prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
|---|---|
Count the sum of all four-digit positive integers that contain only odd digits in their decimal representation. Note that the digits can only be 1, 3, 5, 7, and 9. | 3471875 |
Triangle $ABC$ is isosceles, and $\angle ABC = x^{\circ}$. If the sum of the possible measures of $\angle BAC$ is $240^{\circ}$, find $x$. | 20 |
13 LHS Students attend the LHS Math Team tryouts. The students are numbered $1, 2, \ldots, 13$. Their scores are $s_1, s_2, \ldots, s_{13}$, respectively. There are 5 problems on the tryout, each of which is given a weight, labeled $w_1, w_2, \ldots, w_5$. Each score $s_i$ is equal to the sum of the weights of all prob... | 78 |
Let $ABC$ be an acute triangle with $\angle ABC = 60^{\circ}$. Suppose points $D$ and $E$ are on lines $AB$ and $CB$, respectively, such that $\triangle CDB$ and $\triangle AEB$ are equilateral triangles. Given that the positive difference between the perimeters of $\triangle CDB$ and $\triangle AEB$ is $60$ and $DE = ... | 1625 |
Let $f : \mathbb{N} \to \mathbb{R}$ be a function, satisfying the following condition: for every integer $n > 1$, there exists a prime divisor $p$ of $n$ such that $f(n) = f \left(\frac{n}{p}\right) - f(p)$. If $f(2^{2007}) + f(3^{2008}) + f(5^{2009}) = 2006$, determine the value of $f(2007^2) + f(2008^3) + f(2009^5)$. | 9 |
Let $ABCD$ be a rectangle. Circles with diameters $AB$ and $CD$ meet at points $P$ and $Q$ inside the rectangle such that $P$ is closer to segment $BC$ than $Q$. Let $M$ and $N$ be the midpoints of segments $AB$ and $CD$. If $\angle MPN = 40^\circ$, find the degree measure of $\angle BPC$. | 160 |
The integer $n$ has exactly six positive divisors, which are $1 < a < b < c < d < n$. Let $k = a - 1$. If the $k$-th divisor (according to the above ordering) of $n$ is equal to $(1 + a + b)b$, find the highest possible value of $n$. | 2009 |
Let $\triangle ABC$ have median $CM$ ($M \in AB$) and circumcenter $O$. The circumcircle of $\triangle AMO$ bisects $CM$. Determine the least possible perimeter of $\triangle ABC$ if it has integer side lengths. | 24 |
For every four-digit number, calculate the product of its four digits. Then, sum all of these products together. What is the result? | 4100625 |
A cubic block with dimensions $n \times n \times n$ is composed of $1 \times 1 \times 1$ unit cubes. Determine the smallest value of $n$ such that, after removing the outer two layers of unit cubes, more than half of the original unit cubes remain. | 20 |
Six distinct positive integers are randomly chosen between $1$ and $2011$, inclusive. The probability that some pair of the six chosen integers has a difference that is a multiple of $5$ is $n$ percent. Find $n$. | 100 |
We define a positive integer as **Noble** if it meets the following criteria: 1. It is a composite number. 2. It is not divisible by any prime number greater than $20$. 3. It is not divisible by any perfect cube greater than $1$. Determine the total number of distinct Noble numbers. | 6552 |
Let $a_1$, $a_2$, \ldots$ be an infinite sequence of integers such that $0 \le a_k \le k$ for every positive integer $k$ and such that \[ 2017 = \sum_{k = 1}^\infty a_k \cdot k! \, . \] What is the value of the infinite series $\sum_{k = 1}^\infty a_k$? | 11 |
In triangle $ABC$, let $M$ be the midpoint of $BC$, $H$ be the orthocenter, and $O$ be the circumcenter. Let $N$ be the reflection of $M$ over $H$. Suppose that $OA = ON = 11$ and $OH = 7$. Compute $BC^2$. | 288 |
