prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
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A multiple choice test consists of 100 questions. If a student answers a question correctly, he will get 4 marks; if he answers a question wrongly, he will get \(-1\) mark. He will get 0 mark for an unanswered question. Determine the number of different total marks of the test. (A total mark can be negative.) | 495 |
Triangle $ABC$ satisfies $AB=104$, $BC=112$, and $CA=120$. Let $\omega$ and $\omega_A$ denote the incircle and $A$-excircle of $\triangle ABC$, respectively. There exists a unique circle $\Omega$ passing through $A$ which is internally tangent to $\omega$ and externally tangent to $\omega_A$. Compute the radius of $\Om... | 49 |
Let $\triangle ABC$ be an isosceles triangle with $AB=AC$ and $BC=14$. The triangle is inscribed in a circle $\omega$. Let $D$ be a point on the ray $BC$ such that $CD=6$. The intersection of $AD$ and $\omega$ is point $E$. Given that $AE=7$, find $AC^2$. | 105 |
Find the positive integer $k$ such that the roots of the polynomial $x^3 - 15x^2 + kx - 1105$ are three distinct collinear points in the complex plane. | 271 |
A right regular hexagonal prism has bases $ABCDEF$ and $A'B'C'D'E'F'$ with edges $AA'$, $BB'$, $CC'$, $DD'$, $EE'$, and $FF'$, each perpendicular to both hexagons. The height of the prism is $5$ and the side length of the hexagons is $6$. A plane $P$ passes through points $A$, $C'$, and $E$. The area of the portion of ... | 323 |
How many triples of integers $(a, b, c)$ with $-10 \leq a, b, c \leq 10$ satisfy $a^2 + b^2 + c^2 = (a + b + c)^2$? | 103 |
For positive integers $n$, let $S_n$ be the set of integers $x$ such that $n$ distinct lines, with no three concurrent, can divide a plane into $x$ regions. For example, $S_2 = \{3, 4\}$ because the plane is divided into 3 regions if the two lines are parallel, and 4 regions otherwise. What is the minimum $i$ such that... | 4 |
Evaluate \( \frac{1}{\int _0^{\frac{\pi}{2}} \cos ^{2006}x \cdot \sin 2008 x\, dx} \). | 2007 |
Find the smallest positive integer $k$ such that there is exactly one prime number of the form $kx + 60$ for the integers $0 \le x \le 10$. | 17 |
In a tournament, each player played exactly one game against each of the other players. In each game, the winner was awarded $1$ point, the loser got $0$ points, and each of the two players earned $\frac{1}{2}$ point if the game was a tie. After the completion of the tournament, it was found that exactly half of the po... | 25 |
Let \[ f(x) = \frac{1}{1 - \frac{1}{1-x}} \]. Compute \( f^{2016}(2016) \), where \( f \) is composed upon itself 2016 times. | 2016 |
The integers from 1 to $2008^2$ are written on each square of a $2008 \times 2008$ board. For every row and column, the difference between the maximum and minimum numbers is computed. Let $S$ be the sum of these 4016 numbers. Find the greatest possible value of $S$. | 16184704896 |
Let $ABCDE$ be a regular pentagon, and let $F$ be a point on $\overline{AB}$ with $\angle CDF=55^\circ$. Suppose $\overline{FC}$ and $\overline{BE}$ meet at $G$, and select $H$ on the extension of $\overline{CE}$ past $E$ such that $\angle DHE=\angle FDG$. Find the measure of $\angle GHD$, in degrees. | 19 |
A circle $\omega$ has center $O$ and radius $r$. A chord $BC$ of $\omega$ also has length $r$, and the tangents to $\omega$ at $B$ and $C$ meet at $A$. Ray $AO$ meets $\omega$ at $D$ past $O$, and ray $OA$ meets the circle centered at $A$ with radius $AB$ at $E$ past $A$. Compute the degree measure of $\angle DBE$. | 135 |
The digits $1$, $4$, $9$, and $2$ are each used exactly once to form a $4$-digit number $N$. What is the sum of all possible values of $N$? | 106656 |
Find the sum of all the positive integers that have at most three (not necessarily distinct) prime factors, where the primes come from the set \( \{ 2, 3, 5, 7 \} \). | 1932 |
