prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
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Convex equiangular hexagon $ABCDEF$ has $AB=CD=EF=1$ and $BC = DE = FA = 4$. Congruent and pairwise externally tangent circles $\gamma_1$, $\gamma_2$, and $\gamma_3$ are drawn such that $\gamma_1$ is tangent to side $\overline{AB}$ and side $\overline{BC}$, $\gamma_2$ is tangent to side $\overline{CD}$ and side $\overl... | 14800 |
Let $A, B, C, \ldots, Z$ be 26 nonzero real numbers. Suppose that $T = TNYWR$. Compute the smallest possible value of \[ \left\lceil A^2 + B^2 + \cdots + Z^2 \right\rceil. \] (The notation $\left\lceil x \right\rceil$ denotes the least integer $n$ such that $n \geq x$.) | 5 |
Let $N$ be the number of $2015$-tuples of (not necessarily distinct) subsets $(S_1, S_2, \dots, S_{2015})$ of $\{1, 2, \dots, 2015 \}$ such that the number of permutations $\sigma$ of $\{1, 2, \dots, 2015 \}$ satisfying $\sigma(i) \in S_i$ for all $1 \le i \le 2015$ is odd. Let $k_2, k_3$ be the largest integers such t... | 2030612 |
Let $ABC$ be a triangle with $AB = 2$ and $AC = 3$. Let $H$ be the orthocenter, and let $M$ be the midpoint of $BC$. Let the line through $H$ perpendicular to line $AM$ intersect line $AB$ at $X$ and line $AC$ at $Y$. Suppose that lines $BY$ and $CX$ are parallel. Then $[ABC]^2 = \frac{a + b\sqrt{c}}{d}$ for positive i... | 270382 |
In a triangle $ABC$, the median from $B$ to $CA$ is perpendicular to the median from $C$ to $AB$. If the median from $A$ to $BC$ is $30$, determine $\frac{BC^2 + CA^2 + AB^2}{100}$. | 24 |
Equilateral triangle $ABC$ has an area of $1$. Points $A'$, $B'$, and $C'$ are the midpoints of sides $BC$, $CA$, and $AB$, respectively. Points $A''$, $B''$, and $C''$ are the midpoints of segments $B'C'$, $C'A'$, and $A'B'$, respectively. The area of trapezoid $BB''C''C$ can be expressed as $\frac{m}{n}$, where $m$ a... | 21 |
Solve the following equation in positive numbers: $$ (2a+1)(2a^2+2a+1)(2a^4+4a^3+6a^2+4a+1) = 828567056280801 $$ | 100 |
A canister contains $2\frac{1}{2}$ cups of flour. Greg and Sally have a brownie recipe that calls for $1\frac{1}{3}$ cups of flour. They want to make $1\frac{1}{2}$ recipes of brownies. To the nearest whole percent, what percent of the flour in the canister would they use? | 80 |
In triangle $ABC$, where $AB = 28$, $AC = 36$, and $BC = 32$, let $D$ be the point on segment $BC$ such that $\angle BAD = \angle DAC$. Let $E$ be the unique point such that $DE \parallel AB$ and line $AE$ is tangent to the circumcircle of $\triangle ABC$. Find the length of segment $AE$. | 18 |
A collection of $n$ squares on the plane is called tri-connected if the following criteria are satisfied: 1. All the squares are congruent. 2. If two squares have a point $P$ in common, then $P$ is a vertex of each of the squares. 3. Each square touches exactly three other squares. How many positive integers $n$ are th... | 501 |
For how many integers $n$ with $1 \leq n \leq 2012$ is the product \[ \prod_{k=0}^{n-1} \left( \left( 1 + e^{2 \pi i k / n} \right)^n + 1 \right) \] equal to zero? | 335 |
Find the least positive integer that cannot be represented as \( \frac{2^a - 2^b}{2^c - 2^d} \), where \( a, b, c, \) and \( d \) are positive integers. | 11 |
Given a square $ABCD$ with side length $10$. On sides $BC$ and $AD$ of this square, points $E$ and $F$ are selected such that they form a rectangle $ABEF$. Rectangle $KLMN$ is positioned such that its vertices $K, L, M$, and $N$ lie on segments $CD, DF, FE$, and $EC$, respectively. It is given that rectangles $ABEF$ an... | 5 |
What is the number formed by the last three digits of $1201^{1202}$? | 401 |
A fighting game club has $2024$ members. One day, a game of Smash is played between some pairs of members such that every member has played against exactly $3$ other members. Each match has a winner and a loser. A member will be happy if they won in at least $2$ of the matches. What is the maximum number of happy membe... | 1012 |
