Given the equation $x^2 + y^2 = m^2$ is tangent to the line $x - y = m$, find the possible value(s) of $m$.

Solution: \boxed{m = 0}


Find the number of pairs (x, y) of integers that satisfy the equation x^3 + 4x^2 + x = 18y^3 + 18y^2 + 6y + 3.


The printing machine starts operating at $\text{9:00}\ {\small\text{AM}}$. By $\text{12:00}\ {\small\text{PM}}$, it has completed half of the day's printing job. Find the time when the printing machine will finish the entire job.

Solution: \boxed{3:00 PM}


Samantha gets up at 7:00 a.m., catches the school bus at 8:00 a.m. and arrives home at 5:30 p.m. Given that she has 7 classes that last 45 minutes each, has 45 minutes for lunch, and participates in a 1.5 hour chess club after class, calculate the total number of minutes she spends on the bus.


Suppose that $f(x+2)=5x^2 + 2x + 6$ and $f(x)=ax^2 + bx + c$. Find the value of $a + 2b + 3c$.


Given that all the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ are arranged in a $3\times3$ grid such that if two numbers are consecutive, they occupy squares that share an edge, and the numbers in the four corners add up to $20$, determine the number in the center of the grid.


The sides of a triangle are $40$, $90$, and $100$ units. Given that an altitude is dropped upon the side of length $100$, calculate the length of the larger segment cut off on this side.


Calculate the number of terms in the expansion of $[(a+3b)^{3}(a-3b)^{3}]^{2}$ when fully simplified.


When a certain biased die is rolled, an even number is twice 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.


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


Given a cyclic quadrilateral $ABCD$ inscribed in a circle, with side $AB$ extended beyond $B$ to point $E$. If $\angle BAD = 75^{\circ}$ and $\angle ADC = 83^{\circ}$, find $\angle EBC$.


Given Chloe chooses a real number uniformly at random from the interval $[0, 1000]$ and Laurent chooses a real number uniformly at random from the interval $[0, 2000]$, find the probability that Laurent's number is greater than twice Chloe's number.


Alice has 4 sisters and 6 brothers. Given that Alice's sister Angela has S sisters and B brothers, calculate the product of S and B.


Given Evan and his two older twin sisters' ages are three positive integers whose product is 162, calculate the sum of their ages.


Given that the product of Kiana's older brother, Kiana, and Kiana's younger sister's ages is 72, find the sum of their three ages.


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


A circle with a radius of 8 is inscribed in a rectangle. The ratio of the length of the rectangle to its width is 3:1. Calculate the area of the rectangle.


Cagney can frost a cupcake every 20 seconds, Lacey can frost a cupcake every 30 seconds, and Jamie can frost a cupcake every 15 seconds. Working together, calculate the number of cupcakes they can frost in 10 minutes.


John works on three tasks sequentially. The second task takes twice as long as the first, and the third task takes the same amount of time as the first. He starts the first task at 9:00 AM and completes the second task by 11:30 AM. Determine the completion time of the third task.

Solution: \boxed{12:20 \text{ PM}}


Given that Mark drove the first one-fourth of his trip on a country road, the next 30 miles on a highway, and the remaining one-sixth on a city street, calculate the total distance of Mark's trip in miles.


How many positive odd multiples of $5$ less than $5000$ are perfect squares?


Determine the smallest value of n for which the total cost, after a 5% sales tax, is exactly n dollars.


A woman was born in the nineteenth century and was $x$ years old in the year $x^2$. Find the birth year of the woman.


Given that LeRoy, Bernardo, and Camilla paid $A$, $B$, and $C$ dollars, respectively, with $A < B < C$, calculate the total number of dollars LeRoy must give to Bernardo and Camilla so that they share the costs equally.

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


Given that Mary is $25\%$ older than Sally, and Sally is $45\%$ younger than Danielle, the sum of their ages is $25.5$ years. Express Mary's age on her next birthday as an integer.


Charlie sets out to read a book in a week for her summer project. She reads an average of 40 pages per day for the first two days and an average of 45 pages per day for the next four days. She completes the book by reading 20 pages on the seventh day. Determine the total number of pages in the book.


Given the values $1256, 2561, 5612, 6125$, calculate the sum.


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


Evaluate the expression $(16)^{-2^{-3}}$.


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


Keiko walks around an athletic track that has a similar structure as mentioned before, with straight sides and semicircular ends. The width of the track is $8$ meters. It takes her $48$ seconds longer to walk around the outside edge of the track than around the inside edge. Assuming Keiko walks at a constant speed throughout, determine her speed in meters per second.


Given three positive consecutive integers whose sum is $c = 3a + 3$, compute the average of the next three consecutive integers starting from $c$.

Solution: \boxed{3a + 4}


A fair coin is tossed 4 times. Calculate the probability of getting at least two consecutive heads.


Let $U$ be the set of the $1500$ smallest positive multiples of $5$, and let $V$ be the set of the $1500$ smallest positive multiples of $8$. Calculate the number of elements common to $U$ and $V$.


The real numbers \(a, b, c\) form a geometric sequence and satisfy \(a \leq b \leq c \leq 1\). The quadratic \(bx^2 + cx + a\) has exactly one root. Find this root.


The total area of all the faces of a rectangular solid is $26\text{cm}^2$, and the total length of all its edges is $28\text{cm}$. Determine the length in cm of any one of its interior diagonals.


A circle has (2,2) and (10,8) as the endpoints of its diameter. It intersects the x-axis at a second point. Find the x-coordinate of this point.


In trapezoid $ABCD$, $\overline{AD}$ is perpendicular to $\overline{DC}$, $AD = AB = 5$, and $DC = 10$. Additionally, $E$ is on $\overline{DC}$ such that $\overline{BE}$ is parallel to $\overline{AD}$. Find the area of the parallelogram formed by $\overline{BE}$.


An inverted cone filled with water has a base radius of 15 cm and a height of 15 cm. The water is then transferred into a cylindrical container with a base radius of 30 cm. Determine the height of the water level in the cylinder.


Given the function $f$ that satisfies $f(x+5)+f(x-5) = f(x)$ for all real $x$, determine the smallest positive period $p$ for these functions.


A bug starts crawling on a number line from position -3. It first moves to -7, then turns around and stops briefly at 0 before continuing on to 8. Calculate the total number of units the bug crawls.


Given the set $\{-7, -5, -3, 0, 2, 4, 6\}$, calculate the maximum possible sum of three different numbers where at least one of the numbers must be negative.


How many primes less than $150$ have $3$ as the ones digit?


Given trapezoid $ABCD$, $\overline{AD}$ is perpendicular to $\overline{DC}$, $AD = AB = 4$, and $DC = 8$. Additionally, $E$ is on $\overline{DC}$, and $\overline{BE}$ is parallel to $\overline{AD}$. Furthermore, $E$ is exactly midway between $D$ and $C$. Find the area of $\triangle BEC$.


If rose bushes are spaced about 2 feet apart, calculate the number of bushes needed to surround a circular patio whose radius is 15 feet.


Given the scale model of the Empire State Building has a ratio of $1:50$, calculate the height in feet of the scale model of the building, rounded to the nearest whole number.


Given a square is originally painted blue, each time the square is changed, the middle ninth of each blue square turns red. After four changes, determine the fractional part of the original area of the blue square that remains blue.


Given that set $A$ has $30$ elements, set $B$ has $20$ elements, and set $C$ has $10$ elements, calculate the smallest possible number of elements in $A \cup B \cup C$.


A store sells 9 pencils and 10 notebooks for $\mathdollar 5.35$ and 6 pencils and 4 notebooks for $\mathdollar 2.50$. If a special offer allows buying pencils in packs of 4 for a 10% discount on the per-pencil price, calculate the cost of 24 pencils and 15 notebooks.


Given that a triangular corner with side lengths DB=EB=1.5 is cut from an equilateral triangle ABC of side length 4.5, determine the perimeter of the remaining quadrilateral.


Given \(\frac{\log{a}}{p}=\frac{\log{b}}{q} =\frac{\log{c}}{r} =\log{x}\), where all logarithms are to the same base and \(x \not= 1\). If \(\frac{a^2}{bc}=x^y\), find \(y\).

Solution: \boxed{2p - q - r}


A box contains 35 red balls, 25 green balls, 22 yellow balls, 15 blue balls, 14 white balls, and 12 black balls. Calculate the minimum number of balls that must be drawn from the box without replacement to guarantee that at least 18 balls of a single color will be drawn.


Alice and Bob play a game on a circle divided into 18 equally-spaced points, labeled from 1 to 18. Both start at point 18. Alice moves 7 points clockwise each turn while Bob moves 13 points counterclockwise each turn. Find the minimum number of turns required for Alice and Bob to meet at the same point.


Given the equations of the parabola \( y = x^2 + 2x - 3 \) and the horizontal line \( y = m \), where \( -8<m<5 \), determine the value of \( s \) as \( m \) approaches 0, where \( s = \frac{M(-2m) - M(m)}{2m} \), and \( M(m) \) is the \( x \)-coordinate of the left-most intersection point of the parabola and the line.


Lisa and her father observed a train passing by a crossing, with 10 train cars passing in 15 seconds at a constant speed. The entire train passed the crossing in 3 minutes and 30 seconds. Determine the total number of train cars that passed the crossing.


Two cyclists, C and D, start from Albany towards New York, a distance of 100 miles. Cyclist C travels 5 mph slower than cyclist D. After reaching New York, D turns back immediately and meets C 20 miles from New York. Find the speed of cyclist C.


Moe has a new rectangular lawn measuring 100 feet by 180 feet. He uses a mower with a swath that cuts 30 inches wide, but he overlaps each cut by 6 inches to ensure complete coverage. Moe walks at a speed of 4500 feet per hour while mowing. Calculate the number of hours it will take Moe to mow his entire lawn.


Elmer's new car gives $100\%$ better fuel efficiency, measured in kilometers per liter, than his old car. His new car uses diesel fuel, which is $25\%$ more expensive per liter than the gasoline his old car used. Determine the percent by which Elmer will save money if he uses his new car instead of his old car for a long trip.


Let $Q$ be the product of two numbers: 7,123,456,789 and 23,567,891,234. Determine the number of digits in $Q$.


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


Mr. Thompson has three nephews who visit him regularly. One visits him every six days, another every eight days, and the last one every ten days. All three visited him on January 1, 2020. Calculate the number of days during the year 2020 when Mr. Thompson did not receive a visit from any of his nephews.


Three runners start running simultaneously from the same point on a 400-meter circular track, running clockwise. They maintain constant speeds of 5.0, 5.5, and 6.0 meters per second. Determine the time at which all the runners will be together again.


For how many integers x does a triangle with side lengths 12, 30, and x have all its angles acute?


Consider a grid with dimensions $9 \times 9$, consisting of alternating dark and light squares. Calculate the difference in the number of dark squares compared to light squares.


