prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
|---|---|
Let $p$, $q$, and $r$ be the three roots of the polynomial $x^3 - 2x^2 + 3x - 2023$. Suppose that the polynomial $x^3 + Bx^2 + Mx + T$ has roots $p + q$, $p + r$, and $q + r$ for real numbers $B$, $M$, and $T$. Compute $B - M + T$. | 2006 |
In triangle $ABC$, the angle $C$ is obtuse, $m(\angle A) = 2m(\angle B)$, and the side lengths are integers. Find the smallest possible perimeter of this triangle. | 77 |
Wendy randomly chooses a positive integer less than or equal to $2020$. The probability that the digits in Wendy's number add up to $10$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 107 |
Five people are gathered in a meeting. Some pairs of people shake hands. An ordered triple of people $(A,B,C)$ is called a *trio* if one of the following conditions is true: 1. $A$ shakes hands with $B$, and $B$ shakes hands with $C$, or 2. $A$ does not shake hands with $B$, and $B$ does not shake hands with $C$. If we... | 10 |
Let $p_1 = 2$, $p_2 = 3$, $p_3 = 5$, ... be the sequence of prime numbers. Find the least positive even integer $n$ such that $p_1 + p_2 + p_3 + \ldots + p_n$ is not a prime number. | 8 |
Milo rolls five fair dice, which have 4, 6, 8, 12, and 20 sides respectively, and each one is labeled $1$ to $n$ for the appropriate $n$. How many distinct ways can they roll a full house (three of one number and two of another)? The same numbers appearing on different dice are considered distinct full houses, so $(1,1... | 248 |
In a given sequence $\{S_1,S_2,...,S_k\}$, for terms $n\ge3$, the relationship is defined as: $$ S_n=\sum_{i=1}^{n-1} i\cdot S_{n-i}. $$ For example, if the first two elements are 2 and 3, respectively, the third entry would be $1\cdot3+2\cdot2=7$, and the fourth would be $1\cdot7+2\cdot3+3\cdot2=19$, and so on. Given ... | -2 |
Dexter's Laboratory has $2024$ robots, each with a program set up by Dexter. One day, his naughty sister Dee Dee intrudes and writes an integer in the set $\{1, 2, \dots, 113\}$ on each robot's forehead. Each robot detects the numbers on all other robots' foreheads and guesses its own number based on its program, indiv... | 17 |
Let $k$ be a real number such that the product of real roots of the equation $$X^4 + 2X^3 + (2 + 2k)X^2 + (1 + 2k)X + 2k = 0$$ is $-2013$. Find the sum of the squares of these real roots. | 4027 |
The sequence $x_1, x_2, x_3, \ldots$ is defined by $x_1 = 2022$ and $x_{n+1} = 7x_n + 5$ for all positive integers $n$. Determine the maximum positive integer $m$ such that \[ \frac{x_n(x_n - 1)(x_n - 2) \cdots (x_n - m + 1)}{m!} \] is never a multiple of $7$ for any positive integer $n$. | 404 |
How many lattice points $(v, w, x, y, z)$ does a 5-sphere centered on the origin, with radius 3, contain on its surface or in its interior? | 1343 |
The expression below has six empty boxes. Each box is to be filled in with a number from $1$ to $6$, where all six numbers are used exactly once, and then the expression is evaluated. What is the maximum possible final result that can be achieved? $$\frac{\frac{\square}{\square} + \frac{\square}{\square}}{\frac{\square... | 23 |
Consider the sequence $2, 3, 5, 6, 7, 8, 10, \ldots$ of all positive integers that are not perfect squares. Determine the $2011^{\text{th}}$ term of the sequence. | 2056 |
If $S$ is a set consisting of $12$ elements, what is the maximum number of pairs $(a, b)$ such that $a, b \in S$ and $\frac{b}{a}$ is a prime number? | 20 |
Real numbers $x$, $y$, and $z$ satisfy the following equality: \[4(x+y+z)=x^2+y^2+z^2\] Let $M$ be the maximum of $xy+yz+zx$, and let $m$ be the minimum of $xy+yz+zx$. Find $M+10m$. | 28 |
Let $N = 12!$ and denote by $X$ the set of positive divisors of $N$ other than $1$. A pseudo-ultrafilter $U$ is a nonempty subset of $X$ such that for any $a,b \in X$: \begin{itemize} \item If $a$ divides $b$ and $a \in U$ then $b \in U$. \item If $a,b \in U$ then $\gcd(a,b) \in U$. \item If $a,b \notin U$ then $\opera... | 19 |