If $x$, $y$, and $z$ are real numbers such that $x^2 + z^2 = 1$ and $y^2 + 2y(x + z) = 6$, find the maximum value of $y(z - x)$. | 4 |
Let $1000 \leq n = \text{ABCD}_{10} \leq 9999$ be a positive integer whose digits $\text{ABCD}$ satisfy the divisibility condition: $$1111 \mid (\text{ABCD} + \text{AB} \times \text{CD}).$$ Determine the smallest possible value of $n$. | 1729 |
Let $z$ be a complex number satisfying $(z+\frac{1}{z})(z+\frac{1}{z}+1)=1$. Evaluate $(3z^{100}+\frac{2}{z^{100}}+1)(z^{100}+\frac{2}{z^{100}}+3)$. | 36 |
Five distinct 2-digit numbers are in a geometric progression. Find the middle term. | 36 |
Find the coefficient of $x^7$ in the polynomial expansion of $(1 + 2x - x^2)^4$. | -8 |
You and your friend play a game on a $7 \times 7$ grid of buckets. Your friend chooses $5$ "lucky" buckets by marking an "X" on the bottom that you cannot see. However, he tells you that they either form a vertical or horizontal line of length $5$. To clarify, he will select either of the following sets of buckets: - E... | 9 |
Let $a_n$ be the last digit of the sum of the digits of the number formed by repeating the block $2005$ exactly $n$ times. Calculate $a_1 + a_2 + \dots + a_{2005}$. | 9025 |
Two thousand points are given on a circle. Label one of the points 1. From this point, count 2 points in the clockwise direction and label this point 2. From the point labeled 2, count 3 points in the clockwise direction and label this point 3. Continue this process until the labels $1, 2, 3, \dots, 1993$ are all used.... | 118 |
Frist Campus Center is located $1$ mile north and $1$ mile west of Fine Hall. The area within $5$ miles of Fine Hall that is located north and east of Frist can be expressed in the form $\frac{a}{b} \pi - c$, where $a, b, c$ are positive integers and $a$ and $b$ are relatively prime. Find $a + b + c$. | 30 |
Call a number "Sam-azing" if it is equal to the sum of its digits times the product of its digits. The only two three-digit Sam-azing numbers are $n$ and $n + 9$. Find $n$. | 135 |
Let $T$ be a trapezoid with two right angles and side lengths $4$, $4$, $5$, and $\sqrt{17}$. Two line segments are drawn, connecting the midpoints of opposite sides of $T$ and dividing $T$ into $4$ regions. If the difference between the areas of the largest and smallest of these regions is $d$, compute $240d$. | 120 |
Find the number of integers $n$ for which $\sqrt{\frac{(2020 - n)^2}{2020 - n^2}}$ is a real number. | 90 |
Find the maximal cardinality $|S|$ of the subset $S \subset A=\{1, 2, 3, \dots, 9\}$ given that no two sums $a+b$ where $a, b \in S$ and $a \neq b$ are equal. | 5 |
In a directed graph with $2013$ vertices, there is exactly one edge between any two vertices, and for every vertex, there exists an edge outwards from this vertex. We know that whatever the arrangement of the edges, from every vertex, we can reach $k$ vertices using at most two edges. Find the maximum value of $k$. | 2012 |
For any positive integer $n$, let $r_n$ denote the greatest odd divisor of $n$. Compute the sum: \[ T = r_{100} + r_{101} + r_{102} + \cdots + r_{200} \] | 10025 |
Let $x$, $y$, $z$ be positive integers satisfying $x < y < z$ and $x + xy + xyz = 37$. Find the greatest possible value of $x + y + z$. | 20 |
The diagram below shows a parallelogram $ABCD$ with $AB = 36$ and $AD = 60$. Diagonal $BD$ is perpendicular to side $AB$. Points $E$ and $F$ bisect sides $AD$ and $BC$, respectively. Points $G$ and $H$ are the intersections of $BD$ with $AF$ and $CE$, respectively. Find the area of quadrilateral $EGFH$. | 288 |
A crazy physicist has discovered a new particle called an emon. He starts with two emons in the plane, situated a distance $1$ from each other. He also has a crazy machine which can take any two emons and create a third one in the plane such that the three emons lie at the vertices of an equilateral triangle. After he ... | 108 |