Convex quadrilateral $ABCD$ satisfies the following conditions: \( \angle{CAB} = \angle{ADB} = 30^{\circ} \), \( \angle{ABD} = 77^{\circ} \), \( BC = CD \), and \( \angle{BCD} = n^{\circ} \) for some positive integer \( n \). Compute \( n \). | 68 |
At a party, there were some couples attending. As they arrive, each person gets to talk with all the other persons who are already in the room. During the party, after all the guests arrive, groups of persons form such that: 1. No two persons forming a couple belong to the same group. 2. For each two persons that do no... | 4 |
Math teacher wrote in a table a polynomial $P(x)$ with integer coefficients and said: "Today my daughter has a birthday. If in the polynomial $P(x)$ we have $x=a$ where $a$ is the age of my daughter, then $P(a)=a$ and $P(0)=p$ where $p$ is a prime number such that $p>a$." How old is the daughter of the math teacher? | 1 |
Find the number of integers $n$ such that $1 \le n \le 2017$ and $(n-2)(n-0)(n-1)(n-7)$ is a multiple of $1001$. | 99 |
Let $p, q$ be two-digit numbers, neither of which are divisible by $10$. Let $r$ be the four-digit number formed by placing the digits of $p$ followed by the digits of $q$ in order. As $p$ and $q$ vary, a computer prints $r$ on the screen if $\gcd(p, q) = 1$ and $p + q$ divides $r$. Suppose that the largest number prin... | 13 |
Compute the integer $$\frac{2^{\left(2^5-2\right)/5-1}-2}{5}.$$ | 6 |
Find $100m+n$ if $m$ and $n$ are relatively prime positive integers such that \[ \sum_{\substack{i,j \ge 0 \\ i+j \text{ odd}}} \frac{1}{2^i3^j} = \frac{m}{n}. \] | 504 |
Distinct prime numbers $p$, $q$, and $r$ satisfy the following equation: $$2pqr + 50pq = 7pqr + 55pr = 8pqr + 12qr = A$$ for some positive integer $A$. What is the value of $A$? | 1980 |
A sequence $a_n$ of real numbers satisfies the following conditions: - $a_1 = 1$ - $a_2 = 0$ - $a_n = (S_{n-1} + 1) S_{n-2}$ for all integers $n \geq 3$ where $S_k = a_1 + a_2 + \dots + a_k$ for positive integers $k$. Determine the smallest integer $m > 2$ such that $127$ divides $a_m$. | 10 |
Define the *hotel elevator cubic* as the unique cubic polynomial $P$ for which $P(11) = 11$, $P(12) = 12$, $P(13) = 14$, $P(14) = 15$. What is $P(15)$? | 13 |
Let $ABC$ be a right triangle with hypotenuse $\overline{AC}$ and circumcenter $O$. Point $E$ lies on $\overline{AB}$ such that $AE = 9$, $EB = 3$. Point $F$ lies on $\overline{BC}$ such that $BF = 6$, $FC = 2$. Now suppose $W, X, Y$, and $Z$ are the midpoints of $\overline{EB}$, $\overline{BF}$, $\overline{FO}$, and $... | 12 |
Given that $9 \times 10 \times 11 \times \cdots \times 15 = 32432400$, what is $1 \times 3 \times 5 \times \cdots \times 15$? | 2027025 |
Jenny places 100 pennies on a table, with 30 showing heads and 70 showing tails. She randomly selects 40 pennies and flips them over. This means if a selected penny was showing heads, it will now show tails, and vice versa. What is the expected number of pennies showing heads at the end? | 46 |
Find the number of triples \((a, b, c)\) of positive integers such that \(a + ab + abc = 11\). | 3 |
Let \( \gamma = \alpha \times \beta \) where \[ \alpha = \underbrace{999 \cdots 9}_{2010 \text{ '9's}} \] and \[ \beta = \underbrace{444 \cdots 4}_{2010 \text{ '4's}} \] Find the sum of digits of \( \gamma \). | 18090 |
How many positive five-digit integers have the product of their five digits equal to $900$? | 210 |
For some positive integer $n$, consider the usual prime factorization \[ n = \prod_{i=1}^{k} p_{i}^{e_{i}} = p_1^{e_1}p_2^{e_2}\ldots p_k^{e_k}, \] where $k$ is the number of prime factors of $n$ and $p_{i}$ are the prime factors of $n$. Define $Q(n), R(n)$ by \[ Q(n) = \prod_{i=1}^{k} p_{i}^{p_{i}} \text{ and } R(n) =... | 56 |
Triangles $OAB$, $OBC$, and $OCD$ are isosceles triangles with $\angle OAB = \angle OBC = \angle OCD = 90^\circ$. Find the area of triangle $OAB$ if the area of triangle $OCD$ is 12. | 3 |
A **palindromic table** is a $3 \times 3$ array of letters such that the words in each row and column read the same forwards and backwards. An example of such a table is shown below. \[ \begin{array}{ccc} O & M & O \\ N & M & N \\ O & M & O \end{array} \] How many palindromic tables are there that use only the letters ... | 16 |