Prinstan Trollner and Dukejukem are competing at the game show WASS. Both players spin a wheel which chooses an integer from $1$ to $50$ uniformly at random, and this number becomes their score. Dukejukem then flips a weighted coin that lands heads with probability $\frac{3}{5}$. If he flips heads, he adds $1$ to his s... | 751 |
For all positive integers $k$, define $f(k) = k^2 + k + 1$. Compute the largest positive integer $n$ such that: \[ 2015 \cdot f(1^2) \cdot f(2^2) \cdots f(n^2) \geq \left(f(1) \cdot f(2) \cdots f(n)\right)^2. \] | 44 |
Find the largest positive integer $n \le 2004$ such that $3^{3n+3} - 27$ is divisible by $169$. | 2002 |
Find the greatest possible value of $pq + r$, where $p$, $q$, and $r$ are (not necessarily distinct) prime numbers satisfying $pq + qr + rp = 2016$. | 1008 |
Find the number of nondegenerate triangles whose vertices lie in the set of points $(s, t)$ in the plane such that $0 \leq s \leq 4$, $0 \leq t \leq 4$, where $s$ and $t$ are integers. | 2148 |
Bradley is driving at a constant speed. When he passes his school, he notices that in $20$ minutes he will be exactly $\frac{1}{4}$ of the way to his destination, and in $45$ minutes he will be exactly $\frac{1}{3}$ of the way to his destination. Find the number of minutes it takes Bradley to reach his destination from... | 245 |
Five members of the Lexington Math Team are sitting around a table. Each flips a fair coin. Given that the probability that three consecutive members flip heads is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$. | 43 |
Let \( C = \{ (i,j) | i,j \text{ integers such that } 0 \leq i,j \leq 24 \} \). How many squares can be formed in the plane such that all of their vertices are in \( C \) and their sides are parallel to the \( X \)-axis and \( Y \)-axis? | 4900 |
Given a regular hexagon $ABCDEF$, let point $P$ be the intersection of lines $BC$ and $DE$, and let point $Q$ be the intersection of lines $AP$ and $CD$. If the area of $\triangle QEP$ is equal to $72$, find the area of regular hexagon $ABCDEF$. | 324 |
The Matini company released a special album with the flags of the $12$ countries that compete in the CONCACAM Mathematics Cup. Each postcard envelope contains two flags chosen randomly. Determine the minimum number of envelopes that need to be opened so that the probability of having a repeated flag is $50\%$. | 3 |
Compute the remainder when the largest integer below $\frac{3^{123}}{5}$ is divided by $16$. | 5 |
Say a positive integer $n$ is radioactive if one of its prime factors is strictly greater than $\sqrt{n}$. For example, $2012 = 2^2 \cdot 503$, $2013 = 3 \cdot 11 \cdot 61$, and $2014 = 2 \cdot 19 \cdot 53$ are all radioactive, but $2015 = 5 \cdot 13 \cdot 31$ is not. How many radioactive numbers have all prime factors... | 119 |
For a positive integer $n$, let $f(n)$ be the sum of the positive integers that divide at least one of the nonzero base $10$ digits of $n$. For example, $f(96) = 1 + 2 + 3 + 6 + 9 = 21$. Find the largest positive integer $n$ such that for all positive integers $k$, there is some positive integer $a$ such that $f^k(a) =... | 15 |
Determine the smallest natural number written in the decimal system such that the product of its digits equals $10! = 1 \times 2 \times 3 \times \ldots \times 9 \times 10$. | 45578899 |
The diagram shows a regular pentagon $ABCDE$ and a square $ABFG$. Find the degree measure of $\angle FAD$. | 18 |
Let $ABC$ be a scalene triangle whose side lengths are positive integers. It is called stable if its three side lengths are multiples of 5, 80, and 112, respectively. What is the smallest possible side length that can appear in any stable triangle? | 20 |