Given the expression: \[ A = 3\sum_{m=1}^{n^2}\left(\frac{1}{2} - \{\sqrt{m}\}\right) \] where $n$ is a positive integer. Find the largest integer $k$ such that $n^k$ divides $\lfloor A \rfloor$ (the greatest integer less than or equal to $A$). | 1 |
Let $A$ and $B$ be digits between $0$ and $9$, and suppose that the product of the two-digit numbers $\overline{AB}$ and $\overline{BA}$ is equal to $k$. Given that $k+1$ is a multiple of $101$, find $k$. | 403 |
In triangle $ABC$, $AB=13$, $BC=14$, $CA=15$. Let $\Omega$ and $\omega$ be the circumcircle and incircle of $\triangle ABC$ respectively. Among all circles that are tangent to both $\Omega$ and $\omega$, call those that contain $\omega$ inclusive and those that do not contain $\omega$ exclusive. Let $\mathcal{I}$ and $... | 1558057 |
Let $d$ and $n$ be positive integers such that $d$ divides $n$, $n > 1000$, and $n$ is not a perfect square. The minimum possible value of $|d - \sqrt{n}|$ can be written in the form $a\sqrt{b} + c$, where $b$ is a positive integer not divisible by the square of any prime, and $a$ and $c$ are nonzero integers (not nece... | 38 |
For a real number $\alpha > 0$, consider the infinite real sequence defined by $x_1 = 1$ and \[ \alpha x_n = x_1 + x_2 + \cdots + x_{n+1} \text{ for } n \ge 1. \] Determine the smallest $\alpha$ for which all terms of this sequence are positive reals. | 4 |
Find the smallest prime $p$ such that the sum of the digits of $p$ (in base 10) is a prime number greater than $10$. | 29 |
A finite sequence of positive integers $m_i$ for $i = 1, 2, \ldots, 2006$ is defined such that $m_1 = 1$ and $m_i = 10m_{i-1} + 1$ for $i > 1$. How many of these integers are divisible by $37$? | 668 |
In the $xy$-coordinate plane, the $x$-axis and the line $y=x$ are mirrors. If you shoot a laser beam from the point $(126, 21)$ toward a point on the positive $x$-axis, there are $3$ places you can aim at where the beam will bounce off the mirrors and eventually return to $(126, 21)$. They are $(126, 0)$, $(105, 0)$, a... | 111 |
Find the remainder when $3^{2007}$ is divided by $2007$. | 1620 |
Consider the polynomial \[P(x) = x^3 + 3x^2 + 6x + 10.\] Let its three roots be \(a\), \(b\), and \(c\). Define \(Q(x)\) to be the monic cubic polynomial with roots \(ab\), \(bc\), and \(ca\). Compute \(|Q(1)|\). | 75 |
Let $n = 2^8 \cdot 3^9 \cdot 5^{10} \cdot 7^{11}$. For $k$ a positive integer, let $f(k)$ be the number of integers $0 \le x < n$ such that $x^2 \equiv k^2 \pmod{n}$. Compute the number of positive integers $k$ such that $k \mid f(k)$. | 24 |
How many ordered pairs of positive integers $(m, n)$ satisfy the system: \[ \begin{align*} \gcd (m^3, n^2) & = 2^2 \cdot 3^2, \\ \text{LCM} [m^2, n^3] & = 2^4 \cdot 3^4 \cdot 5^6, \end{align*} \] where $\gcd(a, b)$ and $\text{LCM}[a, b]$ denote the greatest common divisor and least common multiple of $a$ and $b$, respe... | 2 |
Define a sequence $a_i$ as follows: $a_1 = 181$ and for $i \ge 2$, \( a_i = a_{i-1}^2 - 1 \) if $a_{i-1}$ is odd and \( a_i = \frac{a_{i-1}}{2} \) if $a_{i-1}$ is even. Find the least $i$ such that $a_i = 0$. | 105 |
Jimmy invites Kima, Lester, Marlo, Namond, and Omar to dinner. There are nine chairs at Jimmy's round dinner table. Jimmy sits in the chair nearest the kitchen. How many different ways can Jimmy's five dinner guests arrange themselves in the remaining $8$ chairs at the table if Kima and Marlo refuse to be seated in adj... | 5040 |
On January $20$, $2018$, Sally notices that her $7$ children have ages which sum to a perfect square: their ages are $1$, $3$, $5$, $7$, $9$, $11$, and $13$, with $1 + 3 + 5 + 7 + 9 + 11 + 13 = 49$. Let $N$ be the age of the youngest child the next year the sum of the $7$ children's ages is a perfect square on January ... | 218 |
There are $2012$ distinct points in the plane, each of which is to be colored using one of $n$ colors, so that the numbers of points of each color are distinct. A set of $n$ points is said to be multi-colored if their colors are distinct. Determine $n$ that maximizes the number of multi-colored sets. | 61 |