In an isosceles triangle $ABC$ with base $BC$, let $M$ be the midpoint of $BC$. Let $P$ be the intersection of the circumcircle of $\triangle ACM$ with the circle centered at $B$ passing through $M$, such that $P \neq M$. If $\angle BPC = 135^\circ$, then $\frac{CP}{AP}$ can be expressed as $a + \sqrt{b}$ for positive ... | 3 |
The three-digit number $n = CCA$ (in base $10$), where $C \neq A$, is divisible by $14$. How many possible values for $n$ are there? | 7 |
Consider the Harmonic Table: \[ \begin{array}{c@{\hspace{15pt}}c@{\hspace{15pt}}c@{\hspace{15pt}}c@{\hspace{15pt}}c@{\hspace{15pt}}c@{\hspace{15pt}}c} &&&1&&&\\ &&\tfrac{1}{2}&&\tfrac{1}{2}&&\\ &\tfrac{1}{3}&&\tfrac{1}{6}&&\tfrac{1}{3}&\\ \tfrac{1}{4}&&\tfrac{1}{12}&&\tfrac{1}{12}&&\tfrac{1}{4}\\ &&&\vdots&&& \end{arra... | 1944 |
For which $p$ prime numbers, there is an integer root of the polynomial $1 + p + Q(x^1) \cdot Q(x^2) \ldots Q(x^{2p - 2})$ such that $Q(x)$ is a polynomial with integer coefficients? | 2 |
Let $ABCD$ be a convex quadrilateral with $\angle DAB = \angle BDC = 90^\circ$. Let the incircles of triangles $ABD$ and $BCD$ touch $BD$ at $P$ and $Q$, respectively, with $P$ lying in between $B$ and $Q$. If $AD = 999$ and $PQ = 200$, then what is the sum of the radii of the incircles of triangles $ABD$ and $BDC$? | 799 |
Compute $\left|\sum_{i=1}^{2022} \sum_{j=1}^{2022} \cos\left(\frac{ij\pi}{2023}\right)\right|.$ | 1011 |
Let $a_n$ be the least positive integer such that the sum of its digits is $n$. Find $a_1 + a_2 + a_3 + \dots + a_{20}$. | 1074 |
Let $2000 < N < 2100$ be an integer. Suppose the last day of year $N$ is a Tuesday while the first day of year $N+2$ is a Friday. The fourth Sunday of year $N+3$ is the $m$th day of January. What is $m$? | 23 |
Suppose $n$ is an integer $\geq 2$. Determine the first digit after the decimal point in the decimal expansion of the number $\sqrt[3]{n^{3}+2n^{2}+n}$. | 6 |
Daniel writes a list of positive integers, each less than or equal to 10, on a board from top to bottom. Next to each number in Daniel's list, Martin writes the number of times that number appears in Daniel's list, creating a new list of the same length. When Martin's list is read from bottom to top, it matches the lis... | 39 |
Let $L, E, T, M, \text{ and } O$ be digits that satisfy the equation $LEET + LMT = TOOL.$ Given that $O$ has the value of $0$, digits may be repeated, and $L \neq 0$, determine the value of the 4-digit integer $ELMO.$ | 1880 |
Find the number of three-digit numbers such that each of the first two digits is divisible by the third digit. | 138 |
Let $ABC$ be a triangle with circumradius $17$, inradius $4$, circumcircle $\Gamma$, and $A$-excircle $\Omega$. Suppose the reflection of $\Omega$ over line $BC$ is internally tangent to $\Gamma$. Compute the area of $\triangle ABC$. | 128 |
$f(x)$ and $g(x)$ are linear functions such that for all $x$, $f(g(x)) = g(f(x)) = x$. If $f(0) = 4$ and $g(5) = 17$, compute $f(2006)$. | 122 |
Justine has two fair dice, one with sides labeled $1, 2, \ldots, m$ and one with sides labeled $1, 2, \ldots, n.$ She rolls both dice once. If $\frac{3}{20}$ is the probability that at least one of the numbers showing is at most 3, find the sum of all distinct possible values of $m+n$. | 996 |
Let $Q$ be a quadratic polynomial. If the sum of the roots of $Q^{100}(x)$ (where $Q^i(x)$ is defined by $Q^1(x)=Q(x)$, $Q^i(x)=Q(Q^{i-1}(x))$ for integers $i \geq 2$) is $8$ and the sum of the roots of $Q$ is $S$, compute $|\log_2(S)|$. | 96 |
Find the positive integer $n$ such that $32$ is the product of the real number solutions of the equation $x^{\log_2(x^3)-n} = 13$. | 15 |
Find the number of prime factors in the factorization of $12!$, where $12! = 12 \cdot 11 \cdot 10 \cdot 9 \cdot 8 \cdot 7 \cdot 6 \cdot 5 \cdot 4 \cdot 3 \cdot 2 \cdot 1$. | 19 |
