prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
|---|---|
Determine the smallest non-negative integer $n$ such that \( \sqrt{(6n+11)(6n+14)(20n+19)} \in \mathbb{Q} \). | 219 |
Let $ABCD$ be a parallelogram. Let $G$ and $H$ be the feet of the altitudes from $A$ to $CD$ and $BC$, respectively. If $AD = 15$, $AG = 12$, and $AH = 16$, find the length of $AB$. | 20 |
Justin the robot is on a mission to rescue abandoned treasure from a minefield. To do this, he must travel from the point $(0, 0, 0)$ to $(4, 4, 4)$ in three-dimensional space, only taking one-unit steps in the positive $x$, $y$, or $z$-directions. However, the evil David anticipated Justin's arrival and has placed a m... | 10350 |
What is the maximum number of subsets of size $5$, taken from the set $A = \{1, 2, 3, \ldots, 20\}$ such that any two of them share exactly one element? | 16 |
Let $A$ be the greatest possible value of a product of positive integers that sums to $2014$. Compute the sum of all bases and exponents in the prime factorization of $A$. For example, if $A = 7 \cdot 11^5$, the answer would be $7 + 11 + 5 = 23$. | 677 |
A sequence of distinct circles $\omega_1, \omega_2, \cdots$ is inscribed in the parabola $y=x^2$ such that $\omega_n$ and $\omega_{n+1}$ are tangent for all $n$. If $\omega_1$ has a diameter of 1 and touches the parabola at $(0,0)$, find the diameter of $\omega_{1998}$. | 3995 |
A snowman is built on a level plane by placing a ball of radius $6$ on top of a ball of radius $8$, which is on top of a ball of radius $10$, as shown. If the average height above the plane of a point in the snowman is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m + n$. | 61 |
$DEB$ is a chord of a circle such that $DE=3$ and $EB=5$. Let $O$ be the center of the circle. Join $OE$ and extend $OE$ to cut the circle at $C$. Given $EC=1$, find the radius of the circle. | 8 |
In the middle of the school year, $40\%$ of Poolesville magnet students decided to transfer to the Blair magnet, and $5\%$ of the original Blair magnet students transferred to the Poolesville magnet. If the Blair magnet grew from $400$ students to $480$ students, how many students does the Poolesville magnet have after... | 170 |
For $n$ a positive integer, define $f_1(n) = n$. Then for $i$ a positive integer, define $f_{i+1}(n) = f_i(n)^{f_i(n)}$. Determine $f_{100}(75) \pmod{17}$. Justify your answer. | 1 |
There are $4$ pairs of men and women, and all $8$ people are arranged in a row such that in each pair, the woman is somewhere to the left of the man. How many such arrangements are there? | 2520 |
How many ways are there to color the $8$ regions of a three-set Venn Diagram with $3$ colors such that each color is used at least once? Two colorings are considered the same if one can be reached from the other by rotation and/or reflection. | 1248 |
The Dinky is a train connecting Princeton to the outside world. It runs on an odd schedule: the train arrives once every one-hour block at some uniformly random time (once at a random time between $9\text{am}$ and $10\text{am}$, once at a random time between $10\text{am}$ and $11\text{am}$, and so on). One day, Emilia ... | 525 |
We delete the four corners of an $8 \times 8$ chessboard. How many ways are there to place eight non-attacking rooks on the remaining squares? | 21600 |
Let $A_1A_2A_3\ldots A_{12}$ be a regular dodecagon. Find the number of right triangles whose vertices are in the set $\{A_1, A_2, A_3, \ldots, A_{12}\}$. | 60 |
What is the sum of the prime factors of \(20!\)? | 77 |
Determine the sixth digit after the decimal point in the number $(\sqrt{1978} + \lfloor \sqrt{1978} \rfloor)^{20}$. | 9 |
Distinct integers $x, y, z$ satisfy the equation $(x-y)(y-z)(z-x) = x+y+z$. Determine the smallest possible value of $|x+y+z|$. | 54 |
