prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
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Let $A$ be a subset of $\{1,2,\ldots,2020\}$ such that the difference of any two distinct elements in $A$ is not prime. Determine the maximum number of elements in set $A$. | 505 |
Toner Drum and Celery Hilton are both running for president. A total of $2015$ people cast their vote, giving $60\%$ to Toner Drum. Let $N$ be the number of "representative" sets of the $2015$ voters that could have been polled to correctly predict the winner of the election (i.e., more people in the set voted for Drum... | 605 |
A sequence of integers $a_1$, $a_2$, $a_3$, \ldots$ is chosen so that $a_n = a_{n - 1} - a_{n - 2}$ for each $n \ge 3$. What is the sum of the first 2001 terms of this sequence if the sum of the first 1492 terms is 1985, and the sum of the first 1985 terms is 1492? | 986 |
Let $z$ be a complex number. If the equation \[x^3 + (4-i)x^2 + (2+5i)x = z\] has two roots that form a conjugate pair, find the absolute value of the real part of $z$. | 423 |
Consider a cube with a fly standing at each of its vertices. When a whistle blows, each fly moves to a vertex on the same face as the previous one but diagonally opposite to it. After the whistle blows, in how many ways can the flies change position so that there is no vertex with 2 or more flies? | 81 |
Suppose we have $2013$ piles of coins, with the $i$th pile containing exactly $i$ coins. We wish to remove the coins in a series of steps. In each step, we are allowed to take away coins from as many piles as we wish, but we have to take the same number of coins from each pile. We cannot take away more coins than a pil... | 11 |
The one hundred U.S. Senators are standing in a line in alphabetical order. Each senator either always tells the truth or always lies. The $i$th person in line says: "Of the $101-i$ people who are not ahead of me in line (including myself), more than half of them are truth-tellers." How many possibilities are there for... | 101 |
Compute the largest integer $N$ such that one can select $N$ different positive integers, none of which is larger than $17$, and no two of which share a common divisor greater than $1$. | 8 |
This year, February $29$ fell on a Sunday. In which year will February $29$ next fall on a Sunday? | 2032 |
How many 6-tuples \((a_1,a_2,a_3,a_4,a_5,a_6)\) are there such that each of \(a_1,a_2,a_3,a_4,a_5,a_6\) is from the set \(\{1,2,3,4\}\) and the six expressions \[ a_j^2 - a_j a_{j+1} + a_{j+1}^2 \] for \(j = 1,2,3,4,5,6\) (where \(a_7\) is to be taken as \(a_1\)) are all equal to one another? | 40 |
A box contains red, green, blue, and white balls, totaling 111 balls. If you take out 100 balls without looking, there will always be 4 balls of different colors among them. What is the smallest number of balls you must take out without looking to ensure that among them there will always be balls of at least 3 differen... | 88 |
For real numbers $a$, $b$, and $c$, the polynomial $p(x) = 3x^7 - 291x^6 + ax^5 + bx^4 + cx^2 + 134x - 2$ has $7$ real roots whose sum is $97$. Find the sum of the reciprocals of those $7$ roots. | 67 |
Find the sum of all positive integers $B$ such that $(111)_B = (aabbcc)_6$, where $a, b, c$ represent distinct base $6$ digits, with $a \neq 0$. | 237 |
How many integers $n$ satisfy the conditions $1 \leq n \leq 2020$ and $n^n$ is a perfect square? | 1032 |