Eve randomly chooses two **distinct** points on the coordinate plane from the set of all $11^2$ lattice points $(x, y)$ with $0 \le x \le 10$, $0 \le y \le 10$. Then, Anne the ant walks from the point $(0,0)$ to the point $(10, 10)$ using a sequence of one-unit right steps and one-unit up steps. Let $P$ be the number o... | 942 |
For how many positive integers $n \leq 100$ is it true that $10n$ has exactly three times as many positive divisors as $n$ has? | 28 |
Let $X = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ and $A = \{1, 2, 3, 4\}$. Find the number of $4$-element subsets $Y$ of $X$ such that $10 \in Y$ and the intersection of $Y$ and $A$ is not empty. | 74 |
What are the last $3$ digits of $1! + 2! + \cdots + 100!$? | 313 |
Consider the $5 \times 5 \times 5$ equilateral triangular grid as shown: ```asy size(5cm); real n = 5; for (int i = 0; i < n; ++i) { draw((0.5*i,0.866*i)--(n-0.5*i,0.866*i)); } for (int i = 0; i < n; ++i) { draw((n-i,0)--((n-i)/2,(n-i)*0.866)); } for (int i = 0; i < n; ++i) { draw((i,0)--((n+i)/2,(n-i)*0.866)); } ``` E... | 23 |
Let $A_1, A_2, A_3, \ldots , A_{12}$ be the vertices of a regular $12$-gon (dodecagon). Find the number of points in the plane that are equidistant to at least $3$ distinct vertices of this $12$-gon. | 1 |
Let \( p(x) = c_1 + c_2 \cdot 2^x + c_3 \cdot 3^x + c_4 \cdot 5^x + c_5 \cdot 8^x \). Given that \( p(k) = k \) for \( k = 1, 2, 3, 4, 5 \), compute \( p(6) \). | -50 |
Let $BCB'C'$ be a rectangle, let $M$ be the midpoint of $B'C'$, and let $A$ be a point on the circumcircle of the rectangle. Let triangle $ABC$ have orthocenter $H$, and let $T$ be the foot of the perpendicular from $H$ to line $AM$. Suppose that $AM = 2$, $[ABC] = 2020$, and $BC = 10$. Then $AT = \frac{m}{n}$, where $... | 2102 |
Compute the value of the sum: \[ \frac{1}{1 + \tan^3 0^\circ} + \frac{1}{1 + \tan^3 10^\circ} + \frac{1}{1 + \tan^3 20^\circ} + \frac{1}{1 + \tan^3 30^\circ} + \frac{1}{1 + \tan^3 40^\circ} \\ + \frac{1}{1 + \tan^3 50^\circ} + \frac{1}{1 + \tan^3 60^\circ} + \frac{1}{1 + \tan^3 70^\circ} + \frac{1}{1 + \tan^3 80^\circ}... | 5 |
The Tasty Candy Company always puts the same number of pieces of candy into each one-pound bag of candy they sell. Mike bought 4 one-pound bags and gave each person in his class 15 pieces of candy. Mike had 23 pieces of candy left over. Betsy bought 5 one-pound bags and gave 23 pieces of candy to each teacher in her sc... | 302 |
For all integers $0 \leq k \leq 16$, let \[ S_k = \sum_{j=0}^{k} (-1)^j \binom{16}{j}. \] Compute $\max(S_0, S_1, \ldots, S_{16})$. | 6435 |
Find all prime numbers $p$ such that there exist positive integers $q$ and $r$ satisfying the conditions: $p \nmid q$, $3 \nmid q$, and $p^3 = r^3 - q^2$. | 7 |
Find the 2002nd positive integer that is not the difference of two square integers. | 8006 |
Let $a$, $b$, $c$, $d$ be distinct prime numbers such that: - $a + b + c + 18 + d$, - $a + b + c + 18 - d$, - $b + c$, - $c + d$ are also distinct prime numbers. Given that $a + b + c = 2010$, $a$, $b$, $c$, $d \neq 3$, and $d \leq 50$, find the maximum value of the difference between two of these prime numbers. | 2067 |
Alice is standing on the circumference of a large circular room with a radius of $10$. In the center of the room, there is a circular pillar with a radius of $5$ that obstructs Alice's view. The total visible area in the room can be expressed in the form $\frac{m\pi}{n} + p\sqrt{q}$, where $m$ and $n$ are relatively pr... | 206 |
Find the least positive integer $n$ such that the prime factorizations of $n$, $n + 1$, and $n + 2$ each have exactly two factors (as $4$ and $6$ do, but $12$ does not). | 33 |