A collector offers to buy state quarters for 500% of their face value. If Bryden has four state quarters, each with a face value of $0.50, calculate the total amount he will receive from the collector.

Solution: \boxed{10 \text{ dollars}}


What is the sum of the digits of the greatest prime number that is a divisor of $32,767$?


Given $(b_1, b_2, ... b_7)$ be a list of the first 7 even positive integers such that for each $2 \le i \le 7$, either $b_i + 2$ or $b_i - 2$ or both appear somewhere before $b_i$ in the list, determine the number of such lists.


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


Given a set $T$ of 5 integers taken from $\{2,3, \dots, 15\}$ such that if $c$ and $d$ are elements of $T$ with $c<d$, then $d$ is not a divisor of $c$, determine the greatest possible value of an element in $T$.


Find the result when the Greatest Common Divisor (GCD) of $7260$ and $540$ is diminished by $12$, then add $5$ to the result.


Given the first container is $\frac{4}{5}$ full of water and the second container is initially empty, with all the water being poured into the second container filling it to $\frac{2}{3}$ of its capacity, determine the ratio of the volume of the first container to the volume of the second container.


The taxi fare in Metropolis City is $3.00 for the first $\frac{3}{4}$ mile and additional mileage charged at the rate $0.30 for each additional 0.1 mile. You plan to give the driver a $3 tip. Calculate the number of miles you can ride for $15.


First, $a$ is chosen at random from the set $\{1, 2, 3, \ldots, 99, 100\}$, and then $b$ is chosen at random from the same set. The probability that the integer $2^a + 5^b$ has a units digit of $8$ is?
A) $\frac{1}{16}$
B) $\frac{1}{8}$
C) $\frac{3}{16}$
D) $\frac{1}{5}$
E) $0$


Joe had walked one-third of the way from home to school when he realized he was very late. He ran the rest of the way to school. He ran 4 times as fast as he walked. Joe took 9 minutes to walk the one-third distance to school. Calculate the total time it took Joe to get from home to school.


Samantha leaves her house at 7:15 a.m. to catch the school bus, starts her classes at 8:00 a.m., and has 8 classes that last 45 minutes each, a 40-minute lunch break, and spends an additional 90 minutes in extracurricular activities. If she takes the bus home and arrives back at 5:15 p.m., calculate the total time spent on the bus.


Three distinct vertices of a tetrahedron are chosen at random. Find the probability that the plane determined by these three vertices does not pass through the interior of the tetrahedron.


Calculate the product of the 10 factors $\Big(1 - \frac{1}{2}\Big)\Big(1 - \frac{1}{3}\Big)\Big(1 - \frac{1}{4}\Big)\cdots\Big(1 - \frac{1}{11}\Big)$.


A circle with center $O$ is divided into $18$ equal arcs. Points on the circumference are labeled sequentially from $A$ to $R$. Calculate the sum of the degrees in angles $x$ and $y$, where $x$ is the angle at circumference subtended by three arcs, and $y$ is subtended by five arcs.


Suppose the euro is now worth 1.5 dollars. If Marco has 600 dollars and Juliette has 350 euros, find the percentage by which the value of Juliette's money is greater than or less than the value of Marco's money.


Given 18 parking spaces in a row, 14 cars arrive and occupy spaces at random, followed by Auntie Em, who requires 2 adjacent spaces, determine the probability that the remaining spaces are sufficient for her to park.


Let \( m \) be the smallest positive integer such that \( m \) is divisible by 30, \( m^2 \) is a perfect cube, and \( m^3 \) is a perfect square. Determine the number of digits of \( m \).


Given that $\log_8 (y-4) = 1.5$, find the value of $y$.

Solution: \boxed{16\sqrt{2} + 4}


Given a $4 \times 4$ square grid, where each unit square is painted white or black with equal probability and then rotated $180\,^{\circ}$ clockwise, calculate the probability that the grid becomes entirely black after this operation.


Moe uses a mower to cut his rectangular 120-foot by 100-foot lawn. The mower cuts a swath 30 inches wide, but he overlaps each cut by 6 inches to ensure no grass is missed. He walks at the rate of 4000 feet per hour while pushing the mower. Determine the number of hours it will approximately take Moe to mow the entire lawn.


What is the arithmetic mean of the first n positive even integers?


If $|2x - \log z| = 2x + \log z$ where $2x$ and $\log z$ are real numbers, determine $x$ and $z$.
**A)** $x = 0$
**B)** $z = 1$
**C)** $x = 0 \text{ and } z = 1$
**D)** $x(z-1) = 0$
**E)** None of these

Solution: \boxed{x(z-1) = 0}


A large rectangle has area 168 square units and is divided into four rectangles by two segments, with one segment parallel to the width and the other not necessarily parallel to the length. Given three of these resulting rectangles have areas 33, 45, and 20, find the area of the fourth rectangle.


What is the sum of the digits of the greatest prime number that is a divisor of $32,767$?


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


The state income tax where Linda lives is levied at the rate of p% of the first $35000 of annual income plus (p + 3)% of any amount above $35000. Linda noticed that the state income tax she paid amounted to (p + 0.5)% of her annual income. What is her annual income?


Given a regular tetrahedron with edges of length 2, calculate the length of the shortest path from the midpoint of one edge to the midpoint of the opposite edge, subject to a wind multiplier of 1.5.


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


Define a sequence \( \{u_n\} \) such that \( u_1=7 \) and the relationship \( u_{n+1}-u_n = 5n + 2(n-1) \) for \( n=1,2,3,\ldots \). Express \( u_n \) as a polynomial in \( n \) and determine the algebraic sum of its coefficients.


Given a cell phone plan costing $30$ dollars each month, plus $0.10$ dollars for each text message sent, plus $0.12$ dollars for each minute used over $25$ hours, and a $5$ dollar discount for total talking time exceeding $35$ hours, calculate the total cost for a customer who sent $150$ text messages and talked for $36$ hours in a month.


How many subsets containing three different numbers can be selected from the set $\{ 12, 18, 25, 33, 47, 52 \}$ so that the sum of the three numbers is divisible by 3?


Let $s_1$ be the sum of the first $n$ terms of the arithmetic sequence $3, 10, \cdots$ and let $s_2$ be the sum of the first $n$ terms of the arithmetic sequence $5, 8, \cdots$. Determine the number of values of $n$ for which the equation $s_1 = s_2$ is true.


The area of $\triangle ABC$ is $120$. Given that the base of $\triangle ABC$ is $30$ and the lengths of sides $BC$ and $AC$ are $17$ and $25$, calculate the length of the altitude to the base $AB$.


Given the equation $x^{6} + y^2 = 6y$, find the number of ordered pairs of integers $(x, y)$ that satisfy this equation.


Given Aaron needs to read an average of 15 pages per day to complete his book club's challenge over the next 6 days, and that in the first 5 days he reads 18, 12, 23, 10, and 17 pages each day, calculate the number of pages he needs to read on the sixth day to meet the challenge.


Claire holds 2 green and 2 red jelly beans in her hand. Daniel holds 2 green, 3 yellow, and 4 red jelly beans in his hand. Calculate the probability that the colors of the jelly beans picked by Claire and Daniel match.


Let \( M = 57^4 + 4 \cdot 57^3 + 6 \cdot 57^2 + 4 \cdot 57 + 1 \). Calculate the number of positive integers that are factors of \( M \).


What is the largest product that can be formed using two numbers chosen from the set $\{-20, -4, -1, 3, 5, 9\}$?


When the polynomial $x^3 + 3x^2 - 4$ is divided by the polynomial $x^2 + x - 2$, calculate the remainder.


John begins work on four equally time-consuming tasks at 2:00 PM. He finishes the third task at 5:00 PM. Determine the time at which he finishes the fourth and final task.

Solution: \boxed{6:00 \text{ PM}}


What is the hundreds digit of $(30! - 25!)$?


Points A and B lie on a plane with AB = 10. Calculate the number of locations for point C on this plane such that the triangle with vertices A, B, and C forms a right triangle with an area of 20 square units.


Determine the number of quadratic polynomials $ax^2+bx+c$ with real coefficients for which the set of roots matches the set of coefficients, and additionally, one of the coefficients is the arithmetic mean of the other two.


Points A and B lie on a coordinate plane with AB = 10. Determine how many locations for point C exist on this plane such that the triangle with vertices A, B, and C forms a right triangle with an area of 15 square units.


Consider equations of the form \( x^2 + bx + c = 0 \). How many such equations have real roots if coefficients \( b \) and \( c \) are selected from the sets \( B = \{2, 4, 6\} \) and \( C = \{1, 3, 5\} \), respectively?


Consider the sequence $1, -4, 9, -16, 25, -36, \ldots,$ where the $n$th term is $(-1)^{n+1} \cdot n^2$. Calculate the average of the first 100 terms of this sequence.


Josanna's test scores are $92, 78, 84, 76,$ and $88$. She aims to raise her test average by at least $5$ points with her next test. What is the minimum test score she would need on the next test to reach this goal?


Find the number of distinct terms in the expansion of $[(2a+5b)^3(2a-5b)^3]^3$ when simplified.


Given a rectangular racing track consisting of two parallel 100 meters long straight sections and two semicircular ends, where the inner semicircle has a radius of 50 meters and the outer semicircle has a radius of 52 meters, find the runner's average speed in meters per second, given that it takes a runner 40 seconds longer to complete a lap around the outer edge than the inner edge.


Given Eliot's test scores are \(88, 92, 75, 85,\) and \(80\), find the minimum score he would need on his next test to raise his test average by 5 points.


The length of the rectangle PQRS is 8 inches and its width is 6 inches. Diagonal PR is divided into four equal segments by points M, N, and O. Find the area of triangle QMN.


Given the equation $a \cdot b + 45 = 10 \cdot \text{lcm}(a, b) + 18 \cdot \text{gcd}(a, b)$, where $\text{gcd}(a, b)$ denotes the greatest common divisor of $a$ and $b$, and $\text{lcm}(a, b)$ denotes their least common multiple, calculate the number of ordered pairs $(a, b)$ of positive integers that satisfy this equation.


The number of arrangements of the Blue Bird High School chess team with a girl at each end and the three boys in the middle can be found by determining the number of possible permutations.


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


Consider the infinite series $1 - \frac{1}{3} - \frac{1}{9} + \frac{1}{27} - \frac{1}{81} - \frac{1}{243} + \frac{1}{729} - \cdots$. Let \( T \) be the sum of this series. Find \( T \).


A fair 8-sided die is rolled twice. What is the probability that the first number that comes up is greater than or equal to the second number, and the sum of the two rolls is even?


A box contains chips, each of which is red, white, or blue. The number of blue chips is at least one third the number of white chips, and at most one fourth the number of red chips. The number which are white or blue is at least 70. Find the minimum number of red chips.


The integers from $-15$ to $9$, inclusive, are arranged to form a $5$-by-$5$ square such that the sum of the numbers in each row, each column, and each of the main diagonals is the same, calculate the value of this common sum.