Consider a right cylinder with height $5\sqrt{3}$. A plane intersects each of the bases of the cylinder at exactly one point, and the cylindric section (the intersection of the plane and the cylinder) forms an ellipse. Find the product of the sum and the difference of the lengths of the major and minor axes of this ell... | 75 |
Points $A$, $B$, $C$, $D$, and $E$ are located on a plane. It is known that: - $CA = 12$ - $AB = 8$ - $BC = 4$ - $CD = 5$ - $DB = 3$ - $BE = 6$ - $ED = 3$ Find the length of $AE$. | 10 |
Form a square with sides of length $5$. Triangular pieces from the four corners are removed to form a regular octagon. Find the area removed to the nearest integer. | 4 |
Mr. Rose, Mr. Stein, and Mr. Schwartz start at the same point around a circular track and run clockwise. Mr. Stein completes each lap in $6$ minutes, Mr. Rose in $10$ minutes, and Mr. Schwartz in $18$ minutes. How many minutes after the start of the race are the runners at identical points around the track (that is, th... | 45 |
Each square on an $8 \times 8$ checkers board contains either one or zero checkers. The number of checkers in each row is a multiple of $3$, and the number of checkers in each column is a multiple of $5$. Assuming the top left corner of the board is shown below, how many checkers are used in total? | 30 |
Cai and Lai are eating cookies. Their cookies are in the shape of $2$ regular hexagons glued to each other, and the cookies have an area of $18$ units. They each make a cut along the $2$ long diagonals of a cookie; this now makes four pieces for them to eat and enjoy. What is the minimum area among the four pieces? | 3 |
Find the sum of \( \frac{\sigma(n) \cdot d(n)}{ \phi (n)} \) over all positive \( n \) that divide \( 60 \). **Note:** - The function \( d(i) \) outputs the number of divisors of \( i \). - \( \sigma (i) \) outputs the sum of the factors of \( i \). - \( \phi (i) \) outputs the number of positive integers less than or ... | 350 |
Find the integer $n \ge 48$ for which the number of trailing zeros in the decimal representation of $n!$ is exactly $n-48$. | 62 |
The math team at Jupiter Falls Middle School meets twice a month during the summer, and the math team coach, Mr. Fischer, prepares some Olympics-themed problems for his students. One of the problems Joshua and Alexis work on boils down to a system of equations: \[\begin{align*} 2x+3y+3z &= 8,\\ 3x+2y+3z &= 808,\\ 3x+3y... | 10203 |
If $a$ and $b$ are natural numbers such that $a + 13b$ is divisible by $11$ and $a + 11b$ is divisible by $13$, then find the least possible value of $a + b$. | 28 |
Michael's task is to determine how many 7-digit telephone numbers can exist such that all seven digits are distinct, the first four digits are in descending order, and the last four digits are in ascending order. Given a 7-digit number format, consider the number $987-1234$ where the first four digits $(9-8-7-1)$ desce... | 2400 |
There are $256$ players in a tennis tournament ranked from $1$ to $256$, where $1$ is the highest rank and $256$ is the lowest rank. In a match, the player with the higher rank wins with a probability of $\frac{3}{5}$. In each round, players are paired such that the highest-ranked player faces the second-highest, the t... | 103 |
In trapezoid $ABCD$, where $AD \parallel BC$ and $\angle ABC + \angle CDA = 270^{\circ}$, compute $AB^2$ given the following conditions: - $AB \cdot \tan(\angle BCD) = 20$ - $CD = 13$ | 260 |
Let $n$ be an integer between $0$ and $5^5$. Arrange $n$ black stones and $5^5-n$ white stones in a row, and repeat the following operation $5$ times: group the stones in sets of $5$ from left to right, and replace the $5$ stones in each group with one stone of the color that appears more frequently among them. Find th... | 2883 |