In a convex quadrilateral $ABCD$, it is given that $\angle BAD = \angle BCD = 90^\circ$ and $BC = CD$. Let $E$ be the intersection of diagonals $\overline{AC}$ and $\overline{BD}$. Given that $\angle AED = 123^\circ$, find the degree measure of $\angle ABD$. | 78 |
In a triangle $ABC$, points $X$ and $Y$ are on segments $AB$ and $AC$, respectively, such that $AX : XB = 1 : 2$ and $AY : YC = 2 : 1$. If the area of triangle $AXY$ is $10$, what is the area of triangle $ABC$? | 45 |
Construct a geometric figure in a sequence of steps. 1. **Step 1:** Begin with a $4 \times 4$ square. 2. **Step 2:** Attach a $1 \times 1$ square onto each side of the original square such that the new squares are on the outside of the original square, have a side along the side of the original square, and the midpoint... | 285 |
A set of 10 distinct integers $S$ is chosen. Let $M$ be the number of nonempty subsets of $S$ whose elements have an even sum. What is the minimum possible value of $M$? | 511 |
Let us call a positive integer $k$ interesting if the product of the first $k$ primes is divisible by $k$. For example, the product of the first two primes is $2 \cdot 3 = 6$, which is divisible by 2. Hence, 2 is an interesting integer. What is the maximal possible number of consecutive interesting integers? | 3 |
Let $m = 2007^{2008}$. How many natural numbers $n$ are there such that $n < m$ and $n(2n + 1)(5n + 2)$ is divisible by $m$ (i.e., $m \mid n(2n + 1)(5n + 2)$)? | 9 |
Find the maximum of the expression: \[ | \ldots | |x_1 - x_2| - x_3| - \ldots | - x_{2023}| \] where \(x_1, x_2, \ldots, x_{2023}\) are distinct natural numbers between \(1\) and \(2023\). | 2022 |
Find the smallest positive integer $n$ such that $\frac{5^{n+1}+2^{n+1}}{5^n+2^n}>4.99$. | 7 |
A tetrahedron $ABCD$ satisfies the following conditions: the edges $AB$, $AC$, and $AD$ are pairwise orthogonal, $AB=3$, and $CD=\sqrt{2}$. Find the minimum possible value of $$BC^6 + BD^6 - AC^6 - AD^6.$$ | 1998 |
Compute the number of ordered pairs of complex numbers \((u, v)\) such that \(uv = 10\) and such that the real and imaginary parts of \(u\) and \(v\) are integers. | 48 |
Let $s(n)$ denote the sum of digits of a positive integer $n$ in base $10$. If $s(m) = 20$ and $s(33m) = 120$, what is the value of $s(3m)$? | 60 |
A grasshopper jumps on the plane from an integer point (point with integer coordinates) to another integer point according to the following rules: His first jump is of length $\sqrt{98}$, his second jump is of length $\sqrt{149}$, his next jump is of length $\sqrt{98}$, and so on, alternatively. What is the least possi... | 9 |
We have eight light bulbs, placed on the eight lattice points (points with integer coordinates) in space that are $\sqrt{3}$ units away from the origin. Each light bulb can either be turned on or off. These lightbulbs are unstable, however. If two light bulbs that are at most 2 units apart are both on simultaneously, t... | 35 |
Find the remainder when \((1^2+1)(2^2+1)(3^2+1)\dots(42^2+1)\) is divided by \(43\). Your answer should be an integer between \(0\) and \(42\). | 4 |
Circles $C_1$ and $C_2$ intersect at points $X$ and $Y$. Point $A$ is a point on $C_1$ such that the tangent line with respect to $C_1$ passing through $A$ intersects $C_2$ at $B$ and $C$, with $A$ closer to $B$ than $C$, such that $2016 \cdot AB = BC$. Line $XY$ intersects line $AC$ at $D$. If circles $C_1$ and $C_2$ ... | 2017 |
How many numbers between $1$ and $1,000,000$ are perfect squares but not perfect cubes? | 990 |
How many gallons of a solution which is $15\%$ alcohol do we have to mix with a solution that is $35\%$ alcohol to make $250$ gallons of a solution that is $21\%$ alcohol? | 175 |