Determine the number of pairs of regular polygons whose sides are of unit length and the ratio of their interior angles is $4:3$.


Given Miguel read an average of $48$ pages per day for the first four days, an average of $35$ pages per day for the next five days, $28$ pages per day for the subsequent four days, and read the remaining $19$ pages on the final day, calculate the total number of pages in Miguel’s book.


Given a small baseball league team consists of 18 players, each earning a minimum of $20,000, and the total budget for all players' salaries does not exceed $600,000, calculate the maximum possible salary for a single player.


How many sets of three or more consecutive positive integers have a sum of $18$?


Consider a sequence of four-digit integers where each integer has the property that the hundreds, tens, and units digits become, respectively, the thousands, hundreds, and tens digits of the next term, and the hundreds, tens, and units digits of the last term become, respectively, the thousands, hundreds, and tens digits of the first term. Find the largest prime factor that always divides the sum of all the terms in the sequence.


Given the equation $x^4y^4 - 20x^2y^2 + 64 = 0$, find the number of distinct ordered pairs $(x, y)$ where $x$ and $y$ are positive integers.


Given the expression $(1+x^3)(1-x^4+x^5)$ evaluate the expression.

Solution: \boxed{1 + x^3 - x^4 + x^5 - x^7 + x^8}


A circle of radius $r$ is concentric with and outside a regular square of side length $2$. What is the radius $r$ such that the probability that two entire sides of the square are visible from a randomly chosen point on the circle is $1/2$?
A) $\sqrt{2}$
B) $2$
C) $2\sqrt{2}$
D) $4$
E) $4\sqrt{2}$


If \(x+6\) cows give \(x+9\) cans of milk in \(x+4\) days, determine how many days will it take \(x+4\) cows to give \(x+11\) cans of milk.


Evaluate the expression: $\left(2\left(3\left(2\left(3\left(2\left(3 \times (2+1) \times 2\right)+2\right)\times 2\right)+2\right)\times 2\right)+2\right)$.


The greatest prime number that is a divisor of 65,536 is 2, and for 65,535, determine the sum of the digits of the greatest prime number that is a divisor of 65,535.


Calculate the angle formed by the hands of a clock at 3:30, providing the larger of the two possible angles.


A rectangular sheet of metal measures $12$ in. by $16$ in. Squares with side length $x$ inches are cut from each corner, and the sides are then folded to form an open box. Additionally, to ensure the structural integrity of the box, the side length $x$ must not exceed the width of the material divided by $5$. Determine the expression for the volume of the box.

Solution: \boxed{4x^3 - 56x^2 + 192x}


If the digit $5$ is placed after a two-digit number whose tens' digit is $t$ and units' digit is $u$, find the new number.

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


Given positive integers A, B, and C that are co-prime, satisfying the equation $A \log_{100} 5 + B \log_{100} 2 = C$, calculate the sum A + B + C.


Given that the average score of 20 students was 75, and the average score of the 21 students was 76, find Jessica's test score.


A woman buys a property for $150,000 with a goal to achieve a $7\%$ annual return on her investment. She sets aside $15\%$ of each month's rent for maintenance costs, and pays property taxes at $0.75\%$ of the property's value each year. Calculate the monthly rent she needs to charge to meet her financial goals.


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


The base of isosceles triangle $\triangle PQR$ is 30, and its area is 90. Find the length of one of the congruent sides.


Determine the number of positive integers less than $500$ that are divisible by neither $5$ nor $7$.


Determine the number of one-foot cubical blocks Maria uses to construct a fort with exterior dimensions of 15 feet in length, 12 feet in width, and 7 feet in height, given that the floor and the four walls are all 1.5 feet thick.


Samantha has moved to a new house which is 3 blocks east and 4 blocks north of the southeast corner of City Park. Her school is now 4 blocks west and 3 blocks south of the northwest corner of City Park. To get to school, she bikes to the southeast corner of City Park, walks a diagonal path across the park to the northwest corner, and then continues by bike to school. Determine the number of different routes she can take if her route remains as short as possible.


Given the sum $2^{11}+7^{13}$, find the smallest prime number that divides the sum.


Let $R$ denote the set of students owning a rabbit and $G$ denote the set of students owning a guinea pig. Given that $|R|=35$ and $|G|=40$, and $|R|+|G|=50$, determine the number of students owning both a rabbit and a guinea pig.


A pyramid-like stack with a rectangular base containing $6$ apples by $9$ apples is constructed, with each apple above the first level fitting into a pocket formed by four apples below, until no more apples can be fit in a new layer. Determine the total number of apples in the completed stack.


Given $150^8$, determine the total number of positive integer divisors that are perfect squares or perfect cubes, or both.


Let \( r \) represent the result of tripling both the base and exponent of \( a^b \), where \( b \) is non-zero. If \( r \) equals the product of \( a^b \) by \( x^b \), determine the value of \( x \).


A rectangular box has a total surface area of 118 square inches and the sum of the lengths of all its edges is 52 inches. Find the sum of the lengths in inches of all of its interior diagonals.


Cagney can frost a cupcake every 15 seconds, and Lacey can frost a cupcake every 25 seconds. Lacey starts working 30 seconds after Cagney starts. Find the number of cupcakes that they can frost together in 10 minutes.


Given that $\log_b 243$ is a positive integer, how many positive integers $b$ satisfy this condition?


Given the base and height of a triangle are in an arithmetic progression, and the base is \(2a - d\) while the height is \(2a + d\), where \(d\) is the common difference and \(a\) is a positive number, determine the area of the triangle.

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


In the $xy$-plane, the segment with endpoints $(-3,0)$ and $(27,0)$ is the diameter of a circle. A vertical line $x=k$ intersects the circle at two points, and one of the points has a $y$-coordinate of $12$. Find the value of $k$.


Given the numbers $2^{10} + 1$ and $2^{16} + 1$, find the number of perfect cubes between these two values, inclusive.


Given Brianna uses one fourth of her money to buy one half of the books she plans to purchase, and she spends an equal amount in the next transaction to buy the remaining books, determine the fraction of her money that Brianna will have left.


Given the expression $\left[ \sqrt [4]{\sqrt [8]{b^{16}}} \right]^6\left[ \sqrt [8]{\sqrt [4]{b^{16}}} \right]^6$, simplify the expression.


Find the non-zero real value of $x$ that satisfies the equation $(9x)^{18} = (27x)^9$.


Let vertices $A, B, C$, and $D$ form a regular tetrahedron with each edge of length 1 unit. Define point $P$ on edge $AB$ such that $P = tA + (1-t)B$ for some $t$ in the range $0 \leq t \leq 1$ and point $Q$ on edge $CD$ such that $Q = sC + (1-s)D$ for some $s$ in the range $0 \leq s \leq 1$. Determine the minimum possible distance between $P$ and $Q$.


In her youth, Lina could cycle 20 miles in 2 hours, and now she cycles 12 miles in 3 hours. Calculate the difference in her cycling time per mile now compared to her youth.


Given $\log_4{5}=p$ and $\log_5{7}=q$, calculate $\log_{10}{7}$ in terms of $p$ and $q$.


Given that in triangle PQR, PS is the altitude to QR, and the angle ∠QRP is 30°, find the length of PR in terms of PS.


A contractor estimated that one of his two bricklayers would take $8$ hours to build a certain wall and the other $12$ hours. However, he knew from experience that when they worked together, their combined output fell by $12$ bricks per hour. Being in a hurry, he put both men on the job and found that it took them exactly 6 hours to build the wall. Determine the number of bricks in the wall.


Given that two fifths of Jamie's marbles are blue, one third of her marbles are red, and four of them are green, determine the smallest number of yellow marbles Jamie could have.


Emily cycles at a constant rate of 15 miles per hour, and Leo runs at a constant rate of 10 miles per hour. If Emily overtakes Leo when he is 0.75 miles ahead of her, and she can view him in her mirror until he is 0.6 miles behind her, calculate the time in minutes it takes for her to see him.


Given a right circular cone and a sphere have volumes related such that the volume of the cone is one-third that of the sphere, and the radius of the base of the cone is twice the radius of the sphere, determine the ratio of the altitude of the cone to the radius of its base.


Given that $3z$ varies inversely as the square of $w$ and directly as $v$, and given that when $z = 15$ and $w = 2$, $v = 4$, find $z$ when $w = 4$ and $v = 8$.


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


The side length of a cube is 2 units. Four of the eight vertices of this cube are the vertices of a regular tetrahedron. Find the ratio of the surface area of the tetrahedron to the surface area of the cube.


What is the largest quotient that can be obtained using two numbers from the set $\{ -30, -4, 0, 3, 5, 10 \}$?


Given the track has a width of $8$ meters, and it takes Keiko $48$ seconds longer to walk around the outside edge of the track than around the inside edge, determine Keiko's speed in meters per second.

Solution: \boxed{\frac{16\pi}{48} = \frac{\pi}{3}}


Given a positive integer n that has 72 divisors and 5n has 90 divisors, calculate the greatest integer k such that 5^k divides n.


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


Given points $P$ and $Q$ are $8$ units apart in a plane, determine the number of lines containing $P$ and $Q$ that are $4$ units from $P$ and $6$ units from $Q$.


Determine the number of ways a student can schedule four mathematics courses — algebra, geometry, number theory, and statistics — on an 8-period day, given that no two mathematics courses can be scheduled in consecutive periods.


Two candles of the same height are lighted at the same time. The first is consumed in $5$ hours and the second in $4$ hours. Assuming that each candle burns at a constant rate, determine in how many hours after being lighted was the first candle three times the height of the second.


While Jessica and Shawn are fishing 2 miles from the shore, their boat begins to take on water at a constant rate of 12 gallons per minute. The boat will sink if it accumulates more than 50 gallons of water. Jessica starts rowing towards the shore at a constant rate of 3 miles per hour while Shawn bails water out of the boat. Determine the slowest rate, in gallons per minute, at which Shawn can bail if they are to reach the shore without sinking.


Three years ago, Tom was twice as old as his sister Sara, and five years before that, Tom was three times as old as Sara.

Let $t$ and $s$ represent Tom's current age and Sara's current age, respectively. Determine the number of years until the ratio of their ages will be 3:2.


Three eighths of a gallon of paint is used on the first day. One quarter of the remaining paint is used on the second day. Determine the fraction of the original amount of paint available to use on the third day.


A digital watch displays hours and minutes in a 24-hour format. Calculate the largest possible sum of the digits in this display.


The ratio of the width to the length of a rectangle is $3$ : $5$. If the rectangle has a diagonal of length $10d$, find the constant $k$ such that the area of the rectangle can be written as $kd^2$.


Let $r$ be the number that results when both the base and the exponent of $a^b$ are doubled, where $a,b>0$. If $r$ equals the product of $a^b$ and $x^{2b}$ where $x>0$, find $x$.