In a group of $6$ people playing the card game Tractor, all $54$ cards from $3$ decks are dealt evenly to all the players at random. Each deck is dealt individually. Let the probability that no one has at least two of the same card be $X$. Find the largest integer $n$ such that the $n$th root of $X$ is rational. | 54 |
Suppose $P(x)$ is a degree $n$ monic polynomial with integer coefficients such that $2013$ divides $P(r)$ for exactly $1000$ values of $r$ between $1$ and $2013$ inclusive. Find the minimum value of $n$. | 50 |
How many lattice points are exactly twice as close to $(0,0)$ as they are to $(15,0)$? (A lattice point is a point $(a,b)$ such that both $a$ and $b$ are integers.) | 12 |
A circle contains the points \((0, 11)\) and \((0, -11)\) on its circumference and contains all points \((x, y)\) with \(x^2+y^2<1\) in its interior. Compute the largest possible radius of the circle. | 61 |
There are $100$ lightbulbs $B_1, B_2, \ldots, B_{100}$ spaced evenly around a circle in this order. Additionally, there are $100$ switches $S_1, S_2, \ldots, S_{100}$ such that for all $1 \leq i \leq 100$, switch $S_i$ toggles the states of lights $B_{i-1}$ and $B_{i+1}$ (where $B_{101} = B_1$ and $B_0 = B_{100}$). Sup... | 50 |
A box contains $900$ cards numbered from $100$ to $999$. Paulo randomly takes a certain number of cards from the box and calculates the sum of the digits for each card. How many cards does Paulo need to take out of the box to ensure finding at least three cards with the same digit sum? | 53 |
Let $MATH$ be a trapezoid with $MA = AT = TH = 5$ and $MH = 11$. Point $S$ is the orthocenter of $\triangle ATH$. Compute the area of quadrilateral $MASH$. | 62 |
Given a set $I=\{(x_1,x_2,x_3,x_4) \mid x_i \in \{1,2,\cdots,11\}\}$. $A \subseteq I$, satisfying that for any $(x_1,x_2,x_3,x_4), (y_1,y_2,y_3,y_4) \in A$, there exists $i, j \ (1 \leq i < j \leq 4)$, such that $(x_i - x_j)(y_i - y_j) < 0$. Find the maximum value of $|A|$. | 24 |
Positive integers $m$ and $n$ satisfy the equation $$(m + n)(24mn + 1) = 2023.$$ Find the value of $m + n + 12mn$. | 151 |
Sita and Geeta are two sisters. If Sita's age is written after Geeta's age, a four-digit perfect square number is obtained. If the same exercise is repeated after 13 years, another four-digit perfect square number will be obtained. What is the sum of the present ages of Sita and Geeta? | 55 |
How many positive integers $n$, with $n \leq 2007$, yield a solution for $x$ (where $x$ is real) in the equation $\lfloor x \rfloor + \lfloor 2x \rfloor + \lfloor 3x \rfloor = n$? | 1339 |
There is a sequence defined by the following conditions: - $a(2) = 0$ - $a(3) = 1$ - For $n \geq 4$, $a(n) = a\left(\left\lfloor \frac{n}{2} \right\rfloor \right) + a\left(\left\lceil \frac{n}{2} \right\rceil \right)$ Find the value of $a(2014)$. *Note:* - $\left\lfloor \frac{n}{2} \right\rfloor$ is the floor function,... | 34 |
Find the number of positive integers less than or equal to $2017$ that have at least one pair of adjacent digits that are both even. For example, count the numbers $24$, $1862$, and $2012$, but not $4$, $58$, or $1276$. | 738 |
In the number arrangement: \[ \begin{array}{ccccc} \texttt{1}&&&&\\ \texttt{2}&\texttt{3}&&&\\ \texttt{4}&\texttt{5}&\texttt{6}&&\\ \texttt{7}&\texttt{8}&\texttt{9}&\texttt{10}&\\ \texttt{11}&\texttt{12}&\texttt{13}&\texttt{14}&\texttt{15}\\ \vdots&&&& \end{array} \] What is the number that will appear directly below t... | 2073 |
Find the smallest positive value of $36^k - 5^m$, where $k$ and $m$ are positive integers. | 11 |
For every integer $n \ge 1$, the function $f_n : \{ 0, 1, \cdots, n \} \to \mathbb{R}$ is defined recursively by $f_n(0) = 0$, $f_n(1) = 1$, and \[ (n-k) f_n(k-1) + kf_n(k+1) = nf_n(k) \] for each $1 \le k < n$. Let \[ S_N = f_{N+1}(1) + f_{N+2}(2) + \cdots + f_{2N}(N) \] Find the remainder when $\left\lfloor S_{2013} ... | 26 |