Let $a_1 = 11^{11}$, $a_2 = 12^{12}$, $a_3 = 13^{13}$, and \[a_n = |a_{n-1} - a_{n-2}| + |a_{n-2} - a_{n-3}|, \quad n \ge 4.\] Determine $a_{14^{14}}$. | 1 |
A *strip* is the region between two parallel lines. Let $A$ and $B$ be two strips in a plane. The intersection of strips $A$ and $B$ is a parallelogram $P$. Let $A'$ be a rotation of $A$ in the plane by $60^\circ$. The intersection of strips $A'$ and $B$ is a parallelogram with the same area as $P$. Let $x^\circ$ be th... | 150 |
A BWM tree is defined recursively: - An empty tree is a BWM tree of height 0 and size 0. - A nonempty BWM tree consists of a root node and three subtrees, each of which is itself a (possibly empty) BWM tree. The height of the tallest of the subtrees must be at most 2 more than the height of the shortest. - The height o... | 154 |
The diagram below shows a shaded region bounded by two concentric circles where the outer circle has twice the radius of the inner circle. The total boundary of the shaded region has length $36\pi$. Find $n$ such that the area of the shaded region is $n\pi$. | 108 |
Find the number of $10$-tuples $(a_1,a_2,\dots,a_{10})$ of integers such that $|a_1| \leq 1$ and \[a_1^2 + a_2^2 + a_3^2 + \cdots + a_{10}^2 - a_1a_2 - a_2a_3 - a_3a_4 - \cdots - a_9a_{10} - a_{10}a_1 = 2.\] | 3780 |
The polynomial $R(x)$ is the remainder when $x^{2007}$ is divided by $x^2 - 5x + 6$. Express $R(0)$ in the form $ab(a^c - b^c)$. Find the value of $a + c - b$. | 2005 |
Katya Ugolnikova drew several angles, each greater than $10^\circ$, less than $90^\circ$, and expressed as an integer number of degrees. What is the minimum number of drawn angles required to ensure that at least three of them are equal? | 159 |
Three uncoordinated aliens launch a three-day attack on four galaxies. Each day, each of the three aliens chooses a galaxy uniformly at random from the remaining galaxies and destroys it. They make their choice simultaneously and independently, so two aliens could destroy the same galaxy. If the probability that every ... | 1151 |
Find the area $S$ of the region enclosed by the curve $y=\left|x-\frac{1}{x}\right|$ (where $x>0$) and the line $y=2$. | 2 |
Each number in the list $1, 2, 3, \ldots, 10$ is either colored red or blue. Numbers are colored independently, and both colors are equally probable. The expected value of the number of positive integers expressible as a sum of a red integer and a blue integer can be written as $\frac{m}{n}$ for relatively prime positi... | 455 |
Right triangular prism $ABCDEF$ has triangular faces $\triangle ABC$ and $\triangle DEF$, with edges $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$. The angles are $\angle ABC = 90^\circ$ and $\angle EAB = \angle CAB = 60^\circ$. Given that $AE = 2$, the volume of $ABCDEF$ can be expressed as $\frac{m}{n}$, wher... | 5 |
Let $s_n$ be the number of solutions to the equation \(a_1 + a_2 + a_3 + a_4 + b_1 + b_2 = n\), where \(a_1, a_2, a_3,\) and \(a_4\) are elements of the set \(\{2, 3, 5, 7\}\) and \(b_1\) and \(b_2\) are elements of the set \(\{1, 2, 3, 4\}\). Find the number of \(n\) for which \(s_n\) is odd. | 12 |
The $12$-sided polygon below was created by placing three $3 \times 3$ squares with their sides parallel so that vertices of two of the squares are at the center of the third square. Find the perimeter of this $12$-sided polygon. | 24 |
What are the last two digits of $2017^{2017}$? | 77 |
Considering all numbers of the form $n = \left\lfloor \frac{k^3}{2012} \right\rfloor$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$, and $k$ ranges from $1$ to $2012$, how many of these $n$’s are distinct? | 1995 |