Solution: \boxed{2 \sqrt{a}}


Given quadrilateral $\Box FRDS$ with $\triangle FDR$ being a right-angled triangle at point $D$, with side lengths $FD = 3$ inches, $DR = 4$ inches, $FR = 5$ inches, and $FS = 8$ inches, and $\angle RFS = \angle FDR$, find the length of RS.


What is the largest sum of two 3-digit numbers that can be obtained by placing each of the six digits `1, 2, 3, 7, 8, 9` in one of the six boxes in this addition problem?


A sequence of diagrams consists of equilateral triangles, each subsequently divided into smaller equilateral triangles. In the first diagram, there are no shaded triangles. In the second diagram, 1 out of 4 small triangles are shaded. In the third diagram, 4 out of 9 small triangles are shaded. If this pattern of shading continues, determine the fraction of the small triangles in the sixth diagram that would be shaded.


Given that Tom had two containers, and the first was $\frac{3}{5}$ full of oil and the second was empty, then he poured all the oil from the first container into the second container, after which the second container was $\frac{2}{3}$ full of oil. What is the ratio of the volume of the first container to the volume of the second container?


A point P is located 15 inches away from the center of a circle. A secant from P intersects the circle at points Q and R such that the external segment PQ is 11 inches and segment QR is 5 inches. Find the radius of the circle.


Given Alice has $4q + 3$ quarters and Bob has $2q + 8$ quarters. Find the difference in their money in nickels.

Solution: \boxed{10q - 25}


Given that \( \text{rem} \left(\frac{5}{7}, \frac{3}{4}\right) \) must be calculated, determine the value of the remainder.


The sum of six integers is $20$. What is the maximum number of the six integers that can be larger than $15$?


Consider the system of equations where $3x - 4y - 2z = 0$ and $x + 4y - 10z = 0$, with the condition $z \neq 0$. Calculate the numerical value of $\frac{x^2 + 4xy}{y^2 + z^2}$.


Find the area of the irregular quadrilateral formed by the vertices at points (2,1), (4,3), (7,1), and (4,6).


The taxi fare in Metropolis City 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. Calculate the total distance you can ride for $15.


Let $x$, $x$, and $x$ represent the ages of Liam's twin sisters and Liam. If the product of their ages is 72, then find the sum of their ages.


Let $x = -2023$. Find the value of $\bigg| \, ||x|-x|-|x| \, \bigg| - x$.


Al is diagnosed with algebritis and has been prescribed to take one green pill and one pink pill each day for 18 days. Each green pill costs $2 more than each pink pill, and the total cost for all the pills over the 18 days comes to $\textdollar 738$. Given this information, calculate the cost of one green pill.


Approximately how many sheets of the type of paper would be in a stack 10 cm high?


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


Consider a 6-digit palindrome $m$ formed as $\overline{abccba}$ and chosen uniformly at random. Every digit ($a$, $b$, $c$) can be any digit from 0 to 9, except $a$, which cannot be 0 because $m$ is a 6-digit number. What is the probability that both $m$ and $m+11$ are palindromes?
A) $\frac{800}{900}$
B) $\frac{9}{10}$
C) $\frac{7}{9}$
D) $\frac{8}{9}$
E) $\frac{9}{9}$


In a physics lab experiment, a certain fixed natural constant $k$ is measured as $3.56897$ with an accuracy of $\pm 0.00145$. The researcher aims to declare a value of $k$ where every reported digit is significant. The announced value must precisely reflect $k$ when it is rounded to that number of digits. What is the most accurate value that the researcher can announce for $k$?


Evaluate the limit as $x \to \infty$ of the expression $\log_4{(8x-7)} - \log_4{(3x+2)$.

Solution: \boxed{\log_4 \left(\frac{8}{3}\right)}


Given that a line $x = k$ intersects the graph of $y = \log_2 x$ and the graph of $y = \log_2 (x + 6)$ and the distance between the points of intersection is $1$, determine the sum of the integers $a$ and $b$ in the expression $k = a + \sqrt{b}$.


Given that three spinners X, Y, and Z are spun, determine the probability that the sum of the numbers shown is an even number, where spinner X lands on 2, 5, or 7, spinner Y lands on 2, 4, or 6, and spinner Z lands on 1, 2, 3, or 4.


Given $945=a_1+a_2\times2!+a_3\times3!+a_4\times4!+a_5\times5!+\ldots$ where $0 \le a_k \le k$ for all $k$, determine the value of $a_4$.


Alan, Beth, and Chris went on a camping trip and decided to share the expenses equally. Alan paid $110, Beth paid $140, and Chris paid $190. To equalize the cost, Alan gave Chris $a$ dollars and Beth gave Chris $b$ dollars after realizing they had forgotten to split the cost of a $60 picnic they also enjoyed. What is $a-b$?


Given a set of test scores is $[55, 68, 72, 72, 78, 81, 81, 83, 95, 100]$, median score $Q_2 = 79.5$, first quartile $Q_1 = 72$, and third quartile $Q_3 = 83$, determine the number of outliers present in this data set.


Determine the number of positive integer divisors of $202^8$ that are either perfect squares, perfect cubes, or both.


The ratio of the length to the width of a rectangle is $3:2$. If the diagonal of the rectangle is $\sqrt{13}x$, express the area in terms of $kd^2$, where $d$ is the diagonal. Find the value of $k$.


In a school drama club of 150 students, each student can either write, direct, or produce. Every student has at least one skill but no student can do all three. There are 60 students who cannot write, 90 students who cannot direct, and 40 students who cannot produce. Determine the number of students who have exactly two of these skills.


Given that for every dollar David spent on sandwiches, Ben spent $50$ cents more, and David paid $15.00$ less than Ben, calculate the total amount they spent on sandwiches together.


Given that $(-\frac{1}{343})^{-2/3}$, simplify the expression.


Given that a square $S_1$ has an area of $25$, the area of the square $S_3$ constructed by bisecting the sides of $S_2$ is formed by the points of bisection of $S_2$.


If John earned a 'B' on $32$ of the first $40$ tests and needs to earn a 'B' on at least $75\%$ of his $60$ tests, determine the maximum number of remaining tests for which he can earn a grade lower than a 'B'.


Given a number 180, find the sum of its two smallest prime factors.


Yesterday, Han drove 2 hours longer than Ian at an average speed 10 miles per hour faster than Ian. Jan drove 3 hours longer than Ian at an average speed 15 miles per hour faster than Ian. Han drove 120 miles more than Ian. Calculate the difference in the distances driven by Jan and Ian.


Three positive consecutive integers start with $a+1$. Find the average of the next three consecutive integers that start with this new average.

Solution: \boxed{a + 3}


Given the price of a product increased by 30% in January, decreased by 30% in February, increased by 40% in March, and then decreased by y% in April, find the value of y, to the nearest integer, if the price of the product at the end of April is the same as it was at the beginning of January.


If the legs of a right triangle are in the ratio $3:4$, find the ratio of the areas of the two triangles created by dropping an altitude from the right-angle vertex to the hypotenuse.


How many primes less than $100$ have $3$ as the ones digit?


What is the result of $\sqrt{16-8\sqrt{3}} + \sqrt{16+8\sqrt{3}}$?


A parabolic arch has a height of $20$ inches and a span of $50$ inches. Find the height, in inches, of the arch at the point $10$ inches from the center $M$.


Lisa bakes a $24$-inch by $30$-inch pan of brownies and cuts the brownies into pieces that measure $3$ inches by $4$ inches. Calculate the total number of pieces of brownies that the pan contains.


Given that a 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 averaging $50$ miles per gallon for the whole trip, determine the total distance of the trip in miles.


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


Determine the number of points C such that the perimeter of triangle ABC is 60 units and the area of triangle ABC is 144 square units, given that points A and B are 12 units apart.


Given that Jeff spins three spinners labeled $X$, $Y$, and $Z$, where Spinner $X$ has numbers 1, 4, 5, Spinner $Y$ has numbers 1, 2, 3, and Spinner $Z$ has numbers 2, 4, 6, find the probability that the sum of the numbers Jeff gets from spinning $X$, $Y$, and $Z$ is even.


In a coordinate plane, points A and B are 8 units apart. Find the number of points C such that the perimeter of triangle ABC is 40 units and the area of triangle ABC is 80 square units.


Let the sequence $x, 3x+3, 5x+5, \dots$ be in geometric progression. What is the fourth term of this sequence?


Evaluate the value of the expression $3(3(3(3(3x+2)+2)+2)+2)+2$ where $x=5$.


If $m$ men can complete a task in $d$ days, calculate the number of days it will take for $m + r^2$ men to complete the same task.

Solution: \boxed{\frac{md}{m + r^2}}


Calculate the sum of the sequence:
\[3\left(1 - \frac{1}{3}\right) + 4\left(1 - \frac{1}{4}\right) + 5\left(1 - \frac{1}{5}\right) + \cdots + 12\left(1 - \frac{1}{12}\right).\]


The sum of the interior angles of a polygon is (k-2) × 180°. Determine the relationship between the sum of the interior angles and the number of sides k of the polygon.

Solution: \boxed{(k-2) \times 180^\circ \text{ and Increases}}


Given Tamara redesigns her garden with four rows of three $8$-feet by $3$-feet flower beds separated and surrounded by $2$-foot-wide walkways, calculate the total area of the walkways in square feet.


A worker's wage is reduced by 30%. To regain his original wage, what percentage raise is required?


Miki now has 15 oranges and 15 pears of the same size. In her new juicer, Miki can extract 10 ounces of pear juice from 4 pears and 7 ounces of orange juice from 1 orange. She decides to make a juice blend using 12 oranges and 8 pears. Calculate the percentage of the blend that is pear juice.


A basketball league has introduced new rules. Each team now consists of 25 players. A player must earn a minimum of $15,000, and the team's total salary cap is $850,000. Calculate the maximum possible salary for a single player on a team under these new regulations.


The total number of black squares on the nonstandard checkerboard consists of 29 squares on each side.


Let $x=-1000$. Evaluate the expression $\bigg|$ $||x|-x|-|x|$ $\bigg|$ $+x$.


Given a rectangular enclosure that is $15$ feet long, $12$ feet wide, and $6$ feet high with the floor and walls each $1.5$ feet thick, calculate the number of one-foot cubical blocks needed to complete the enclosure.


The ratio of $a$ to $b$ in the arithmetic sequence $a, y, b, 3y$ is what fraction.


Given the equations $x^2 + kx + 12 = 0$ and $x^2 - kx + 12 = 0$. If, when the roots of the equations are suitably listed, each root of the second equation is $7$ more than the corresponding root of the first equation, determine the value of $k$.


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 $n$.


$\triangle ABC$ has a right angle at $C$ and $\angle A = 30^\circ$. If $BD$ ($D$ in $\overline{AC}$) is the bisector of $\angle ABC$, then find the measure of $\angle BDC$.


Determine the sum of the digits of the greatest prime number that is a divisor of $2^{13} - 1$.