Compute the following expression: \[ \sum_{i=0}^{\frac{q-1}{2}}\left\lfloor\frac{ip}{q}\right\rfloor + \sum_{j=0}^{\frac{p-1}{2}}\left\lfloor\frac{jq}{p}\right\rfloor \] where \( p = 51 \) and \( q = 81 \). | 1001 |
Three circles are tangent to each other, as shown in the figure. The area of the region of the outer circle that is not covered by the two inner circles is $2 \pi$. Determine the length of the segment $PQ$. | 4 |
A $3 \times 6$ grid is filled with the numbers in the list $\{1,1,2,2,3,3,4,4,5,5,6,6,7,7,8,8,9,9\}$ according to the following rules: 1. Both the first three columns and the last three columns contain the integers 1 through 9. 2. No numbers appear more than once in a given row. Let $N$ be the number of ways to fill th... | 13 |
Define a sequence $\{a_{i}\}$ by $a_{1}=3$ and $a_{i+1}=3^{a_{i}}$ for $i \geq 1.$ Which integers between $0$ and $99$ inclusive occur as the last two digits in the decimal expansion of infinitely many $a_{i}$? | 87 |
What is the remainder when $13^{16} + 17^{12}$ is divided by $221$? | 1 |
The perfect square game is played as follows: player 1 says a positive integer, then player 2 says a strictly smaller positive integer, and so on. The game ends when someone says 1; that player wins if and only if the sum of all numbers said is a perfect square. What is the sum of all $n$ such that, if player 1 starts ... | 9 |
Patty is picking peppermints off a tree. They come in two colors: red and white. She picks fewer than $100$ total peppermints but at least one of each color. The white flavor is stronger, so she prefers red to white. Thus, she always picks fewer white peppermints than ten times the number of reds. How many different co... | 4446 |
For a positive integer $n$, let $\omega(n)$ denote the number of positive prime divisors of $n$. Find the smallest positive integer $k$ such that $2^{\omega(n)} \leq k \sqrt[4]{n}$ for all $n \in \mathbb{N}$. | 5 |
Marie is painting a $4 \times 4$ grid of identical square windows. Initially, they are all orange, but she wants to paint $4$ of them black. How many ways can she do this up to rotation and reflection? | 252 |
Let $a$, $b$, $c$ be positive integers and $p$ be a prime number. Assume that \[ a^n(b+c)+b^n(a+c)+c^n(a+b) \equiv 8 \pmod{p} \] for each nonnegative integer $n$. Let $m$ be the remainder when $a^p+b^p+c^p$ is divided by $p$, and $k$ the remainder when $m^p$ is divided by $p^4$. Find the maximum possible value of $k$. | 399 |
The non-negative real numbers $x, y, z$ satisfy the equality $x + y + z = 1$. Determine the highest possible value of the expression $E(x, y, z) = (x + 2y + 3z)(6x + 3y + 2z)$. | 8 |
Find the smallest $x \in \mathbb{N}$ for which $\frac{7x^{25}-10}{83}$ is an integer. | 69 |
Consider the $2015$ integers $n$, from $1$ to $2015$. Determine for how many values of $n$ it is verified that the number $n^3 + 3^n$ is a multiple of $5$. | 403 |
The only prime factors of an integer $n$ are 2 and 3. If the sum of the divisors of $n$ (including itself) is $1815$, find $n$. | 648 |
At a strange party, each person knew exactly $22$ others. 1. For any pair of people $X$ and $Y$ who knew one another, there was no other person at the party that they both knew. 2. For any pair of people $X$ and $Y$ who did not know each other, there were exactly six other people that they both knew. How many people we... | 100 |
How many polynomials of degree exactly $5$ with real coefficients map the set \( \{1, 2, 3, 4, 5, 6\} \) to a permutation of itself? | 718 |
Given a positive integer $n$, let $s(n)$ denote the sum of the digits of $n$. Compute the largest positive integer $n$ such that: $$n = s(n)^2 + 2s(n) - 2.$$ | 397 |