Let $A_0A_1 \dots A_{11}$ be a regular $12$-gon inscribed in a circle with diameter $1$. For how many subsets $S \subseteq \{1,\dots,11\}$ is the product $\prod_{s \in S} A_0A_s$ equal to a rational number? (The empty product is declared to be $1$.) | 192 |
The squares of an $8 \times 8$ board are colored alternately black and white. A rectangle consisting of some of the squares of the board is called important if its sides are parallel to the sides of the board and all its corner squares are colored black. The side lengths can be anything from $1$ to $8$ squares. On each... | 100 |
A sequence \((a_n)\) of real numbers is defined by \(a_0=1\), \(a_1=2015\) and for all \(n \geq 1\), we have: \[ a_{n+1} = \frac{n-1}{n+1}a_n - \frac{n-2}{n^2+n}a_{n-1}. \] Calculate the value of \[ \frac{a_1}{a_2} - \frac{a_2}{a_3} + \frac{a_3}{a_4} - \frac{a_4}{a_5} + \ldots + \frac{a_{2013}}{a_{2014}} - \frac{a_{201... | 3021 |
Let $S(n)$ be the sum of the digits of the positive integer $n$. Determine $S(S(S(2003^{2003})))$. | 8 |
A subset $B$ of the set of the first $100$ positive integers has the property that no two elements of $B$ sum to $125$. What is the maximum possible number of elements in $B$? | 62 |
Let $a$, $b$, $c$, and $d$ be prime numbers such that $a > 3b > 6c > 12d$ and $a^2 - b^2 + c^2 - d^2 = 1749$. Determine all possible values of $a^2 + b^2 + c^2 + d^2$. | 1999 |
Ethan Song and Bryan Guo are playing an unfair game of rock-paper-scissors. In any game, Ethan has a \( \frac{2}{5} \) chance to win, \( \frac{2}{5} \) chance to tie, and \( \frac{1}{5} \) chance to lose. How many games is Ethan expected to win before losing? | 2 |
Let $ABC$ be a triangle with $BC = 7$, $AB = 5$, and $AC = 8$. Let $M$ and $N$ be the midpoints of sides $AC$ and $AB$, respectively, and let $O$ be the circumcenter of $\triangle ABC$. Let $BO$ and $CO$ meet $AC$ and $AB$ at $P$ and $Q$, respectively. If $MN$ meets $PQ$ at $R$ and $OR$ meets $BC$ at $S$, then the valu... | 240607 |
Dragoons take up $1 \times 1$ squares in the plane with sides parallel to the coordinate axes such that the interiors of the squares do not intersect. A dragoon can fire at another dragoon if the difference in the $x$-coordinates of their centers and the difference in the $y$-coordinates of their centers are both at mo... | 168 |
Let $\pi$ be a permutation of the numbers from $1$ through $2012$. What is the maximum possible number of integers $n$ with $1 \le n \le 2011$ such that $\pi(n)$ divides $\pi(n + 1)$? | 1006 |
Consider the graph in 3-space of \[0 = xyz(x + y)(y + z)(z + x)(x - y)(y - z)(z - x).\] This graph divides 3-space into \(N\) connected regions. What is \(N\)? | 48 |
The integers $1$ through $1000$ are arranged on the circumference of a circle in natural order. Starting with $1$, every fifteenth number (i.e., $1, 16, 31, \ldots$) is marked. The marking continues until an already marked number is reached. How many numbers will remain unmarked? | 800 |
Find the largest positive integer $k$ for which there exists a convex polyhedron $\mathcal{P}$ with 2022 edges, which satisfies the following properties: - The degrees of the vertices of $\mathcal{P}$ don’t differ by more than one, and - It is possible to color the edges of $\mathcal{P}$ with $k$ colors such that for e... | 2 |
Compute the number of positive integers $n$ satisfying the inequalities: \[ 2^{n-1} < 5^{n-3} < 3^n. \] | 5 |