What is the greatest number of consecutive integers whose sum is $48?$


Elena builds a rectangular enclosure using one-foot cubical blocks that measures 15 feet in length, 13 feet wide, and 6 feet high. Both the floor and the ceiling, as well as the four walls, are all one foot thick. Calculate the number of blocks used in the construction of this enclosure.


Elena intends to buy 7 binders priced at $\textdollar 3$ each. Coincidentally, a store offers a 25% discount the next day and an additional $\textdollar 5$ rebate for purchases over $\textdollar 20$. Calculate the amount Elena could save by making her purchase on the day of the discount.


Evaluate the expression \(\sqrt{\frac{4}{3} \left( \frac{1}{15} + \frac{1}{25} \right)}\).


Given $(729^{\log_3 2187})^{\frac{1}{3}}$, evaluate the expression.


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


What is the smallest whole number larger than the perimeter of any triangle with a side of length $7$ and a side of length $24$?


Given that all of Maria's telephone numbers have the form $555-abc-defg$, where $a$, $b$, $c$, $d$, $e$, $f$, and $g$ are distinct digits, in increasing order, and none is either $0$, $1$, or $9$, calculate the total number of different telephone numbers Maria can have.


What is the minimum possible product of three different numbers from the set $\{-9, -5, -3, 0, 4, 6, 8\}$?


A rectangular garden that is $14$ feet wide and $19$ feet long is paved with $2$-foot square pavers. Given that a bug walks from one corner to the opposite corner in a straight line, determine the total number of pavers the bug visits, including the first and the last paver.


Solution: \boxed{4ab^2 + 4b}


The ratio of the length to the width of a rectangle is $5$ : $2$. If the rectangle has a diagonal of length $13$ units, find the constant $k$ such that the area of the rectangle can be expressed as $kd^2$.


Solution: \boxed{-\frac{3}{5} - \frac{1}{5}i}


For a set of three distinct lines in a plane, there are exactly $M$ distinct points that lie on two or more of these lines. What is the sum of all possible values of $M$?


Find the minimum value of the function $f(x) = 2x^2 + 4x + 6 + 2\sqrt{x}$, assuming $x \geq 0$.


Given that the base-ten representation for $15!$ is $1,307,674,H00,M76,T80$, determine the sum of the digits $H$, $M$, and $T$.


If $x$ cows give $x+2$ cans of milk in $x+1$ days, determine how many cows are needed to give $x+4$ cans of milk in $x+3$ days.


Points $B$ and $C$ lie on $\overline{AD}$. The length of $\overline{AB}$ is $3$ times the length of $\overline{BD}$, and the point $C$ is the midpoint of $\overline{AD}$. Determine the fraction of the length of $\overline{AD}$ that is occupied by $\overline{BC}$.


Given that Mrs. Germain has $15$ students, Mr. Newton has $12$ students, and Mrs. Young has $9$ students taking the AMC $8$, and $3$ students from Mrs. Young's class are also in Mr. Newton's class, determine the total number of students at Euclid Middle School who are taking the contest.


Given integers between $2000$ and $9999$ have four distinct digits, calculate the total count.


The sum of 30 consecutive odd integers is 7,500. Find the value of the largest of these 30 consecutive integers.


A frog makes $4$ jumps, each exactly $0.5$ meters long. Determine the probability that the frog's final position is no more than $0.5$ meters from its starting position.


Determine the number of points $C$ that exist such that the perimeter of $\triangle ABC$ is $60$ units and the area of $\triangle ABC$ is $144$ square units, given that points $A$ and $B$ are $12$ units apart.


Given a triangle with a side of length 7 and a side of length 21, what is the smallest whole number larger than its perimeter?


A rectangular barn with a roof is 12 yd. wide, 15 yd. long and 7 yd. high. Calculate the total number of sq. yd. to be painted inside and outside on the walls and the ceiling, and only on the outside of the roof but not on the floor.


Given that Spinner 1 has sectors numbered 1, 2, 3 and Spinner 2 has sectors numbered 4, 5, 6, determine the probability that the sum of the numbers on which the spinners land is prime.


The mean, median, and unique mode of the positive integers 2, 3, 4, 8, 8, 9, 10, and $y$ are all equal. What is the value of $y$?


Given that a room initially requires 2 gallons of paint, on the first day one-fourth of the paint is used, and on the second day one-half of the remaining paint is used, and an additional half gallon is needed to finish painting, calculate the total amount of additional paint needed.


Calculate the fourth term in the expansion of \((\frac{a}{x} + \frac{x}{a^3})^7\).


A sphere with center $O$ has radius $8$. A triangle with sides of length $13$, $13$, and $10$ is situated in space so that each of its sides is tangent to the sphere. Calculate the distance between $O$ and the plane determined by the triangle.


Amanda divided her hiking trip into four segments. She hiked the first one-fourth of the total distance to a stream and then went another $25$ kilometers through a forest. The next segment involved crossing a mountain range, which took up another one-sixth of the total trip, before finishing the last portion on a plain. If the plain's segment was twice the length of the forest path, derive the length of Amanda's entire hiking trip.


Packaged snacks are sold in containers of 5, 10, and 20 units. Find the minimum number of containers needed to acquire exactly 85 units of snacks.


Given in isosceles trapezoid $ABCD$, sides $AB$ and $CD$ are bases with $AB < CD$, the height of the trapezoid from $AB$ to $CD$ is 6 units, $AB$ is 10 units and $CD$ is 18 units, find the perimeter of $ABCD$.

Solution: \boxed{28 + 4\sqrt{13}}


If a wooden cube of side length \( n \) units is entirely painted blue on all six faces and then divided into \( n^3 \) smaller cubes of unit volume each, and exactly one-third of the total number of faces of these unit cubes are blue, what is \( n \)?


Given a $24$-inch by $30$-inch pan of brownies, cut into pieces that measure $3$ inches by $4$ inches. Calculate the number of pieces of brownie the pan contains.


Given square $ABCD$ has side lengths of 4, segments $CM$ and $CN$ divide the square's area into three equal parts. Each triangle formed by these segments and square's sides shares one vertex $C$. Calculate the length of segment $CM$.


Jasmine now has access to three sizes of shampoo bottles: a small bottle that holds 25 milliliters, a medium bottle that holds 75 milliliters, and a large bottle that holds 600 milliliters. Determine the minimum total number of bottles Jasmine must buy to almost completely fill a large bottle, using small bottles for the remaining volume.


Given that in triangle ABC, segment AD divides the side BC into segments BD and DC such that $BD + DC = p$ and $BD \cdot DC = q^2$, find the polynomial whose roots are $BD$ and $DC$ under these conditions.

Solution: \boxed{x^2 - px + q^2}


Given that Mr. and Mrs. Delta are deciding on a name for their baby such that its monogram (first, middle, and last initials) will be in alphabetical order with no letter repeated, and the middle initial must be 'M', calculate the number of such monograms possible.


At a certain park, 20% of the birds are pigeons, 30% are parrots, 15% are peacocks, and 35% are sparrows. Calculate what percent of the birds that are not sparrows are pigeons.


Walter was one-third as old as his grandmother at the end of $1998$. Find Walter's age at the end of $2003$, given that the sum of the years in which they were born was $3860$.


Linda plans to participate in a mini-triathlon. She averages 1 mile per hour swimming for a 1/8-mile stretch and 8 miles per hour running for a 2-mile stretch. She aims to finish the mini-triathlon in 1.5 hours. Determine her required average speed in miles per hour for the 8-mile bike ride.


Pablo drove at an average speed of 60 km/h and then stopped for 30 minutes. After the stop, he drove at an average speed of 90 km/h. Altogether, he drove 270 km in a total trip time of 4 hours including the stop. Which equation could be used to solve for the time $t$ in hours that he drove before his stop?

Solution: \boxed{60t + 90\left(\frac{7}{2} - t\right) = 270}


A ferry boat starts its service to an island at 10 AM and ends at 4 PM, shuttling tourists every hour. On the first trip at 10 AM, there were 120 tourists, and on each subsequent trip, the number of tourists decreased by 2 from the previous trip. Calculate the total number of tourists transported by the ferry that day.


Four fair dice are tossed at random. What is the probability that the numbers turned up can be arranged to form an arithmetic progression with common difference one or two?


A circle of radius 6 is inscribed in a rectangle. The ratio of the length of the rectangle to its width is 3:1. Find the area of the rectangle.


Determine the values of $a, b$, and $c$ that satisfy the equality $\sqrt{a^2 - \frac{b}{c}} = a - \sqrt{\frac{b}{c}}$, where $a, b$, and $c$ are positive integers.

Solution: \boxed{b = a^2c}


Given the in-store price for a laptop is $\textdollar 599.99$, the store offers a 5% discount. Additionally, an online advertisement offers the same laptop for five equal payments of $\textdollar 109.99$ plus a one-time shipping charge of $\textdollar 19.99$. Calculate the amount saved by purchasing the laptop online.


Points $A$ and $B$ are on a circle with radius $7$ and $AB=8$. Points $C$ and $E$ are the midpoints of the minor and major arcs $AB$, respectively. Find the length of the line segment $AC$.

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


Given a right circular cone and a sphere with the same radius $r$, 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.


Given a circle is inscribed in a triangle with side lengths $9, 12$, and $15$, let the segments of the side of length $9$, made by a point of tangency, be $p$ and $q$, with $p < q$. Calculate the ratio $p:q$.


Given the line whose equation is $y = 2x + 3$, and the points $(p, q)$ and $(r, s)$ satisfy $s = 2r + 6$, find the distance between $(p, q)$ and $(r, s)$ in terms of $p$ and $r$.

Solution: \boxed{\sqrt{5(r - p)^2 + 12(r - p) + 9}}


Lucia needs to save 35 files onto disks, each with a capacity of 1.6 MB. 5 of the files are 0.9 MB each, 10 of the files are 0.8 MB each, and the rest are 0.5 MB each. Calculate the smallest number of disks needed to store all 35 files.


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


Charlie and Dana start their new jobs on the same day. Charlie's schedule is 4 work-days followed by 2 rest-days. Dana's schedule is 9 work-days followed by 1 rest-day. Determine how many times both have rest-days on the same day in their first 1200 days of work.


Given the quadratic equation $ax^2 + bx + c = 0$, if one root is triple the other, determine the relationship among the coefficients $a,\,b,\,c$.

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


Given that Frank the flamingo takes $60$ equal steps to walk between consecutive beacons, and Peter the penguin covers the same distance in $15$ equal slides, and the $31$st beacon is exactly half a mile ($2640$ feet) from the first beacon, find the difference in length, in feet, between one of Peter's slides and one of Frank's steps.


Determine the number of integers $x$ that satisfy $x^6 - 52x^3 + 51 < 0$.


Alice has taken three exams, each worth a maximum of 100 points and her scores are 85, 76, and 83. What is the lowest possible score she could achieve on one of her remaining two exams to still reach an average of 80 for all five exams?