Let $A$ and $B$ be two finite sets such that there are exactly $144$ sets which are subsets of $A$ or subsets of $B$. Find the number of elements in $A \cup B$. | 8 |
On equilateral $\triangle{ABC}$, point D lies on BC a distance 1 from B, point E lies on AC a distance 1 from C, and point F lies on AB a distance 1 from A. Segments AD, BE, and CF intersect in pairs at points G, H, and J, which are the vertices of another equilateral triangle. The area of $\triangle{ABC}$ is twice the... | 30 |
Let $\sigma(n)$ represent the number of positive divisors of $n$. Define $s(n)$ as the number of positive divisors of $n+1$ such that for every divisor $a$, $a-1$ is also a divisor of $n$. Find the maximum value of $2s(n) - \sigma(n)$. | 2 |
The sequence of integers $a_1, a_2, \dots$ is defined as follows: - $a_1 = 1$. - For $n > 1$, $a_{n+1}$ is the smallest integer greater than $a_n$ such that $a_i + a_j \neq 3a_k$ for any $i, j, k$ from $\{1, 2, \dots, n+1\}$, where $i, j, k$ are not necessarily different. Define $a_{2004}$. | 4507 |
For $n$ measured in degrees, let $T(n) = \cos^2(30^\circ -n) - \cos(30^\circ -n)\cos(30^\circ +n) + \cos^2(30^\circ +n)$. Evaluate $4\sum^{30}_{n=1} n \cdot T(n).$ | 1395 |
Let $P(x)$ be a polynomial with positive integer coefficients and $P(0) > 1$. The product of all the coefficients is $47$ and the remainder when $P(x)$ is divided by $(x - 3)$ is $9887$. What is the degree of $P(x)$? | 8 |
Let $a_1$, $a_2$, $a_3$, $a_4$, and $a_5$ be real numbers satisfying \begin{align*} a_1a_2 + a_2a_3 + a_3a_4 + a_4a_5 + a_5a_1 & = 20,\\ a_1a_3 + a_2a_4 + a_3a_5 + a_4a_1 + a_5a_2 & = 22. \end{align*} Then the smallest possible value of $a_1^2 + a_2^2 + a_3^2 + a_4^2 + a_5^2$ can be expressed as $m + \sqrt{n}$, where $... | 2105 |
Find the area of the $MNRK$ trapezoid with the lateral side $RK = 3$ if the distances from the vertices $M$ and $N$ to the line $RK$ are $5$ and $7$, respectively. | 18 |
Find the least positive integer $n$ such that the polynomial $P(X) = \sqrt{3} \cdot X^{n+1} - X^n - 1$ has at least one root of modulus $1$. | 10 |
Determine the minimal number of operations needed to calculate $x^{2006}$ using only multiplication and division of any two given or already obtained powers of the number $x$. For example, $x \cdot x = x^{2}$, $x^{2} \cdot x^{2} = x^{4}$, $x^{4} : x = x^{3}$, etc. | 14 |
Let set $S$ contain all positive integers that are one less than a perfect square. Find the sum of all powers of $2$ that can be expressed as the product of two (not necessarily distinct) members of $S$. | 64 |
Let $f(x,y)$ be a function defined for all pairs of nonnegative integers $(x, y)$, such that $f(0,k) = f(k,0) = 2^k$ and \[ f(a,b) + f(a+1,b+1) = f(a+1,b) + f(a,b+1) \] for all nonnegative integers $a, b$. Determine the number of positive integers $n \leq 2016$ for which there exist two nonnegative integers $a, b$ such... | 65 |
Suppose that the following equations hold for positive integers $x$, $y$, and $n$, where $n > 18$: \[ \begin{align*} x + 3y & \equiv 7 \pmod{n} \\ 2x + 2y & \equiv 18 \pmod{n} \\ 3x + y & \equiv 7 \pmod{n} \end{align*} \] Compute the smallest nonnegative integer $a$ such that $2x \equiv a \pmod{n}$. | 20 |
Let $\triangle ABC$ be a triangle with $AB = 4$, $BC = 6$, and $CA = 5$. Let the angle bisector of $\angle BAC$ intersect $BC$ at the point $D$ and the circumcircle of $\triangle ABC$ again at the point $M \neq A$. The perpendicular bisector of segment $DM$ intersects the circle centered at $M$ passing through $B$ at t... | 36 |
Let $a$ and $b$ be positive integers such that $1 \leq a < b \leq 100$. If there exists a positive integer $k$ such that $ab \mid a^k + b^k$, we say that the pair $(a, b)$ is good. Determine the number of good pairs. | 96 |
Let $f$ be a function from the set $X = \{1,2, \dots, 10\}$ to itself. Call a partition $(S, T)$ of $X$ $f$-balanced if for all $s \in S$, we have $f(s) \in T$, and for all $t \in T$, we have $f(t) \in S$. A partition $(S, T)$ is a pair of subsets $S$ and $T$ of $X$ such that $S \cap T = \emptyset$ and $S \cup T = X$. ... | 961 |