Vasya has $n$ candies of several types, where $n > 145$. It is known that for any group of at least 145 candies, there is a type of candy which appears exactly 10 times. Find the largest possible value of $n$. | 160 |
Call a positive integer $n \geq 2$ "junk" if there exist two distinct $n$-digit binary strings $a_1a_2\cdots a_n$ and $b_1b_2\cdots b_n$ such that: - $a_1 + a_2 = b_1 + b_2$, - $a_{i-1} + a_i + a_{i+1} = b_{i-1} + b_i + b_{i+1}$ for all $2 \leq i \leq n-1$, - $a_{n-1} + a_n = b_{n-1} + b_n$. Find the number of junk pos... | 672 |
Let $p$ be a constant number such that $0 < p < 1$. Evaluate \[ \sum_{k=0}^{2004} \frac{p^k (1-p)^{2004-k}}{\int_0^1 x^k (1-x)^{2004-k} \, dx} \] | 2005 |
The three roots of the cubic $30x^3 - 50x^2 + 22x - 1$ are distinct real numbers between $0$ and $1$. For every nonnegative integer $n$, let $s_n$ be the sum of the $n$th powers of these three roots. What is the value of the infinite series \[ s_0 + s_1 + s_2 + s_3 + \dots \] ? | 12 |
Let $ABCD$ be a parallelogram with an area of 160. The diagonals $AC$ and $BD$ intersect at point $E$. Point $P$ is on the line segment $\overline{AE}$ such that $EC = 4EP$. If the line $DP$ intersects $AB$ at point $F$, find the area of quadrilateral $BFPC$. | 62 |
If $\phi$ is the Golden Ratio, we know that $\frac{1}{\phi} = \phi - 1$. Define a new positive real number, called $\phi_d$, where $\frac{1}{\phi_d} = \phi_d - d$ (so $\phi = \phi_1$). Given that $\phi_{2009} = \frac{a + \sqrt{b}}{c}$, where $a$, $b$, $c$ are positive integers, and the greatest common divisor of $a$ an... | 4038096 |
In $\triangle ABC$, $AB = AC$. Its circumcircle, $\Gamma$, has a radius of 2. Circle $\Omega$ has a radius of 1 and is tangent to $\Gamma$, $\overline{AB}$, and $\overline{AC}$. The area of $\triangle ABC$ can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a, b, c$, where $b$ is squarefree and $\gcd (a, c... | 339 |
Zang is at the point $(3,3)$ in the coordinate plane. Every second, he can move one unit up or one unit right, but he may never visit points where the $x$ and $y$ coordinates are both composite. In how many ways can he reach the point $(20, 13)$? | 210 |
Find the sum of all positive integers $n$ with the following properties: - $n$ is not divisible by any primes larger than $10$. - For some positive integer $k$, the positive divisors of $n$ are ordered as: \[1 = d_1 < d_2 < d_3 < \cdots < d_{2k} = n.\] - The divisors of $n$ satisfy the condition: \[d_1 + d_2 + \cdots +... | 416 |
Let $\clubsuit(x)$ denote the sum of the digits of the positive integer $x$. For example, $\clubsuit(8) = 8$ and $\clubsuit(123) = 1 + 2 + 3 = 6$. For how many two-digit values of $x$ is $\clubsuit(\clubsuit(x)) = 3$? | 10 |
Let $a$ and $b$ be complex numbers such that $(a+1)(b+1)=2$ and $(a^2+1)(b^2+1)=32.$ Compute the sum of all possible values of $(a^4+1)(b^4+1).$ | 1160 |
Let $S$ be the set of all subsets of $\{2,3,\ldots,2016\}$ with size $1007$, and for a nonempty set $T$ of numbers, let $f(T)$ be the product of the elements in $T$. Determine the remainder when \[ \sum_{T\in S}\left(f(T)-f(T)^{-1}\right)^2\] is divided by $2017$. Note: For $b$ relatively prime to $2017$, $b^{-1}$ is t... | 2014 |
Find the smallest positive prime that divides $n^2 + 5n + 23$ for some integer $n$. | 17 |
Farmer John has a flexible fence of length $L$ and two straight walls that intersect at a corner perpendicular to each other. He knows that if he doesn't use any walls, he can enclose a maximum possible area of $A_0$. When he uses one of the walls or both walls, he gets a maximum area of $A_1$ and $A_2$, respectively. ... | 4000 |