In the diagram, $\overline{AB}$ and $\overline{CD}$ are diameters of a circle with center $O$, with $\overline{AB} \perp \overline{CD}$. Chord $\overline{DF}$ intersects $\overline{AB}$ at $E$. If $DE = 8$ and $EF = 4$, calculate the area of the circle.


Given the function $x \spadesuit y = (x+y)^2(x-y)$, calculate the value of $2 \spadesuit (3 \spadesuit 6)$.


In a sign pyramid where a cell receives a "+" if the two cells below it have the same sign and a "-" otherwise, determine the number of ways to fill the five cells in the bottom row to produce a "+" at the top of a new pyramid with an increased number of levels.


If $y$ cows give $y+2$ cans of milk in $y+1$ days, calculate how many days will it take $y+4$ cows to give $y+10$ cans of milk.


Find the value of A when the minimum of $\sin\frac{A}{2} - \cos\frac{A}{2}$ is attained.


Consider the repeating decimal number $3.71717171\ldots$. When written in the fraction form and reduced to the lowest terms, find the sum of the numerator and denominator.


Given the sales of $110, 90, 50, 130, 100, and 60$ dollars for January, February, March, April, May, and June, respectively, calculate the average monthly sales in dollars, accounting for a 20% discount on all candy sales in June.


Given that there are 12 students in Mr. Gauss' class, 10 students in Ms. Euler's class, and 7 students in Mr. Fibonacci's class, with one student counted twice, calculate the total number of distinct mathematics students at Euclid Middle School taking the AMC 8 this year.


The centers of two circles are $50$ inches apart. The smaller circle has a radius of $7$ inches and the larger one has a radius of $10$ inches. Find the length of the common internal tangent.


A merchant buys goods at 30% off the list price. He desires to mark the goods so that he can give a discount of 20% on the marked price and still clear a profit of 30% on the selling price. What percent of the list price must he mark the goods?


In a modified sequence of numbers, the first number is $1$, and for all $n \geq 2$, the product of the first $n$ numbers in the sequence is $(n+1)^3$. Find the sum of the third and the fifth numbers in this sequence.


Given the probability that the red ball is tossed into bin k is $3^{-k}$, and the probability that the green ball is tossed into bin k is $2^{-k}$ for k = 1,2,3,..., find the probability that the red ball is tossed into a higher-numbered bin than the green ball.


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


Positive integers $a$ and $b$ are each less than $8$. Find the smallest possible value for $3 \cdot a - 2 \cdot a \cdot b$.


Determine the smallest positive integer $n$ such that $n$ is divisible by $36$, $n^2$ is a perfect cube, and $n^3$ is a perfect square.


What is the least possible value of $(xy+1)^2+(x-y)^2$ for real numbers $x$ and $y$?


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


A corner of a tiled floor is shown. If the entire floor is tiled in this way and each of the four corners looks like this one, determine the fraction of the tiled floor that is made of darker tiles, considering that the repeating pattern is a $4 \times 4$ grid and the top $2 \times 2$ area contains 3 dark tiles.


Given that Shauna takes five tests, each worth a maximum of 120 points, and her scores on the first three tests are 86, 102, and 97, in order to average 90 for all five tests, calculate the lowest score she could earn on one of the other two tests.


What is the maximum value of n for which there is a set of distinct positive integers k_1, k_2, ..., k_n such that k_1^2 + k_2^2 + ... + k_n^2 = 2500?


Four fair dice are tossed at random. What is the probability that the four numbers turned up can be arranged to form an arithmetic progression with a common difference of one?


Given a list of $3000$ positive integers with a unique mode that occurs exactly $15$ times, determine the least number of distinct values that can occur in the list.


Given a basketball player made 7 baskets during a game, with each basket worth 1, 2, or 3 points, determine the number of different numbers that could represent the total points scored by the player.


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


What percent of the birds that were not swans were sparrows?


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}}


A 5x5 block of calendar dates is given. First, the order of the numbers in the second and the fifth rows are reversed. Then, the numbers on each diagonal are added. Calculate the positive difference between the two diagonal sums.


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


Given a rectangle with sides of length 3 cm and 4 cm, construct a new rectangle where one dimension equals the diagonal of the original rectangle, and the other dimension equals the sum of the original rectangle's sides. Determine the area of the new rectangle.


Given a polynomial function \( f \) of degree \( \ge 1 \) such that \( f(x^2) = [f(x)]^3 \) and \( f(f(x)) = [f(x)]^2 \), determine the number of possible polynomial functions \( f \).


Given $d$ is a digit, determine the number of values of $d$ for which $3.0d05 > 3.005$.


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


Determine the median length of the names of 25 individuals recorded in a survey.


Given that \(x\) is a real number, find the least possible value of \((x+2)(x+3)(x+4)(x+5)+3033\).


The doughnut machine is turned on at 9:00 AM and finishes one fourth of the day's job by 12:00 PM, determine the time it will complete the job.

Solution: \boxed{9:00 PM}


A basketball player made 7 baskets during a game. Each basket was worth either 1, 2, or 3 points. Calculate how many different numbers could represent the total points scored by the player.


Circle $C$ has a radius of $144$. Circle $D$ has an integer radius $s < 144$ and remains internally tangent to circle $C$ as it rolls once around the circumference of circle $C$. The circles have the same points of tangency at the beginning and end of circle $D$'s trip. Find the number of possible values of $s$.


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


Given that Bag A contains the chips labeled 0, 1, 3, and 5, and Bag B contains the chips labeled 0, 2, 4, and 6, determine the number of different values that are possible for the sum of the two numbers on the chips drawn from each bag.


Given that $OPQR$ is a rectangle where $O$ is the origin and point $Q$ has coordinates $(4,2)$, calculate the coordinates for a point $T$ on the x-axis so that the area of triangle $PQT$ is half the area of rectangle $OPQR$.


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

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


Emily a cell phone plan costs $30 each month, including unlimited calls up to the first 50 hours, and each additional hour is charged at $15. Additionally, each text message costs $0.10, but after the first 150 messages, the cost per message doubles. In February, Emily sent 200 text messages and talked for 52 hours. Calculate the total amount Emily had to pay.


Mrs. Anna Quick needs to pick up her friend from the airport every Friday. The airport is a certain distance from her home. If she drives at an average speed of 50 miles per hour, she arrives 5 minutes late, and if she drives at an average speed of 75 miles per hour, she arrives 5 minutes early. Determine the speed at which Mrs. Anna Quick should drive to arrive exactly on time.


A rectangular picture is framed with a border three inches wide on all sides. The picture itself measures $12$ inches in height and $15$ inches in width. Calculate the total area of the border.


Given the initial percentage of girls in the group is $50\%$, let $g$ represent the number of girls initially in the group. Shortly thereafter, three girls leave and three boys join the group, and then $40\%$ of the group are girls. Calculate the initial number of girls in the group.


Given that a positive integer consists of three digits whose product equals 30, calculate the number of 3-digit positive integers.


Given that a cell phone plan costs $25$ dollars each month, plus $0.10$ dollars per text message sent, plus $0.15$ dollars for each minute used over $25$ hours, and an extra $2$ dollars per GB for data exceeding $3$ GB, calculate the total cost for Jackie in February given that she sent $200$ text messages and talked for $26.5$ hours, using $4.5$ GB of data.


Given that a water tower stands 60 meters high and contains 150,000 liters of water, and a model of the tower holds 0.15 liters, determine the height of Liam's model tower.


If one side of a triangle is $15$ inches and the opposite angle is $45^{\circ}$, calculate the diameter of the circumscribed circle.

Solution: \boxed{15\sqrt{2} \text{ inches}}


Two pitchers, one with a capacity of 800 mL and the other with 500 mL, contain apple juice. The first pitcher is 1/4 full and the second pitcher is 3/8 full. Water is added to fill each pitcher completely, then both pitchers are poured into one large container. What fraction of the mixture in the large container is apple juice?


Find the number of pairs (m, n) of integers which satisfy the equation $m^3 + 10m^2 + 11m + 2 = 81n^3 + 27n^2 + 3n - 8$.


A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, with the parallel sides of the trapezoid having lengths $20$ meters and $30$ meters. Determine the fraction of the yard occupied by the flower beds.


A pitcher is filled to five-sixths of its capacity with orange juice. The juice is then poured equally into 3 cups. What is the percent of the total capacity of the pitcher that each cup received?

Solution: \boxed{27.78 \%}


Given that a rectangle $R$ has dimensions $a$ and $b$ with $a < b$, determine the number of possible values of $x$ and $y$ such that the perimeter is half of $R$'s, and the area is half of $R$'s.


Given a 16-slice pizza, Anna eats 2 slices and shares another slice equally with Ben and John, her two cousins. Determine the fraction of the pizza that Anna eats.


Given circle $O$, point $C$ is on one side of diameter $\overline{AB}$, and point $D$ is on the opposite side. If $\angle AOC = 40^{\circ}$, and $\angle DOB = 30^{\circ}$, calculate the ratio of the area of the smaller sector $COD$ to the area of the circle.


Given the expression $\left(\frac{x^2+2}{x^2}\right)\left(\frac{y^2+2}{y^2}\right)+\left(\frac{x^2-2}{y^2}\right)\left(\frac{y^2-2}{x^2}\right)$, with $xy \not= 0$, simplify the expression.

Solution: \boxed{2 + \frac{8}{x^2y^2}}


For how many positive integers $m$ is $\frac{180}{m^2 - 3}$ a positive integer?


Given that a group initially consists of $30\%$ girls, and after three girls leave and three boys join, the group consists of $25\%$ girls, determine the initial number of girls in the group.


Given that the sum of the interior angles of a convex polygon is 2797 degrees, find the combined degree measure of the two forgotten angles that Ben initially missed in his calculations.


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


How many $3$-digit positive integers have digits whose product equals $30$?


Given Jeremy's father drives him to school under normal conditions in 30 minutes, and on a particularly clear day, he can drive 15 miles per hour faster in 18 minutes, calculate the distance to school in miles.


Given that a ticket to a school concert costs a whole number of dollars, and a group of 11th graders buys tickets costing a total of $72, and a group of 12th graders buys tickets costing a total of $90, find the number of possible values for the cost of a ticket.


A positive integer divisor of $10!$ is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.


Lucas's apartment consists of 2 rooms, each 15 feet long, 12 feet wide, and 10 feet high. Doorways and windows, which will not be painted, occupy 75 square feet in total per room. Calculate the total number of gallons of paint Lucas needs to buy if he wants to apply 2 coats of paint, given that each gallon of paint covers 350 square feet for one coat.


Given that $36$ students are participating in the summer training program, with equal numbers of undergraduate and graduate students on the coding team, and $20\%$ of the undergraduate students and $25\%$ of the graduate students are on the coding team, determine the number of undergraduate students in the program.


Karl's car uses a gallon of gas every 30 miles, and his gas tank holds 16 gallons when it is full. One day, Karl started with a full tank of gas, drove 420 miles, bought 10 gallons of gas, and continued driving until his gas tank was three-quarters full. Determine the total distance Karl drove that day.