Let the complex number $z = \cos\frac{1}{1000} + i \sin\frac{1}{1000}$. Find the smallest positive integer $n$ such that $z^n$ has an imaginary part that exceeds $\frac{1}{2}$. | 524 |
Find all prime numbers $p$ for which there exists an integer $n$ such that there are no integers $x, y$ satisfying $x^3 + y^3 \equiv n \pmod{p}$ (i.e., not all residues are the sum of two cubes). For example, for $p = 7$, one could set $n = \pm 3$ since $x^3, y^3 \equiv 0, \pm 1 \pmod{7}$, thus $x^3 + y^3 \equiv 0, \pm... | 7 |
Suppose that $P$ is a polynomial with integer coefficients such that $P(1) = 2$, $P(2) = 3$, and $P(3) = 2016$. If $N$ is the smallest possible positive value of $P(2016)$, find the remainder when $N$ is divided by $2016$. | 2013 |
Determine all real numbers $q$ for which the equation $x^4 - 40x^2 + q = 0$ has four real solutions that form an arithmetic progression. | 144 |
Camila creates a pattern to write the following numbers: \[ 2, 4 \\ 5, 7, 9, 11 \\ 12, 14, 16, 18, 20, 22 \\ 23, 25, 27, 29, 31, 33, 35, 37 \\ \ldots \] Following the same pattern, what is the sum of the numbers in the tenth row? | 3840 |
How many numbers appear in both the arithmetic sequence $10, 16, 22, 28, \ldots, 1000$ and the arithmetic sequence $10, 21, 32, 43, \ldots, 1000$? | 16 |
Let $X$ be a point inside convex quadrilateral $ABCD$ such that $\angle AXB + \angle CXD = 180^{\circ}$. Given $AX = 14$, $BX = 11$, $CX = 5$, $DX = 10$, and $AB = CD$, find the sum of the areas of $\triangle AXB$ and $\triangle CXD$. | 90 |
In a circle, let $AB$ and $BC$ be chords, with $AB = \sqrt{3}$, $BC = 3\sqrt{3}$, and $\angle ABC = 60^\circ$. Find the length of the circle chord that bisects the angle $\angle ABC$. | 4 |
Evaluate $2000^3 - 1999 \cdot 2000^2 - 1999^2 \cdot 2000 + 1999^3$ | 3999 |
Let $a_1$, $a_2$, \dots, $a_{2015}$ be a sequence of positive integers in $[1,100]$. Call a nonempty contiguous subsequence of this sequence "good" if the product of the integers in it leaves a remainder of $1$ when divided by $101$. In other words, it is a pair of integers $(x, y)$ such that $1 \le x \le y \le 2015$ a... | 19320 |
Suppose $A \subseteq \{0,1,\dots,29\}$. It satisfies that for any integer $k$ and any two members $a,b \in A$ (where $a$ and $b$ are allowed to be the same), $a + b + 30k$ is always not the product of two consecutive integers. Please find $A$ with the largest possible cardinality. | 10 |
Compute the minimum value of \[ \frac{x^4 + 2x^3 + 3x^2 + 2x + 10}{x^2 + x + 1} \] where \( x \) can be any real number. | 6 |
On the whiteboard, the numbers are written sequentially: $1 \ 2 \ 3 \ 4 \ 5 \ 6 \ 7 \ 8 \ 9$. Andi has to paste a $+$ (plus) sign or $-$ (minus) sign in between every two successive numbers, and compute the value. Determine the least odd positive integer that Andi can't get from this process. | 43 |
The base-7 representation of the number \( n \) is \( \overline{abc}_{(7)} \), and the base-9 representation of the number \( n \) is \( \overline{cba}_{(9)} \). What is the decimal (base-10) representation of \( n \)? | 248 |
For a positive integer $n$, define $d(n)$ to be the sum of the digits of $n$ when written in binary. For example, $d(13) = 1 + 1 + 0 + 1 = 3$. Let \[ S = \sum_{k=1}^{2020} (-1)^{d(k)} k^3. \] Determine $S$ modulo $2020$. | 1990 |
Sixteen dots are arranged in a 4x4 grid. The distance between any two dots is defined as the minimum number of horizontal and vertical steps along the grid lines needed to travel from one dot to the other. For instance, two adjacent dots are a distance of 1 apart, and two dots at opposite corners of the grid are a dist... | 11 |
Suppose $P(x)$ is a quadratic polynomial with integer coefficients satisfying the identity \[ P(P(x)) - P(x)^2 = x^2 + x + 2016 \] for all real $x$. What is $P(1)$? | 1010 |