Let $\{a_n\}_{n \geq 1}$ be an increasing and bounded sequence. Calculate the limit: $$\lim_{{n \to \infty}} \left( 2a_n - a_1 - a_2 \right) \left( 2a_n - a_2 - a_3 \right) \cdots \left( 2a_n - a_{n-2} - a_{n-1} \right) \left( 2a_n - a_{n-1} - a_1 \right).$$ | 0 |
The positive integers $a, b, c$ satisfy the following conditions: 1. $\text{gcd}(a, b, c) = 1$ 2. $\text{gcd}(a, b + c) > 1$ 3. $\text{gcd}(b, c + a) > 1$ 4. $\text{gcd}(c, a + b) > 1$ Determine the smallest possible value of $a + b + c$. | 30 |
Let $m$ and $n$ be positive integers such that $\gcd(m,n)=1$ and $$\sum_{k=0}^{2020} (-1)^k {2020 \choose k} \cos(2020\cos^{-1}(\tfrac{k}{2020}))=\frac{m}{n}.$$ Suppose $n$ is written as the product of a collection of (not necessarily distinct) prime numbers. Compute the sum of the members of this collection. (For exam... | 209601 |
Triangle $ABC$ has sides $AB = 4$, $AC = 5$, and $BC = 6$. An angle bisector is drawn from angle $A$ and meets $BC$ at point $M$. Find the nearest integer to $100 \frac{AM}{CM}$. | 100 |
Farmer James invents a new currency such that for every positive integer $n \le 6$, there exists an $n$-coin worth $n!$ cents. Furthermore, he has exactly $n$ copies of each $n$-coin. An integer $k$ is said to be \textit{nice} if Farmer James can make $k$ cents using at least one copy of each type of coin. How many pos... | 210 |
Let $L_n$ be the least common multiple of the integers $1, 2, \dots, n$. For example, $L_{10} = 2,520$ and $L_{30} = 2,329,089,562,800$. Find the remainder when $L_{31}$ is divided by $100,000$. | 46800 |
Find all three-digit numbers such that the square of the number is equal to the sum of their digits raised to the power of 5. | 243 |
Let $ABC$ be an equilateral triangle with side length $10$. A square $PQRS$ is inscribed in it, with $P$ on $AB$, $Q$ and $R$ on $BC$, and $S$ on $AC$. If the area of the square $PQRS$ is $m + n\sqrt{k}$ where $m$ and $n$ are integers and $k$ is a prime number, then determine the value of $\sqrt{\frac{m+n}{k^2}}$. | 10 |
A kite is inscribed in a circle with center $O$ and radius $60$. The diagonals of the kite meet at a point $P$, and $OP$ is an integer. The minimum possible area of the kite can be expressed in the form $a\sqrt{b}$, where $a$ and $b$ are positive integers and $b$ is squarefree. Find $a+b$. | 239 |
Let $A$, $B$, and $C$ be unique collinear points such that $AB = BC = \frac{1}{3}$. Let $P$ be a point that lies on the circle centered at $B$ with radius $\frac{1}{3}$ and the circle centered at $C$ with radius $\frac{1}{3}$. Find the measure of angle $\angle PAC$ in degrees. | 30 |
Determine the largest possible number of groups one can compose from the integers $1, 2, 3, \ldots, 19, 20$, so that the product of the numbers in each group is a perfect square. Each group may contain exactly one number, in which case the product equals this number, and each number must be in exactly one group. | 8 |
56 lines are drawn on a plane such that no three of them are concurrent. If the lines intersect at exactly 594 points, what is the maximum number of them that could have the same slope? | 44 |
Let $n$ be a positive integer such that $12n^2 + 12n + 11$ is a four-digit number with all four digits equal. Determine the value of $n$. | 21 |
Suppose there are $160$ pigeons and $n$ holes. The $1$st pigeon flies to the $1$st hole, the $2$nd pigeon flies to the $4$th hole, and so on, such that the $i$th pigeon flies to the $(i^2 \mod n)$th hole, where $k \mod n$ is the remainder when $k$ is divided by $n$. What is the minimum $n$ such that there is at most on... | 326 |