Given that $b = 2$, evaluate the expression $\frac{3b^{-1} + \frac{b^{-1}}{3} + 3}{b + 1}$.


Calculate the number of terms in the arithmetic sequence $17, 21, 25, \dots, 101, 105$ such that an additional term $49$ also belongs to the sequence.


Three years ago, Tom was four times as old as his brother Jerry, and five years before that, Tom was five times as old as Jerry. Determine how many years from now the ratio of their ages will be $3 : 1$.


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


Points $A$ and $B$ are on a circle of radius $7$ and $\overline{AB}=10$. Point $C$ is the midpoint of the minor arc $AB$. What is the length of the line segment $AC$.

Solution: \boxed{\sqrt{98 - 28\sqrt{6}}}


Given the stronger Goldbach conjecture, for the even number 130, find the largest possible difference between two distinct primes that add up to 130.


Given every 8-digit whole number is a possible telephone number except those that begin with 0 or 1, or end in 9, determine the fraction of these telephone numbers that begin with 9 and end with 0.


Define a sequence $\{b_n\}$ such that $b_1=3$ and $b_{n+1}=b_n + 2n + 1$ where $n \geq 1$. Find the value of $b_{100}$.


If Lin drove 100 miles on the highway and 20 miles on a forest trail, where she drove four times as fast on the highway as on the forest trail, and the time taken to drive on the forest trail was 40 minutes, calculate the total time taken for the entire trip.


Given that each student scored at least $70$ points on a $120$-point test, seven students scored a perfect $120$, and the mean score of the class was $85$, calculate the smallest possible number of students in the class.


How many different four-digit numbers can be formed by rearranging the four digits in $3008$?


Given that $a * b$ means $4a - 2b$, find the value of $x$ if $3 * (6 * x) = -2$.


Given rectangle ABCD, 30% of its area overlaps with square EFGH. Square EFGH shares 40% of its area with rectangle ABCD. If AD equals one-tenth of the side length of square EFGH, determine the ratio of AB to AD.


Given that the perimeter of a rectangle is 60 units, and the side lengths are integers, find the difference between the largest and smallest possible areas of the rectangle.


Cagney can frost a cupcake every 18 seconds and Lacey can frost a cupcake every 40 seconds. Lacey starts working 1 minute after Cagney starts. Calculate the number of cupcakes that they can frost together in 6 minutes.


Given $A, B$, and $C$ are pairwise disjoint sets of animals, the average weights of animals in sets $A, B, C, A \cup B, A \cup C$, and $B \cup C$ are $45, 35, 55, 40, 50$, and $45$ respectively. Find the average weight of the animals in set $A \cup B \cup C$.


Triangle $ABC$ has a right angle at $B$. Point $D$ is the foot of the altitude from $B$, $AD=5$, and $DC=3$. Calculate the area of $\triangle ABC$.


Given that the car ran exclusively on its battery for the first 50 miles, used gasoline at a rate of 0.03 gallons per mile for the rest of the trip, and averaged 50 miles per gallon for the entire trip, calculate the total length of the trip in miles.


Expand and simplify the expression $(1+x^3)(1-x^4)$.

Solution: \boxed{1 + x^3 - x^4 - x^7}


Given that Brianna used one fourth of her money to buy one fourth of the CDs, determine the fraction of her money that she will have left after she buys all the CDs.


If \(y\) cows produce \(y+2\) cans of milk in \(y+3\) days, determine the number of days it will take \(y+4\) cows to produce \(y+6\) cans of milk.


Consider two concentric circles where the width of the track formed by these circles is 15 feet. If the diameter of the smaller circle is 25 feet, calculate how long the outer circle's circumference is compared to the inner circle.


A magician has 3 red chips and 3 green chips in a hat. Chips are drawn randomly one at a time without replacement, until all 3 of the red chips are drawn or until all 3 green chips are drawn. Determine the probability that all the 3 red chips are drawn first.


Given the equation $4^{x+1} - 8 \cdot 2^{x+1} - 2^{x} + 8 = 0$, determine the number of real numbers $x$ that satisfy this equation.


Given that a positive integer is a perfect square, an even multiple of 5, and less than 2500, determine the number of such positive even multiples of 5.


What is the smallest integer larger than $(\sqrt{5}+\sqrt{3})^4$?


Given the equation $\frac{X}{8}=\frac{8}{Y}$, determine the number of ordered pairs of positive integers $(X, Y).$


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


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


The number of positive integers less than $1200$ that are divisible by neither $6$ nor $8$ is what?


Given that bricklayer Brenda would take $8$ hours to build a chimney alone, and bricklayer Brandon would take $12$ hours to build it alone, when they work together they reduce their combined output by $12$ bricks per hour, and that they complete the chimney in $6$ hours, calculate the total number of bricks in the chimney.


How many 7-digit palindromes can be formed using the digits 1, 1, 2, 2, 2, 4, 4?


The average age of the family members is $22$, the father is $50$ years old, and the average age of the mother and children is $15$. Given that the average age of all the family members is $22$, determine the number of children in the Alvez family.


Given that Square $ABCD$ has side length $5$, point $M$ is chosen on side $AB$ so that $\angle AMD = \angle CMD$, calculate the degree measure of $\angle AMD$.


Given that there are $n > 15$ students in a class with an average score of 10 and a subgroup of 15 students with an average score of 16, determine the average score of the remaining students in terms of $n$.

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


A rectangular box has a total surface area of 142 square inches and the sum of the lengths of all its edges is 60 inches. Find the sum of the lengths in inches of all of its interior diagonals.


A regular hexagonal estate with a scale of 300 miles to 1 inch has a side length of 6 inches on the map. What is the area of this estate in square miles?


If Leyla has a $3 \times 5$ index card and shortens the length of one side of this card by $2$ inches, the area becomes $9$ square inches, find the area of the card if she instead shortens the length of the other side by $2$ inches, keeping the first side's reduction as well.


Starting with the display "1," calculate the fewest number of keystrokes needed to reach "300" using the keys [+1] and [x2].


Given that $\log_8 x = 3.5$, solve for the value of $x$.


The sum of two numbers is 3 times their difference. If the larger number is denoted by x and the smaller one by y, express the ratio x/y as a simplified fraction.


If ten students from Delta school worked for 4 days, six students from Echo school worked for 6 days, eight students from Foxtrot school worked for 3 days, and three students from Golf school worked for 10 days, and the total payment made for the students' work was $1233, with each student receiving the same amount for a day's work, calculate the total amount earned by the students from Echo school altogether.

Solution: \boxed{341.45\text{ dollars}}


Given two numbers, their difference, their sum, and their product are to one another as $1:9:40$. Calculate the product of the two numbers.


$\triangle ABC$ has a right angle at $C$ and $\angle A = 30^{\circ}$. Given that $BD$ ($D$ in $\overline{AC}$) is the bisector of $\angle ABC$, determine $\angle BDC$.


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


Given the number $873$, express it in the form $873 = b_1 + b_2 \times 2! + b_3 \times 3! + \ldots + b_n \times n!$, where $0 \le b_k \le k$, and solve for $b_4$.


Given the sets of consecutive integers $\{1\}$,$\{2, 3\}$,$\{4,5,6\}$,$\{7,8,9,10\}$,$\; \cdots \;$, where each set contains one more element than the preceding one, and where the first element of each set is one more than the last element of the preceding set, let $S_n$ be the sum of the elements in the nth set. Find the value of $S_{15}$.


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


A 4x4x4 cube is constructed from 64 standard dice, where opposite dice faces sum to 7. Determine the smallest possible sum of all visible values on the 6 faces of this larger cube.


Given a rectangle where the perimeter is $p$ and the diagonal is $d$, calculate the area of the rectangle.

Solution: \boxed{\frac{p^2 - 4d^2}{8}}


Using the calculator with only two keys [+1] and [x2], starting from "1", determine the fewest number of keystrokes needed to reach a final display of "250".


Cassandra sets her watch to the correct time at noon. At the actual time of 1:00 PM, she notices that her watch reads 12:58 and 12 seconds. Assuming that her watch loses time at a constant rate, determine the actual time when her watch first reads 8:00 PM.

Solution: \boxed{8:14:51 \text{ PM}}


If the area of $\triangle ABC$ is $100$ square units and the geometric mean between sides $AB$ and $AC$ is $15$ inches, find $\sin A$.


Determine the relationship between the arithmetic mean of $x$, $y$, and $z$ and their geometric mean.

Solution: \boxed{\frac{x+y+z}{3} \geq \sqrt[3]{xyz}}


Given the numbers $1357$, $3571$, $5713$, and $7135$, calculate their sum.


Given the expression $\left[(a + 2b)^3 (a - 2b)^3\right]^2$, determine the number of distinct terms when the expression is simplified.


Given that Paul owes Paula $50$ cents and has access to $5$-cent, $10$-cent, $20$-cent, and $25$-cent coins, determine the difference between the largest and the smallest number of coins he can use to pay her.


Given points B and C lie on AD, and the length of AB is 3 times the length of BD, and the length of AC is 7 times the length of CD, determine the fraction of the length of AD that is represented by the length of BC.


Determine the number of different total scores an athlete could achieve with 8 attempts, where each attempt results in either a 2-pointer or a 3-pointer.


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


In the number $86549.2047$, calculate the ratio of the value of the place occupied by the digit 6 to the value of the place occupied by the digit 2.


Given that Jessie moves from 0 to 24 in six steps, and travels four steps to reach point x, then one more step to reach point z, and finally one last step to point y, calculate the value of y.


The population of Hypothetical Town at one time was a perfect square. Later, with an increase of $150$, the population was one more than a perfect square. Now, with an additional increase of $150$, the population is again a perfect square.


Given that Abby and Bridget are seated at a circular table with 6 seats, with Abby and Bridget occupying one of the 6 seats, calculate the probability that they are seated next to each other.


How many distinct triangles can be drawn using three of the dots below as vertices, where the dots are arranged in a grid of 2 rows and 4 columns?


Moe has a new, larger lawn which is a rectangular area of $120$ feet by $200$ feet. His mower has a swath width of $30$ inches and he overlaps each swath by $6$ inches. Moe walks at a pace of $4000$ feet per hour while mowing. Calculate the time it will take him to mow the entire lawn.


Tom buys either a $60$-cent apple or a $90$-cent orange each day over a seven-day week, and his total expenditure for the week was a whole number of dollars. Find the number of oranges he bought.


Given that the mean of three numbers is $20$ more than the least of the numbers and $25$ less than the greatest, and the median of the three numbers is $10$, calculate their sum.

Solution: \boxed{The three numbers are a = -5, b = 10, and c = 40. Their sum is: -5 + 10 + 40 = 45}


Evaluate the expression $1 - \frac{1}{1 + \sqrt{5}} + \frac{1}{1 - \sqrt{5}}$.