Consider all positive integers written in a row: $123456789101112131415\ldots$ Find the $206788$-th digit from the left. | 7 |
Let $f(x) = 2^x + 3^x$. For how many integers $1 \leq n \leq 2020$ is $f(n)$ relatively prime to all of $f(0), f(1), \dots, f(n-1)$? | 11 |
Eight football teams play matches against each other such that no two teams meet twice and no three teams play all three possible matches. What is the largest possible number of matches? | 16 |
Compute the smallest positive integer $a$ for which $$\sqrt{a + \sqrt{a + \ldots}} - \frac{1}{a + \frac{1}{a+\ldots}} > 7$$ | 43 |
For a positive integer $n$, let $s(n)$ be the sum of the digits of $n$. If $n$ is a two-digit positive integer such that $\frac{n}{s(n)}$ is a multiple of $3$, compute the sum of all possible values of $n$. | 162 |
The number $16^4 + 16^2 + 1$ is divisible by four distinct prime numbers. Compute the sum of these four primes. | 264 |
Let $\triangle ABC$ be a triangle with circumcenter $O$ and orthocenter $H$. Let $D$ be a point on the circumcircle of $ABC$ such that $AD \perp BC$. Suppose that $AB = 6$, $DB = 2$, and the ratio $\frac{\text{area}(\triangle ABC)}{\text{area}(\triangle HBC)}=5$. Then, if $OA$ is the length of the circumradius, then $O... | 29 |
Find $k$ such that $k\pi$ is the area of the region of points in the plane satisfying $$\frac{x^2+y^2+1}{11} \le x \le \frac{x^2+y^2+1}{7}.$$ | 18 |
Evaluate the integer part of the number \[ H = \sqrt{1 + 2011^2 + \frac{2011^2}{2012^2}} + \frac{2011}{2012}. \] | 2012 |
We consider positive integers $n$ having at least six positive divisors. Let the positive divisors of $n$ be arranged in a sequence $(d_i)_{1\le i\le k}$ with \[ 1 = d_1 < d_2 < \dots < d_k = n \quad (k \ge 6). \] Find all positive integers $n$ such that \[ n = d_5^2 + d_6^2. \] | 500 |
David has a collection of 40 rocks, 30 stones, 20 minerals, and 10 gemstones. An operation consists of removing three objects, no two of the same type. What is the maximum number of operations he can possibly perform? | 30 |
Find the number of lattice points that the line $19x + 20y = 1909$ passes through in Quadrant I. | 5 |
Find all prime numbers \( p \) such that \( p^2 \) can be written as the sum of cubes of two positive integers, i.e., \( p^2 = a^3 + b^3 \) where \( a \) and \( b \) are positive integers. | 3 |
64 numbers (not necessarily distinct) are placed on the squares of a chessboard such that the sum of the numbers in every $2\times2$ square is 7. What is the sum of the four numbers in the corners of the board? | 7 |
A subset $S$ of the set of integers from 0 to 99 is said to have property $A$ if it is impossible to fill a crossword puzzle with 2 rows and 2 columns using numbers from $S$ (where 0 is written as 00, 1 as 01, and so on). Determine the maximal number of elements in the set $S$ that has property $A$. | 25 |
Let $ABCD$ be a square with side length $4$. Points $E$ and $F$ are chosen on sides $BC$ and $DA$, respectively, such that $EF = 5$. Find the sum of the minimum and maximum possible areas of trapezoid $BEDF$. | 16 |
Which digit must be substituted instead of the star so that the following large number \( \underbrace{66\ldots66}_{2023} \star \underbrace{55\ldots55}_{2023} \) is divisible by \( 7 \)? | 6 |
Let $N$ be a 3-digit number with three distinct non-zero digits. We say that $N$ is mediocre if it has the property that when all six 3-digit permutations of $N$ are written down, the average is $N$. For example, $N = 481$ is mediocre, since it is the average of \{418, 481, 148, 184, 814, 841\}. Determine the largest m... | 629 |
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