Let $n$ be the smallest positive integer such that the remainder of $3n + 45$, when divided by $1060$, is $16$. Find the remainder of $18n + 17$ upon division by $1920$. | 1043 |
Suppose $\overline{a_1a_2\ldots a_{2009}}$ is a $2009$-digit integer such that for each $i = 1, 2, \ldots, 2007$, the $2$-digit integer $\overline{a_ia_{i+1}}$ contains $3$ distinct prime factors. Find $a_{2008}$. (Note: $\overline{xyz\ldots}$ denotes an integer whose digits are $x, y, z, \ldots$.) | 6 |
One day while Tony plays in the backyard of the Kubik's home, he wonders about the width of the backyard, which is in the shape of a rectangle. A row of trees spans the width of the back of the yard by the fence, and Tony realizes that all the trees have almost exactly the same diameter, and the trees look equally spac... | 82 |
Evaluate the following sum: $$ \frac{1}{\log_2\left(\frac{1}{7}\right)} + \frac{1}{\log_3\left(\frac{1}{7}\right)} + \frac{1}{\log_4\left(\frac{1}{7}\right)} + \frac{1}{\log_5\left(\frac{1}{7}\right)} + \frac{1}{\log_6\left(\frac{1}{7}\right)} - \frac{1}{\log_7\left(\frac{1}{7}\right)} - \frac{1}{\log_8\left(\frac{1}{7... | 1 |
Let $a$ and $b$ be two positive integers satisfying the equation: \[ 20\sqrt{12} = a\sqrt{b}. \] Compute the sum of all possible distinct products $ab$. | 10800 |
Find the smallest positive integer $n$ such that $107n$ has the same last two digits as $n$. | 50 |
The area of parallelogram $ABCD$ is $51\sqrt{55}$ and $\angle DAC$ is a right angle. If the side lengths of the parallelogram are integers, what is the perimeter of the parallelogram? | 90 |
The polynomials $P_{n}(x)$ are defined by $P_{0}(x)=0$, $P_{1}(x)=x$, and \[P_{n}(x)=xP_{n-1}(x)+(1-x)P_{n-2}(x) \quad n\geq 2\] For every natural number $n\geq 1$, find all real numbers $x$ satisfying the equation $P_{n}(x)=0$. | 0 |
In the following diagram (not to scale), $A$, $B$, $C$, $D$ are four consecutive vertices of an 18-sided regular polygon with center $O$. Let $P$ be the midpoint of $AC$ and $Q$ be the midpoint of $DO$. Find $\angle OPQ$ in degrees. | 30 |
The remainder when $x^{100} - x^{99} + \cdots - x + 1$ is divided by $x^2 - 1$ can be written in the form $ax + b$. Find $2a + b$. | -49 |
Numbers $1, \frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{2001}$ are written on a blackboard. A student erases two numbers $x, y$ and writes down the number $x + y + xy$ instead. Determine the number that will be written on the board after $2000$ such operations. | 2001 |
Find the largest integer $d$ that divides all three numbers $abc$, $bca$, and $cab$, where $a$, $b$, and $c$ are some nonzero and mutually different digits. | 54 |
Find the number of positive integers $n$ not greater than 2017 such that $n$ divides $20^n + 17k$ for some positive integer $k$. | 1899 |
For every positive integer $n$, let $T_n = \frac{n(n+1)}{2}$ be the $n^{\text{th}}$ triangular number. What is the $2018^{\text{th}}$ smallest positive integer $n$ such that $T_n$ is a multiple of 1000? | 1009375 |
Let $v$, $w$, $x$, $y$, and $z$ be five distinct integers such that $45 = v \times w \times x \times y \times z$. What is the sum of the integers? | 5 |
Define $f(x) = x^2 - 45x + 21$. Find the sum of all positive integers $n$ with the following property: there is exactly one integer $i$ in the set $\{1, 2, \ldots, n\}$ such that $n$ divides $f(i)$. | 2592 |
For a positive integer $n$, let $A(n)$ be the remainder when $n$ is divided by $11$. Define $T(n) = A(1) + A(2) + A(3) + \ldots + A(n)$. Find the value of $A(T(2021))$. | 3 |
Let $ABC$ be a triangle with $\angle BAC = 90^\circ$. Construct the square $BDEC$ such that $A$ and the square are on opposite sides of $BC$. Let the angle bisector of $\angle BAC$ intersect the sides $[BC]$ and $[DE]$ at $F$ and $G$, respectively. If $|AB|=24$ and $|AC|=10$, calculate the area of quadrilateral $BDGF$. | 338 |