Find the least natural number $n$ such that, if the set $\{1,2, \ldots, n\}$ is arbitrarily divided into two non-intersecting subsets, then one of the subsets contains 3 distinct numbers such that the product of two of them equals the third. | 96 |
Let $m$ and $n$ be positive integers such that $m^4 - n^4 = 3439$. What is the value of $mn$? | 90 |
In a triangle $ABC$, $\angle BAC = 90^{\circ}$. Let $D$ be the point on $BC$ such that $AB + BD = AC + CD$. Suppose $BD : DC = 2:1$. If $\frac{AC}{AB} = \frac{m + \sqrt{p}}{n}$, where $m, n$ are relatively prime positive integers and $p$ is a prime number, determine the value of $m+n+p$. | 34 |
How many sequences of real numbers $a_1, a_2, \ldots, a_9$ satisfy \[ |a_1 - 1| = |a_2 - a_1| = \cdots = |a_9 - a_8| = |1 - a_9| = 1? \] | 252 |
Find the number of 5-digit numbers that each contains the block '15' and is divisible by 15. | 479 |
Let $p=2017$ be a prime. Find the remainder when \[ \left\lfloor \frac{1^p}{p} \right\rfloor + \left\lfloor \frac{2^p}{p} \right\rfloor + \left\lfloor \frac{3^p}{p} \right\rfloor + \cdots + \left\lfloor \frac{2015^p}{p} \right\rfloor \] is divided by $p$. Here $\lfloor \cdot \rfloor$ denotes the greatest integer functi... | 1010 |
Start with a three-digit positive integer $A$. Obtain $B$ by interchanging the two leftmost digits of $A$. Obtain $C$ by doubling $B$. Obtain $D$ by subtracting $500$ from $C$. Given that $A + B + C + D = 2014$, find $A$. | 344 |
Five students take a test where any integer score from $0$ to $100$ is possible. What is the largest possible difference between the median and the mean of the scores? | 40 |
Let $C$ denote the curve $y^2 = \frac{x(x+1)(2x+1)}{6}$. The points $(\frac{1}{2}, a)$, $(b, c)$, and $(24, d)$ lie on $C$ and are collinear, and $ad < 0$. Given that $b$ and $c$ are rational numbers, find $100b^2 + c^2$. | 101 |
Leticia has a $9 \times 9$ board. She says that two squares are "friends" if they share a side, if they are at opposite ends of the same row, or if they are at opposite ends of the same column. Every square has $4$ friends on the board. Leticia will paint every square one of three colors: green, blue, or red. In each s... | 486 |
Let $S$ be the set of all positive integers from 1 through 1000 that are not perfect squares. What is the length of the longest, non-constant, arithmetic sequence that consists of elements of $S$? | 333 |
Find the area of the figure on the coordinate plane bounded by the straight lines $x = 0$, $x = 2$, and the graphs of the functions $y = \sqrt{x^3 + 1}$ and $y = -\sqrt[3]{x^2 + 2x}$. | 6 |
How many ordered pairs of integers $(x, y)$ satisfy the equation: $$2x^2 + y^2 + xy = 2(x + y)$$ | 4 |
Compute the smallest positive integer $n$ such that $2016^n$ does not divide $2016!$. | 335 |
Find the sum of all three-digit numbers (written in base $10$) such that the leading digit is the sum of the other two digits. Express your answer in base $10$. | 34815 |
Find the number of positive integers $n$ such that the highest power of $7$ dividing $n!$ is $8$. | 7 |
Let $a$ and $b$ be nonzero real numbers such that $\frac{1}{3a} + \frac{1}{b} = 2011$ and $\frac{1}{a} + \frac{1}{3b} = 1$. What is the quotient when $a + b$ is divided by $ab$? | 1509 |
Consider the function $f(x) = 5x^4 - 12x^3 + 30x^2 - 12x + 5$. Let $f(x_1) = p$, where $x_1$ and $p$ are non-negative integers, and $p$ is prime. Find with proof the largest possible value of $p$. | 5 |
Find the largest positive integer $n$ not divisible by $10$ which is a multiple of each of the numbers obtained by deleting two consecutive digits (neither of them in the first or last position) of $n$. | 9999 |