Find the maximum value of an integer $B$ such that for every 9 distinct natural numbers with the sum of $2023$, there must exist a sum of 4 of these numbers that is greater than or equal to $B$. | 910 |
For any positive integer $n$, let \( a_n = \sum_{k=1}^{\infty} \left\lfloor \frac{n+2^{k-1}}{2^k} \right\rfloor \), where \( \left\lfloor x \right\rfloor \) is the largest integer that is equal to or less than \( x \). Determine the value of \( a_{2015} \). | 2015 |
Let $a$, $b$, $c$ be positive reals for which \[ (a+b)(a+c) = bc + 2 \\ (b+c)(b+a) = ca + 5 \\ (c+a)(c+b) = ab + 9 \] If $abc = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $100m+n$. | 4532 |
Let $ABCD$ be a rectangle inscribed in circle $\Gamma$, and let $P$ be a point on the minor arc $AB$ of $\Gamma$. Suppose that $PA \cdot PB = 2$, $PC \cdot PD = 18$, and $PB \cdot PC = 9$. The area of rectangle $ABCD$ can be expressed as $\frac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers an... | 21055 |
For each nonnegative integer $n$, we define $A_n = 2^{3n} + 3^{6n+2} + 5^{6n+2}$. Find the greatest common divisor of the numbers $A_0, A_1, \ldots, A_{1999}$. | 7 |
Let $ABC$ be a triangle with $AB = 26$, $BC = 51$, and $CA = 73$. Let $O$ be an arbitrary point in the interior of $\triangle ABC$. Lines $\ell_1$, $\ell_2$, and $\ell_3$ pass through $O$ and are parallel to $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$, respectively. The intersections of $\ell_1$, $\ell_2$, an... | 280 |
The roots of the polynomial $x^4 - 4ix^3 + 3x^2 - 14ix - 44$ form the vertices of a parallelogram in the complex plane. What is the area of the parallelogram? | 12 |
Let $n, m$ be positive integers of different parity, and $n > m$. Find all integers $x$ such that $\frac{x^{2^n} - 1}{x^{2^m} - 1}$ is a perfect square. | 0 |
A cardboard box in the shape of a rectangular parallelepiped is to be enclosed in a cylindrical container with a hemispherical lid. If the total height of the container from the base to the top of the lid is $60$ centimeters and its base has a radius of $30$ centimeters, find the volume of the largest box that can be c... | 64000 |
Let $x_1 = \sqrt{10}$ and $y_1 = \sqrt{3}$. For all $n \geq 2$, let \[ x_n = x_{n-1} \sqrt{77} + 15y_{n-1}, \\ y_n = 5x_{n-1} + y_{n-1} \sqrt{77}. \] Find $x_5^6 + 2x_5^4 - 9x_5^4y_5^2 - 12x_5^2y_5^2 + 27x_5^2y_5^4 + 18y_5^4 - 27y_5^6$. | 4608 |
Positive integer $n$ when divided by $3$ gives a remainder $a$, when divided by $5$ gives a remainder $b$, and when divided by $7$ gives a remainder $c$. Find the remainder when dividing $n$ by $105$ if $4a + 3b + 2c = 30$. | 89 |
Garfield and Odie are situated at $(0,0)$ and $(25,0)$, respectively. Suddenly, Garfield and Odie dash in the direction of the point $(9, 12)$ at speeds of $7$ and $10$ units per minute, respectively. During this chase, the minimum distance between Garfield and Odie can be written as $\frac{m}{\sqrt{n}}$ for relatively... | 159 |
Find the number of positive integers less than 101 that cannot be written as the difference of two squares of integers. | 25 |
You have four fair $6$-sided dice, each numbered $1$ to $6$ (inclusive). If all four dice are rolled, the probability that the product of the rolled numbers is prime can be written as $\frac{a}{b}$, where $a$ and $b$ are relatively prime. What is $a+b$? | 109 |
The diagram below shows a large triangle with area $72$. Each side of the triangle has been trisected, and line segments have been drawn between these trisection points parallel to the sides of the triangle. Find the area of the shaded region. | 48 |
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