Let $k \ll n$ denote that $k < n$ and $k \mid n$. Let $f: \{1, 2, \ldots, 2013\} \rightarrow \{1, 2, \ldots, M\}$ be such that, if $n \leq 2013$ and $k \ll n$, then $f(k) \ll f(n)$. What is the least possible value of $M$? | 1024 |
A natural number $k > 1$ is called \textit{good} if there exist natural numbers \(a_1 < a_2 < \cdots < a_k\) such that \(\frac{1}{\sqrt{a_1}} + \frac{1}{\sqrt{a_2}} + \cdots + \frac{1}{\sqrt{a_k}} = 1\). Let \(f(n)\) be the sum of the first \(n\) \textit{good} numbers, \(n \geq 1\). Find the sum of all values of \(n\) ... | 18 |
Consider a function $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for any two positive integers $x, y$, the equation $f(x f(y)) = y f(x)$ holds. Find the smallest possible value of $f(2007)$. | 18 |
Each positive integer $n \geq 1$ is assigned the number $p_n$, which is the product of all its non-zero digits. For example, $p_6 = 6$, $p_{32} = 6$, $p_{203} = 6$. Let $S = p_1 + p_2 + p_3 + \dots + p_{999}$. Find the largest prime that divides $S$. | 103 |
Michelle is at the bottom-left corner of a $6 \times 6$ lattice grid, at $(0,0)$. The grid also contains a pair of one-time-use teleportation devices at $(2,2)$ and $(3,3)$. The first time Michelle moves to one of these points, she is instantly teleported to the other point and the devices disappear. If she can only mo... | 280 |
For every pair of real numbers $0 < a < b < 1$, we define sequences $\{x_n\}_{n \ge 0}$ and $\{y_n\}_{n \ge 0}$ by $x_0 = 0$, $y_0 = 1$, and for each integer $n \ge 1$: \[ x_n = (1 - a) x_{n - 1} + a y_{n - 1}, \\ y_n = (1 - b) x_{n - 1} + b y_{n - 1}. \] The supermean of $a$ and $b$ is the limit of $\{x_n\}$ as $n$ ap... | 307 |
A **Beaver-number** is a positive 5-digit integer whose digit sum is divisible by 17. Call a pair of **Beaver-numbers** differing by exactly $1$ a **Beaver-pair**. The smaller number in a **Beaver-pair** is called an **MIT Beaver**, while the larger number is called a **CIT Beaver**. Find the positive difference betwee... | 79200 |
Let $ABC$ be a triangle. Point $P$ lies in the interior of $\triangle ABC$ such that $\angle ABP = 20^\circ$ and $\angle ACP = 15^\circ$. Compute $\angle BPC - \angle BAC$. | 35 |
The centers of the three circles \(A\), \(B\), and \(C\) are collinear with the center of circle \(B\) lying between the centers of circles \(A\) and \(C\). Circles \(A\) and \(C\) are both externally tangent to circle \(B\), and the three circles share a common tangent line. Given that circle \(A\) has radius \(12\) a... | 147 |
Find the smallest possible value of $x + y$ where $x, y \ge 1$ and $x$ and $y$ are integers that satisfy $x^2 - 29y^2 = 1$. | 11621 |
How many integers between $2$ and $100$ inclusive cannot be written as $m \cdot n$, where $m$ and $n$ have no common factors and neither $m$ nor $n$ is equal to $1$? | 35 |
A group of $25$ friends were discussing a large positive integer. The first friend said, "It can be divided by $1$." The second friend said, "It can be divided by $2$." The third friend said, "And by $3$." The fourth friend added, "And by $4$." This pattern continued until everyone had made such a comment. If exactly $... | 787386600 |
An triangle with coordinates $(x_1,y_1)$, $(x_2, y_2)$, $(x_3,y_3)$ has centroid at $(1,1)$. The ratio between the lengths of the sides of the triangle is $3:4:5$. Given that \[x_1^3+x_2^3+x_3^3=3x_1x_2x_3+20\] and \[y_1^3+y_2^3+y_3^3=3y_1y_2y_3+21,\] the area of the triangle can be expressed as $\frac{m}{n}$, where $m... | 107 |
Define the infinite products: \[ A = \prod_{i=2}^{\infty} \left(1-\frac{1}{n^3}\right) \text{ and } B = \prod_{i=1}^{\infty}\left(1+\frac{1}{n(n+1)}\right). \] If \(\frac{A}{B} = \frac{m}{n}\) where \(m,n\) are relatively prime positive integers, determine \(100m+n\). | 103 |