Solution: \boxed{1 - \frac{\sqrt{5}}{2}}


Carlos took $60\%$ of a whole pie. Maria then took one quarter of the remainder. Find the remaining portion of the whole pie.


Assume that x is a positive real number. Simplify the expression \( \sqrt[5]{x\sqrt[4]{x}} \).


The circle having $(0,0)$ and $(10,8)$ as the endpoints of a diameter intersects the $x$-axis at a second point. Find the $x$-coordinate of this point.


Given an average population of the towns in the Region of Maplefield is halfway between $4,800$ and $5,300$, calculate the total population of all $25$ towns.


A farmer plans to fence a rectangular garden using 60 meters of fencing material. One side of the garden borders a river, and thus does not require fencing. If the length of the garden alongside the river is twice as long as its width, calculate the area of the garden.


While Sophia is driving her car on a highway, she notices Sam riding his motorbike in the same direction 1 mile ahead of her. After overtaking him, she can continue to see him in her rearview mirror until he is 1 mile behind her. Sophia drives at a constant speed of 20 miles per hour, and Sam rides at a constant speed of 14 miles per hour. Calculate the time in minutes that Sophia can see Sam.


Let C represent the number of cows, D represent the number of ducks, and H represent the number of chickens.  If the number of legs was 20 more than twice the number of heads, then $2C+2D+2H=2(H+C+D)+20$. Also, if the ducks and chickens together were twice the number of cows, then $D+H=2C$.


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


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


A bag contains $\frac{3}{4}$ red kernels and $\frac{1}{4}$ green kernels. Only $\frac{3}{5}$ of the red kernels will pop, whereas $\frac{3}{4}$ of the green ones will pop. Determine the probability that the kernel selected was red.


Bella has two tanks. Initially, the first tank is $\frac{3}{4}$ full of oil and the second tank is completely empty. Bella pours all the oil from the first tank into the second tank, afterward, the second tank is $\frac{2}{5}$ full of oil. Determine the ratio of the volume of the first tank to the volume of the second tank.


Given the expression $\left(\frac{x^4+1}{x^2}\right)\left(\frac{y^4+1}{y^2}\right)+\left(\frac{x^4-1}{y^2}\right)\left(\frac{y^4-1}{x^2}\right)$, where $xy \not= 0$, simplify the expression.

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


A rectangular box has a total surface area of 166 square inches and the sum of the lengths of all its edges is 64 inches. Find the sum of the lengths in inches of all of its interior diagonals.


Given that Teresa's age is a prime number, and at least half of the students guessed too low, three students guessed exactly 43, and the incorrect guesses are off by at least two from Teresa's age, find Teresa's age.


Suppose the estimated $40$ billion dollar cost to send a scientific team to the moon is shared equally by $200$ million people in a country. Calculate each person's share.


Given that the probability of drawing a red card from a deck is initially $\frac{1}{4}$ and changes to $\frac{1}{6}$ after adding $6$ black cards, determine the original number of cards in the deck.


The polygon enclosed by solid lines consists of 5 congruent squares joined edge-to-edge in a cross shape. One more congruent square is attached to an edge at one of the twelve positions indicated. Determine the number of the resulting polygons that can be folded to form a cube.


Given that $E(n)$ denotes the sum of the even digits of $n$, find the sum of $E(1) + E(2) + \cdots + E(999)$.


Given that old license plates consist of a letter followed by three digits and new license plates consist of four letters followed by four digits, determine the ratio of the number of possible new license plates to the number of possible old license plates.


Express the number $700$ in a factorial base of numeration, that is, $700=a_1+a_2\times2!+a_3\times3!+a_4\times4!+ \ldots a_n \times n!$ where $0 \le a_k \le k,$ for each $k$. Find the coefficient $a_4$.


The sum of the numbers on each pair of opposite faces is equal, and the middle number in this range is the largest number on one of the faces, where the numbers on the faces are consecutive whole numbers between 15 and 20. Calculate the sum of all the numbers on this cube.


Find the sum of all prime numbers between $1$ and $120$ that are simultaneously $1$ greater than a multiple of $3$ and $1$ less than a multiple of $5$.


Jeremy had exactly enough money to buy 24 posters when he visited a store, but he found a special deal: buy one poster at regular price and get the second poster for half off. Determine the total number of posters Jeremy could buy under this sale.


A rectangular park is to be fenced on three sides using a 150-meter concrete wall as the fourth side. Fence posts are to be placed every 15 meters along the fence, including at the points where the fence meets the concrete wall. Calculate the minimal number of posts required to fence an area of 45 m by 90 m.


Two circles are associated with a regular pentagon $ABCDE$. The first circle is tangent to $\overline{AB}$ and the extended sides, while the second circle is tangent to $\overline{DE}$ and also the extended sides. What is the ratio of the area of the second circle to that of the first circle?


Find the number of distinct points in the $xy$-plane common to the graphs of $(x+2y-7)(2x-y+4)=0$ and $(x-2y+3)(4x+3y-18)=0$.


Lilian has two older twin sisters, and the product of their three ages is 162. Find the sum of their three ages.


Let $\frac {42x - 37}{x^2 - 4x + 3} = \frac {N_1}{x - 1} + \frac {N_2}{x - 3}$ be an identity in $x$. Find the numerical value of $N_1N_2$.


In a regional athletics competition, $275$ sprinters participate in a $100-$meter dash. The track can accommodate $8$ runners at a time, and in each race, the top 2 runners qualify for the next round while the others are eliminated. Determine the number of races required to declare the overall winner.


A pair of standard $6$-sided dice is rolled to determine the side length of a square. What is the probability that the numerical value of the area of the square is less than the numerical value of the perimeter?


Given a two-digit number whose tens' digit is $a$ and units' digit is $b$, find the expression for the new number formed when the digit $2$ is placed after it.


Given that the perimeter of rectangle PQRS is 40 cm, find the maximum value of the diagonal PQ in centimeters.


Given the number $7.47474747\ldots$, express it as a fraction in lowest terms and find the sum of the numerator and denominator.


Find the largest number by which the expression $n^3 - n - 6$ is divisible for all possible integral values of $n$.


Given Orvin had enough money to buy 24 balloons at the regular price, and he can buy another balloon at half off for each balloon purchased at the regular price, determine the maximum number of balloons Orvin can now purchase.


In an $h$-meter race, Sunny is exactly $2d$ meters ahead of Windy when Sunny finishes the race. The next time they race, Sunny sportingly starts $2d$ meters behind Windy, who is at the starting line. Both runners run at the same constant speed as they did in the first race. Calculate the distance that Sunny is ahead when Sunny finishes the second race.


Given that the speed of sound is $1100$ feet per second, estimate, to the nearest quarter-mile, how far Charlie was from the flash of lightning, given that fifteen seconds passed between the lightning flash and the sound of thunder.


A baseball league consists of two four-team divisions. Each team plays every other team in its division N games. Each team plays every team in the other division M games with $N>2M$ and $M>6$. Each team plays a $92$ game schedule. Determine the number of games that a team plays within its own division.


Given that Lucy's odometer initially showed the palindrome $123321$ and after driving for $4$ hours it displayed another palindrome, calculate her average speed during these $4$ hours.

Solution: \boxed{275 \text{ mph}}


A circle of radius 3 is centered at point $A$. An equilateral triangle with side length 6 has a vertex at $A$. Find the difference between the area of the region that lies inside the circle but outside the triangle and the area of the region that lies inside the triangle but outside the circle.

Solution: \boxed{9(\pi - \sqrt{3})}


A triangle-shaped grid of points is labeled with vertices A, B, C, and an additional point D, the centroid, inside the triangle. Determine the number of non-congruent triangles that can be formed by selecting vertices from these four points.


John took $80\%$ of a whole pizza. Emma took one fourth of the remainder. Calculate the portion of the whole pizza that was left.


A farmer needs to fence off a completely rectangular grazing area measuring 48 meters by 72 meters. Fence posts are to be installed every 8 meters around the perimeter. What is the minimum number of fence posts required?


If $x+2$ cows give $x+4$ cans of milk in $x+3$ days, determine the number of days it will take $x+5$ cows to give $x+9$ cans of milk.

Solution: \boxed{\frac{(x+9)(x+2)(x+3)}{(x+5)(x+4)}}


A square with integer side length is cut into 12 squares, 9 of which have an area of 1 and 3 of which have an area of 4 each. Calculate the smallest possible side length of the original square.


Jamie's smartphone battery lasts for 20 hours if the phone is not used but left on, and 4 hours if used constantly. After being on for 10 hours, with 90 minutes of use, how many more hours will the battery last if it is not used but left on?


Determine the value of the product $\left(1-\frac{1}{3^{2}}\right)\left(1-\frac{1}{4^{2}}\right)...\left(1-\frac{1}{12^{2}}\right)$.


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


Given that the distance EG along Elm Street is 8 miles, the height EF from point F to Elm Street is 4 miles, another triangle EFG' is within triangle EFG where G' is on the line FG, EG' is 4 miles, and the height EF' from F to Elm Street is 2 miles. Determine the area of the triangular plot EFG, excluding the area of triangle EFG'.


Numbers from 1 through 81 are written sequentially on a 9 by 9 grid checkerboard, where each successive row continues the sequence from the previous row. Calculate the sum of the numbers in the four corners of this 9x9 checkerboard.


Triangle XYZ has a right angle at Y. Point W is the foot of the altitude from Y, XW=4, and WZ=9. Find the area of triangle XYZ.


The number obtained from the last two nonzero digits of $100!$ is equal to $n$. What is $n$?


Country $X$ has $40\%$ of the world's population and $60\%$ of the world's wealth. Country $Y$ has $20\%$ of the world's population but $30\%$ of its wealth. Country $X$'s top $50\%$ of the population owns $80\%$ of the wealth, and the wealth in Country $Y$ is equally shared among its citizens. Determine the ratio of the wealth of an average citizen in the top $50\%$ of Country $X$ to the wealth of an average citizen in Country $Y$.


If $2b$ men can lay $3f$ bricks in $c$ days, calculate the number of days it will take $4c$ men to lay $6b$ bricks.


Ahn chooses a three-digit integer, subtracts it from 300, triples the result, and then adds 50. What is the largest number Ahn can obtain?


What is the sum of the two smallest prime factors of the number $540$?


How many pairs $(m,n)$ of integers satisfy the equation $(m-2)(n-2)=4$?


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


Given the store's sales tax rate is 8%, the original price of the jacket is $120, and the promotional discount is 25%, calculate the difference between the total prices computed by Pete and Polly.


Determine how many hours it will take Carl to mow the lawn, given that the lawn measures 120 feet by 100 feet, the mower's swath is 30 inches wide with an overlap of 6 inches, and Carl walks at a rate of 4000 feet per hour.


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


Given a square initially painted black, with $\frac{1}{2}$ of the square black and the remaining part white, determine the fractional part of the original area of the black square that remains black after six changes where the middle fourth of each black area turns white.


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


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