Ryan chooses five subsets $S_1, S_2, S_3, S_4, S_5$ of the set $\{1, 2, 3, 4, 5, 6, 7\}$ such that $|S_1| = 1$, $|S_2| = 2$, $|S_3| = 3$, $|S_4| = 4$, and $|S_5| = 5$. Moreover, for all $1 \le i < j \le 5$, either $S_i \cap S_j = S_i$ or $S_i \cap S_j = \emptyset$ (in other words, the intersection of $S_i$ and $S_j$ is... | 11760 |
There is a positive integer $s$ such that there are $s$ solutions to the equation \[ 64\sin^2(2x) + \tan^2(x) + \cot^2(x) = 46 \] in the interval \( (0, \frac{\pi}{2}) \), all of the form \( \frac{m_k}{n_k}\pi \), where $m_k$ and $n_k$ are relatively prime positive integers, for $k = 1, 2, 3, \ldots, s$. Find \((m_1 + ... | 100 |
Let $f(x)$ be a continuous function over the real numbers such that for every integer $n$, $f(n) = n^2$ and $f(x)$ is linear over the interval $[n, n + 1]$. There exists a unique two-variable polynomial $g$ such that $g(x, \lfloor x \rfloor) = f(x)$ for all $x$. Compute $g(20, 23)$. Here, $\lfloor x \rfloor$ is defined... | 388 |
Find the greatest positive integer $n$ such that $n$ is divisible by all positive integers whose cube is not greater than $n$. | 420 |
Consider all pairs of points $(a,b,c)$ and $(d,e,f)$ in the 3-D coordinate system with the condition $ad + be + cf = -2023$. What is the least positive integer that can be the distance between such a pair of points? | 90 |
Find the smallest integer $n \geq 1$ such that the equation: $$a^2 + b^2 + c^2 - nd^2 = 0$$ has $(0, 0, 0, 0)$ as the unique solution. | 7 |
Three people, John, Macky, and Rik, play a game of passing a basketball from one to another. Find the number of ways of passing the ball starting with Macky and reaching Macky again at the end of the seventh pass. | 42 |
Let $P(x)$ be a real quadratic trinomial such that for all $x \in \mathbb{R}$, the inequality $P(x^3 + x) \geq P(x^2 + 1)$ holds. Find the sum of the roots of $P(x)$. | 4 |
As in the following diagram, square $ABCD$ and square $CEFG$ are placed side by side (i.e. $C$ is between $B$ and $E$ and $G$ is between $C$ and $D$). If $CE = 14$, $AB > 14$, compute the minimal area of $\triangle AEG$. ```asy size(120); defaultpen(linewidth(0.7)+fontsize(10)); pair D2(real x, real y) { pair P = (x,y)... | 98 |
Find the smallest possible value of $n$ such that $n+2$ people can stand inside or on the border of a regular $n$-gon with side length $6$ feet where each pair of people are at least $6$ feet apart. | 10 |
The sequence of nonnegative integers $F_0, F_1, F_2, \dots$ is defined recursively as $F_0 = 0$, $F_1 = 1$, and $F_{n+2} = F_{n+1} + F_n$ for all integers $n \geq 0$. Let $d$ be the largest positive integer such that, for all integers $n \geq 0$, $d$ divides $F_{n+2020} - F_n$. Compute the remainder when $d$ is divided... | 638 |
How many ordered pairs of integers \((a,b)\) satisfy all of the following inequalities? \[ \begin{align*} & a^2 + b^2 < 16 \\ & a^2 + b^2 < 8a \\ & a^2 + b^2 < 8b \end{align*} \] | 6 |
In triangle $ABC$, $AB = 100$, $BC = 120$, and $CA = 140$. Points $D$ and $F$ lie on $\overline{BC}$ and $\overline{AB}$, respectively, such that $BD = 90$ and $AF = 60$. Point $E$ is an arbitrary point on $\overline{AC}$. Denote the intersection of $\overline{BE}$ and $\overline{CF}$ as $K$, the intersection of $\over... | 91 |
The function $f(n)$ is defined on the positive integers and takes non-negative integer values. It is given that $f(2) = 0$, $f(3) > 0$, and $f(9999) = 3333$. For all positive integers $m$ and $n$, the following condition holds: \[ f(m+n) - f(m) - f(n) = 0 \text{ or } 1. \] Determine the value of $f(1982)$. | 660 |
The \textit{arithmetic derivative} $D(n)$ of a positive integer $n$ is defined via the following rules: \begin{itemize} \item $D(1) = 0$; \item $D(p) = 1$ for all primes $p$; \item $D(ab) = D(a)b + aD(b)$ for all positive integers $a$ and $b$. \end{itemize} Find the sum of all positive integers $n$ below $1000$ satisfy... | 31 |
Let $f(n)$ denote the sum of the distinct positive integer divisors of $n$. Evaluate: \[ f(1) + f(2) + f(3) + f(4) + f(5) + f(6) + f(7) + f(8) + f(9). \] | 69 |
