his lunch break, determine the equation that could be used to solve for the time t in hours that John drove before his lunch break.

Solution: \boxed{60t + 90\left(\frac{7}{2} - t\right) = 300}


Let $n$ be the smallest positive integer such that $n$ is divisible by $30$, $n^2$ is a perfect cube, and $n^3$ is a perfect square. What is the number of digits of $n$?


If a medium jar can hold $50$ milliliters of spices and a larger pantry container can hold $825$ milliliters of spices, and at most one jar might not be transferred successfully, calculate the minimum number of medium jars Harry must prepare to fill a large pantry container.


Find the equation of the straight line passing through the point (1,3) and perpendicular to the line 2x-6y-8=0.

Solution: \boxed{y + 3x - 6 = 0}


Given the expressions $k(a+bc)$ and $(a+b)(a+c)$ are equal, where $k$ is a constant, and the condition $a + b + c = 1$ holds, determine the value of $k$.


In a survey with 45 people preferring Apples, 75 people preferring Bananas, 30 people preferring Cherries, and 50 people preferring Dragonfruit, calculate the percentage of surveyed people who preferred Bananas.


Given the expression $\log_4{16} \div \log_{4}{\frac{1}{16}} + \log_4{32}$, simplify the expression.


Given that the students in sixth grade run an average of $18$ minutes per day, seventh grade run an average of $20$ minutes per day and eighth grade run an average of $12$ minutes per day, and there are three times as many sixth graders as seventh graders and the number of eighth graders is half the number of seventh graders, and seventh graders spend an extra $5$ minutes on a stretching routine, determine the average number of minutes spent on physical activities per day by these students.


Find the area of the rectangular region defined by the vertical lines $x = 2e$ and $x = -3f$, and the horizontal lines $y = 4g$ and $y = -5h$, where $e, f, g,$ and $h$ are positive numbers.

Solution: \boxed{8eg + 10eh + 12fg + 15fh}


Given the sum of two numbers is S, if 5 is added to each number and then each of the resulting numbers is tripled, determine the sum of the final two numbers.

Solution: \boxed{3S + 30}


Consider all triangles ABC with AB = AC, and point D on AC such that BD ⊥ AC. Given conditions are AC and CD are integers, and BD^2 = 68. Find the smallest possible value of AC when CD is a prime number less than 10.


Solution: \boxed{\frac{45}{9 + x}}


Let $T_1$ be a triangle with side lengths $20, 21,$ and $29$. For $n \ge 1$, if $T_n = \triangle ABC$ and $D, E,$ and $F$ are the points of tangency of the incircle of $\triangle ABC$ to the sides $AB, BC$, and $AC,$ respectively, then $T_{n+1}$ is a triangle with side lengths $AD, BE,$ and $CF,$ if it exists. What is the perimeter of the last triangle in the sequence $( T_n )$.


Consider the quadratic equation $2x^2 - 5x + m = 0$. Find the value of $m$ such that the sum of the roots of the equation is maximized while ensuring that the roots are real and rational.


Given medians AD and BE of triangle ABC are perpendicular, AD=6, and BE=9. Compute the area of triangle ABC.


The probability that a randomly chosen divisor of $25!$ is odd.


Find the arithmetic mean of $\frac{x + 2a}{x}$ and $\frac{x - 3a}{x}$, where $x \neq 0$ and $a$ is a constant.

Solution: \boxed{1 - \frac{a}{2x}}