Let $a, b, c$ be real numbers satisfying: \[ 3ab + 2 = 6b, \quad 3bc + 2 = 5c, \quad 3ca + 2 = 4a. \] Let $\mathbb{Q}$ denote the set of all rational numbers. Given that the product $abc$ can take two values $\frac{r}{s} \in \mathbb{Q}$ and $\frac{t}{u} \in \mathbb{Q}$, both in their lowest terms, find $r + s + t + u$. | 18 |
Since this is the $6$th Greek Math Olympiad and the year is $1989$, can you find the last two digits of $6^{1989}$? | 96 |
Show that the number \( 4\sin\frac{\pi}{34}\left(\sin\frac{3\pi}{34}+\sin\frac{7\pi}{34}+\sin\frac{11\pi}{34}+\sin\frac{15\pi}{34}\right) \) is an integer and determine it. | 1 |
Suppose $a$ and $b$ are positive integers for which $8a^a b^b = 27a^b b^a$. Find $a^2 + b^2$. | 117 |
Determine the value of $q(q(q(2000^{2000})))$, where $q(n)$ represents the sum of the digits of a natural number $n$. | 4 |
There are $1000$ rooms in a row along a long corridor. Initially, the first room contains $1000$ people, and the remaining rooms are empty. Each minute, the following happens: for each room containing more than one person, someone in that room decides it is too crowded and moves to the next room. All these movements ar... | 61 |
Given that nonzero real numbers $a, b, c, d$ satisfy $a^b = c^d$ and $\frac{a}{2c} = \frac{b}{d} = 2$, compute $\frac{1}{c}$. | 16 |
Let $N = 30^{2015}$. Find the number of ordered 4-tuples of integers $(A, B, C, D) \in \{1, 2, \ldots, N\}^4$ (not necessarily distinct) such that for every integer $n$, $An^3 + Bn^2 + 2Cn + D$ is divisible by $N$. | 2 |
The diagram shows a circle with radius $24$ that contains two smaller circles, each with a radius of $12$. These two circles are tangent to each other and to the larger circle. There is also a smallest circle that is tangent to the three other circles. Determine the radius of this smallest circle. | 8 |
**Problem 1.** The least common multiple of the numbers $a$ and $b$ is $88$. If $a^2 + b^2 = 2000$, find the value of $(a + b)$. | 52 |
Given a strictly increasing sequence $a_1 < a_2 < \dots$ of positive integers such that each term $a_k$ is divisible either by 1005 or 1006, but neither term is divisible by 97. Find the least possible value of the maximal difference of consecutive terms $a_{i+1} - a_i$. | 2010 |
Compute the number of positive integer divisors of $2121$ with a units digit of $1$. | 4 |
Let $\eta(m)$ be the product of all positive integers that divide $m$, including $1$ and $m$. If $\eta(\eta(\eta(10))) = 10^n$, compute $n$. | 450 |
Two math students play a game with $k$ sticks. Alternating turns, each player chooses a number from the set \{1, 3, 4\} and removes exactly that number of sticks from the pile. If the pile has fewer sticks than the chosen number, the player must choose a number that can be removed. The winner is the player who takes th... | 71 |
In an isosceles triangle $ABC$ with an angle $\angle A = 20^\circ$ and base $BC = 12$, a point $E$ on the side $AC$ is chosen such that $\angle ABE = 30^\circ$. Additionally, a point $F$ is on the side $AB$ such that $EF = FC$. Find the length of $FC$. | 12 |
Marisa has two identical cubical dice labeled with the numbers $\{1, 2, 3, 4, 5, 6\}$. However, the two dice are not fair, meaning that they can land on each face with different probabilities. Marisa rolls the two dice and calculates their sum. Given that the sum is $2$ with probability $0.04$, and $12$ with probabilit... | 28 |
A cuboctahedron is a solid with 6 square faces and 8 equilateral triangle faces, with each edge adjacent to both a square and a triangle. Suppose the ratio of the volume of an octahedron to a cuboctahedron with the same side length is $r$. Find $100r^2$. | 4 |
Let $A$ be the area of the largest semicircle that can be inscribed in a quarter-circle of radius $1$. Compute $\frac{120A}{\pi}$. | 15 |