Let $a$, $b$, $c$ be the roots of the polynomial equation $x^3 - 7x^2 - 6x + 5 = 0$. Compute $(a+b)(a+c)(b+c)$. | -37 |
A semicircle with a radius of $5$ and a quarter circle with a radius of $8$ are tangent to each other and are situated inside a square as shown in the figure. Find the length of the part of the common tangent that is enclosed within the same square.  = 28$, where $\sigma(n)$ is the sum of the divisors of $n$, including $n$. | 12 |
A quadrilateral is given where the lengths of two sides are $1$ and $4$. Additionally, a diagonal of length $2$ divides the quadrilateral into two isosceles triangles. Find the perimeter of this quadrilateral. | 11 |
Consider the $10$-digit number $M = 9876543210$. We obtain a new $10$-digit number from $M$ according to the following rule: we can choose one or more disjoint pairs of adjacent digits in $M$ and interchange the digits in these chosen pairs, keeping the remaining digits in their own places. For example, from $M = 9\und... | 88 |
Consider an integer $n \ge 4$ and a sequence of real numbers $x_1, x_2, x_3, \ldots, x_n$. An operation consists of eliminating all numbers not having the rank of the form $4k + 3$, thus leaving only the numbers $x_3, x_7, x_{11}, \ldots$. For example, the sequence $4, 5, 9, 3, 6, 6, 1, 8$ produces the sequence $9, 1$.... | 683 |
Let $O$ be the center of the circumcircle of triangle $ABC$. We know that $AB = 1$ and $AO = AC = 2$. Points $D$ and $E$ lie on extensions of sides $AB$ and $AC$ beyond points $B$ and $C$ respectively, such that $OD = OE$ and $BD = \sqrt{2} \cdot EC$. Find $OD^2$. | 4 |
Determine the greatest natural number $n$ such that for every set $S$ of 2015 different integers, there exist two subsets of $S$ (each subset can have at least one element and the subsets do not need to be disjoint) such that the sum of the elements in each subset is divisible by $n$. | 2015 |
Let triangle $ABC$ have side lengths $AB = 13$, $BC = 14$, $AC = 15$. Let $I$ be the incenter of $\triangle ABC$. The circle centered at $A$ with radius $AI$ intersects the circumcircle of $\triangle ABC$ at points $H$ and $J$. Let $L$ be a point that lies on both the incircle of $\triangle ABC$ and line $HJ$. If the m... | 17 |
Let $ABC$ be a triangle with $\angle BAC = 117^\circ$. The angle bisector of $\angle ABC$ intersects side $AC$ at $D$. Suppose $\triangle ABD \sim \triangle ACB$. Compute the measure of $\angle ABC$, in degrees. | 42 |
In an equilateral triangle $ABC$, the midpoint of $\overline{BC}$ is $M$. If the circumcircle of triangle $MAB$ has an area of $36\pi$, find the perimeter of the triangle. | 36 |
Find how many multiples of 360 are of the form $\overline{ab2017cd}$, where $a$, $b$, $c$, $d$ are digits, with $a > 0$. | 20 |
The area of the region bound by the equations $y = 2\cos(4x)$, $y = \sin(2x) + 10$, $x=0$, and $x = 5\pi$ can be expressed as $x\pi$ for some value of $x$. Evaluate $x$. | 50 |
The equation $$ (x-1)(x-2)\cdots(x-2016) = (x-1)(x-2)\cdots(x-2016) $$ is written on the board, with $2016$ linear factors on each side. What is the least possible value of $k$ for which it is possible to erase exactly $k$ of these $4032$ linear factors so that at least one factor remains on each side and the resulting... | 2016 |
Find the natural number $A$ such that there are $A$ integer solutions to $x+y \geq A$ where $0 \leq x \leq 6$ and $0 \leq y \leq 7$. | 10 |
In triangle $ABC$, point $F$ is on segment $AB$ such that $CF$ bisects $\angle ACB$. Points $D$ and $E$ are on line $CF$ such that lines $AD$ and $BE$ are perpendicular to $CF$. Let $M$ be the midpoint of $AB$. If $ME = 13$, $AD = 15$, and $BE = 25$, find $AC + CB$. | 104 |
Seongcheol has $3$ red shirts and $2$ green shirts. He cannot distinguish between the red shirts or between the green shirts. In how many ways can he arrange them in a row in his closet such that the two green shirts are not adjacent? | 6 |