Let $ABC$ be a triangle with $AB = 20$ and $AC = 18$. $E$ is on segment $AC$ and $F$ is on segment $AB$ such that $AE = AF = 8$. Let $BE$ and $CF$ intersect at $G$. Given that $AEGF$ is cyclic, then $BC = m\sqrt{n}$ for positive integers $m$ and $n$ such that $n$ is not divisible by the square of any prime. Compute $10... | 305 |
After Jennifer walked $r$ percent of the way from her home to the store, she turned around and walked home, got on her bicycle, and bicycled to the store and back home. Jennifer bicycles two and a half times faster than she walks. Find the largest value of $r$ so that returning home for her bicycle was not slower than ... | 60 |
There are $n$ sticks which have distinct integer lengths. Suppose that it's possible to form a non-degenerate triangle from any $3$ distinct sticks among them. It's also known that there are sticks of lengths $5$ and $12$ among them. What is the largest possible value of $n$ under such conditions? | 6 |
If Percy rolls a fair six-sided die until he rolls a $5$, what is his expected number of rolls, given that all of his rolls are prime? | 3 |
Consider a triangle $ABC$ and let $M$ be the midpoint of the side $BC$. Suppose $\angle MAC = \angle ABC$ and $\angle BAM = 105^{\circ}$. Find the measure of $\angle ABC$. | 30 |
What is the maximum number of distinct integers in a row such that the sum of any 11 consecutive integers is either 100 or 101? | 22 |
Let $\triangle ABC$ have $\angle ABC=67^{\circ}$. Point $X$ is chosen such that $AB = XC$, $\angle XAC=32^\circ$, and $\angle XCA=35^\circ$. Compute $\angle BAC$ in degrees. | 81 |
Start by writing the integers $1, 2, 4, 6$ on the blackboard. At each step, write the smallest positive integer $n$ that satisfies both of the following properties on the board: 1. $n$ is larger than any integer currently on the board. 2. $n$ cannot be written as the sum of two distinct integers already on the board. F... | 388 |
Compute the number of nonnegative integer triples \((x, y, z)\) which satisfy the inequality: \[ 4x + 2y + z \leq 36 \] | 1330 |
How many pairs of positive integers $(a, b)$ satisfy $\frac{1}{a} + \frac{1}{b} = \frac{1}{2004}$? | 45 |
In parallelogram $ABCD$, the longer side is twice the shorter side. Let $XYZW$ be the quadrilateral formed by the internal bisectors of the angles of $ABCD$. If the area of $XYZW$ is $10$, find the area of $ABCD$. | 40 |
Concentric circles $\Omega_1$ and $\Omega_2$ with radii $1$ and $100$, respectively, are drawn with center $O$. Points $A$ and $B$ are chosen independently at random on the circumferences of $\Omega_1$ and $\Omega_2$, respectively. Denote by $\ell$ the tangent line to $\Omega_1$ passing through $A$, and denote by $P$ t... | 10004 |
Find the three-digit number that has the greatest number of different divisors. | 840 |
The corners of a $2$-dimensional room in the shape of an isosceles right triangle are labeled $A$, $B$, $C$ where $AB = BC$. Walls $BC$ and $CA$ are mirrors. A laser is shot from $A$, hits off each of the mirrors once, and lands at a point $X$ on $AB$. Let $Y$ be the point where the laser hits off $AC$. If $\frac{AB}{A... | 131 |
Let $S_0 = \varnothing$ denote the empty set, and define $S_n = \{ S_0, S_1, \dots, S_{n-1} \}$ for every positive integer $n$. Find the number of elements in the set \[ (S_{10} \cap S_{20}) \cup (S_{30} \cap S_{40}). \] | 30 |
What is the largest positive integer $n < 1000$ for which there is a positive integer $m$ satisfying \[ \text{lcm}(m,n) = 3m \times \gcd(m,n)? \] | 972 |
Three rods of lengths $1396$, $1439$, and $2018$ millimeters have been hinged from one tip on the ground. What is the smallest value for the radius of the circle passing through the other three tips of the rods in millimeters? | 311 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.