Cyclic quadrilateral $ABCD$ has $AC \perp BD$, $AB + CD = 12$, and $BC + AD = 13$. Find the greatest possible area of $ABCD$. | 36 |
Laurie loves multiplying numbers in her head. One day, she decides to multiply two 2-digit numbers $x$ and $y$ such that $x \leq y$ and the two numbers collectively have at least three distinct digits. Unfortunately, she accidentally remembers the digits of each number in the opposite order (for example, instead of rem... | 28 |
Triangle $ABC$ has side lengths $AC=3$, $BC=4$, and $AB=5$. Let $R$ be a point on the incircle $\omega$ of $\triangle ABC$. The altitude from $C$ to $\overline{AB}$ intersects $\omega$ at points $P$ and $Q$. Then, the greatest possible area of $\triangle PQR$ is $\frac{m\sqrt{n}}{p}$, where $m$ and $p$ are relatively p... | 43 |
The real numbers $a_0, a_1, \dots, a_{2013}$ and $b_0, b_1, \dots, b_{2013}$ satisfy the recurrence relations: \[ a_{n} = \frac{1}{63} \sqrt{2n+2} + a_{n-1} \quad \text{and} \quad b_{n} = \frac{1}{96} \sqrt{2n+2} - b_{n-1} \] for every integer $n = 1, 2, \dots, 2013$. Given the initial conditions $a_0 = b_{2013}$ and $... | 671 |
In a triangle $ABC$, $\angle C = 2\angle A$. A point $D$ is marked on the side $AC$ such that $\angle ABD = \angle DBC$. Knowing that $AB = 10$ and $CD = 3$, what is the length of the side $BC$? | 7 |
In the convex and cyclic quadrilateral $ABCD$, we have $\angle B = 110^{\circ}$. The intersection of $AD$ and $BC$ is $E$, and the intersection of $AB$ and $CD$ is $F$. If the perpendicular from $E$ to $AB$ intersects the perpendicular from $F$ to $BC$ on the circumcircle of the quadrilateral at point $P$, what is $\an... | 70 |
Find the least positive integer that cannot be represented as $\frac{2^a-2^b}{2^c-2^d}$ for some positive integers $a, b, c, d$. | 11 |
Let $n \geq 3$ be an integer. Mario draws $20$ lines in the plane, such that no two lines are parallel. For each equilateral triangle formed by three of these lines, Mario receives three coins. For each isosceles and non-equilateral triangle formed by three of these lines, Mario receives one coin. What is the maximum n... | 380 |
Let a rad number be a palindrome such that the square root of the sum of its digits is irrational. Find the number of $4$-digit rad numbers. | 79 |
A number $N$ has 2009 positive factors. What is the maximum number of positive factors that $N^2$ could have? | 13689 |
Let $ABC$ be an equilateral triangle. Let $P$ and $S$ be points on $AB$ and $AC$, respectively, and let $Q$ and $R$ be points on $BC$ such that $PQRS$ is a rectangle. If $PQ = \sqrt{3} \cdot PS$ and the area of $PQRS$ is $28\sqrt{3}$, what is the length of $PC$? | 14 |
\( \triangle ABC \) has \( AB = 5 \), \( BC = 12 \), and \( AC = 13 \). A circle is inscribed in \( \triangle ABC \), and \( MN \) tangent to the circle is drawn such that \( M \) is on \( \overline{AC} \), \( N \) is on \( \overline{BC} \), and \( \overline{MN} \parallel \overline{AB} \). The area of \( \triangle MNC ... | 43 |
Find the number of ordered tuples \((C,A,M,B)\) of non-negative integers such that \[ C! + C! + A! + M! = B! \] | 7 |
Let $a, b, c > 1$ be positive real numbers such that $a^{\log_b c} = 27$, $b^{\log_c a} = 81$, and $c^{\log_a b} = 243$. Then the value of $\log_3{abc}$ can be written as $\sqrt{x} + \sqrt{y} + \sqrt{z}$ for positive integers $x, y,$ and $z$. Find $x + y + z$. | 47 |
Let $O$ be a circle with diameter $AB = 2$. Circles $O_1$ and $O_2$ have centers on $\overline{AB}$ such that $O$ is tangent to $O_1$ at $A$ and to $O_2$ at $B$, and $O_1$ and $O_2$ are externally tangent to each other. The minimum possible value of the sum of the areas of $O_1$ and $O_2$ can be written in the form $\f... | 3 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.