Let $\overline{abc}$ be a three-digit number with nonzero digits such that $a^2 + b^2 = c^2$. What is the largest possible prime factor of $\overline{abc}$? | 29 |
Let $\overline{CH}$ be an altitude of $\triangle ABC$. Let $R$ and $S$ be the points where the circles inscribed in the triangles $ACH$ and $BCH$ are tangent to $\overline{CH}$. If $AB = 1995$, $AC = 1994$, and $BC = 1993$, then $RS$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Fi... | 997 |
Let $a > 1$ be a positive integer. The sequence of natural numbers $\{a_n\}_{n \geq 1}$ is defined such that $a_1 = a$ and for all $n \geq 1$, $a_{n+1}$ is the largest prime factor of $a_n^2 - 1$. Determine the smallest possible value of $a$ such that the numbers $a_1, a_2, \ldots, a_7$ are all distinct. | 46 |
How many triples \((a,b,c)\) of positive integers strictly less than 51 are there such that \(a+b+c\) is divisible by \(a\), \(b\), and \(c\)? | 221 |
For a positive integer $n$ not divisible by $211$, let $f(n)$ denote the smallest positive integer $k$ such that $n^k - 1$ is divisible by $211$. Find the remainder when $\sum_{n=1}^{210} nf(n)$ is divided by $211$. | 48 |
Hapi, the god of the annual flooding of the Nile, is preparing for this year's flooding. The shape of the channel of the Nile can be described by the function $y = \frac{-1000}{x^2+100}$, where the $x$ and $y$ coordinates are in meters. The current depth of the river is $5$ meters. Hapi plans to increase the water leve... | 40 |
How many positive integers $k$ are there such that $\frac{k}{2013}(a+b) = \text{lcm}(a,b)$ has a solution in positive integers $(a,b)$? | 1006 |
Let $x_1, x_2, \dots, x_{42}$ be real numbers such that $5x_{i+1} - x_i - 3x_ix_{i+1} = 1$ for each $1 \le i \le 42$, with $x_1 = x_{43}$. Find the product of all possible values for $x_1 + x_2 + \dots + x_{42}$. | 588 |
For a positive integer $n$, let $a_n$ be the integer consisting of $n$ digits of $9$ followed by the digits $488$. For example, $a_3 = 999488$ and $a_7 = 9999999488$. Find the value of $n$ such that $a_n$ is divisible by the highest power of $2$. | 6 |
Determine the least possible value of $f(1998)$, where $f:\mathbb{N} \to \mathbb{N}$ is a function such that for all $m,n \in \mathbb{N}$, \[ f\left( n^{2}f(m)\right) = m\left( f(n)\right)^{2}. \] | 120 |
In a $4 \times 4$ table, we place $k$ blocks such that: 1. Each block covers exactly 2 cells. 2. Each cell is covered by at least one block. 3. If we remove a block, there is at least one cell that is not covered. Find the maximum value of $k$. Note: The blocks can overlap. | 12 |
Find the number of ordered pairs of integers $(a, b)$ that satisfy the inequality: \[ 1 < a < b+2 < 10. \] | 28 |
Let $S$ be a subset of $\{1,2,\dots,2017\}$ such that for any two distinct elements in $S$, both their sum and product are not divisible by seven. Compute the maximum number of elements that can be in $S$. | 865 |
Triangle $ABC$ has $AB = BC = 10$ and $CA = 16$. The circle $\Omega$ is drawn with diameter $BC$. $\Omega$ meets $AC$ at points $C$ and $D$. Find the area of triangle $ABD$. | 24 |
Consider trapezoid $ABCD$ with $AB \parallel CD$, where $AB = 5$ and $CD = 9$. Additionally, $\angle C = 15^\circ$ and $\angle D = 75^\circ$. Let $M_1$ be the midpoint of $AB$ and $M_2$ be the midpoint of $CD$. What is the distance $M_1M_2$? | 2 |
The decimal digits of a natural number $A$ form an increasing sequence (from left to right). Find the sum of the digits of $9A$. | 9 |
Maryssa, Stephen, and Cynthia played a game. Each of them independently privately chose one of Rock, Paper, and Scissors at random, with all three choices being equally likely. Given that at least one of them chose Rock and at most one of them chose Paper, the probability that exactly one of them chose Scissors can be ... | 916 |
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