A point $(x, y)$ in the first quadrant lies on a line with intercepts $(a, 0)$ and $(0, b)$, where $a, b > 0$. Rectangle $M$ has vertices $(0, 0)$, $(x, 0)$, $(x, y)$, and $(0, y)$, while rectangle $N$ has vertices $(x, y)$, $(x, b)$, $(a, b)$, and $(a, y)$. What is the ratio of the area of $M$ to that of $N$? | 1 |
In the MTRP district, there are $10$ cities. Bob the builder wants to construct roads between these cities such that there is exactly one unique path between any two cities. The government has provided him with a budget of Rs. $20$, and each road requires a positive integer amount (in Rs.) to build. In how many ways ca... | 0 |
Let $ABCDEF$ be a regular hexagon, and let $P$ be a point inside quadrilateral $ABCD$. If the area of triangle $PBC$ is $20$, and the area of triangle $PAD$ is $23$, compute the area of hexagon $ABCDEF$. | 189 |
For the NEMO, Kevin needs to compute the product: \[ 9 \times 99 \times 999 \times \cdots \times 999999999. \] Kevin takes exactly $ab$ seconds to multiply an $a$-digit integer by a $b$-digit integer. Compute the minimum number of seconds necessary for Kevin to evaluate the expression together by performing eight such ... | 870 |
Compute $\left\lceil\sum_{k=2018}^{\infty}\frac{2019!-2018!}{k!}\right\rceil$. The notation $\left\lceil x\right\rceil$ denotes the least integer $n$ such that $n\geq x$. | 2019 |
A tailor met a tortoise sitting under a tree. When the tortoise was the tailor's age, the tailor was only a quarter of his current age. When the tree was the tortoise's age, the tortoise was only a seventh of its current age. If the sum of their ages is now $264$, how old is the tortoise? | 77 |
What is the largest positive integer $n$ for which there are no positive integers $a, b$ such that $8a + 11b = n$? | 88 |
Evaluate $\int_1^4 \frac{x-2}{(x^2+4)\sqrt{x}}\,dx.$ | 0 |
(MON 1) Find the number of five-digit numbers with the following properties: there are two pairs of digits such that digits from each pair are equal and are next to each other, digits from different pairs are different, and the remaining digit (which does not belong to any of the pairs) is different from the other digi... | 1944 |
Given that a duck found that $5 - 2\sqrt{3}i$ is one of the roots of the polynomial $-259 + 107x - 17x^2 + x^3$, what is the sum of the real parts of the other two roots? | 12 |
Triangle $\triangle ABC$ has circumcenter $O$ and incircle $\gamma$. Suppose that $\angle BAC = 60^\circ$ and $O$ lies on $\gamma$. If \[ \tan B \tan C = a + \sqrt{b} \] for positive integers $a$ and $b$, compute $100a+b$. | 408 |
On a $10 \times 10$ chessboard, several knights are placed. In any $2 \times 2$ square, there is at least one knight. What is the smallest number of cells these knights can threaten? (Note: A knight does not threaten the square on which it stands, but it does threaten the squares on which other knights are standing.) | 50 |
For real numbers $x_1, x_2, x_3, \dots, x_{333} \in [-1, \infty)$, let $S_k = \sum_{i = 1}^{333} x_i^k$ for each $k$. If $S_2 = 777$, compute the least possible value of $S_3$. | 999 |
The function $f: \mathbb{N} \to \mathbb{N}_0$ is defined such that: - $f(2) = 0$ - $f(3) > 0$ - $f(6042) = 2014$ - For all $m, n \in \mathbb{N}$, $f(m + n) - f(m) - f(n) \in \{0, 1\}$ Determine $f(2014)$. Note: $\mathbb{N}_0 = \{0, 1, 2, \ldots\}$ | 671 |
A square is inscribed in a circle with a radius of $6$. A quarter circle is inscribed in the square, as depicted in the diagram below. Given that the area of the region inside the circle but outside the quarter circle is $n\pi$ for some positive integer $n$, determine the value of $n$. [asy] size(5 cm); draw((0,0)--(2,... | 18 |
Find the number of ordered triples of positive integers $(a, b, c)$ such that \[6a + 10b + 15c = 3000.\] | 4851 |
Let $a, b, c, d$ be four positive integers such that $a > b > c > d$. Given that $ab + bc + ca + d^2 \mid (a + b)(b + c)(c + a)$. Find the minimal value of $\Omega(ab + bc + ca + d^2)$. Here $\Omega(n)$ denotes the number of prime factors $n$ has. For example, $\Omega(12) = 3$. | 3 |
Triangle $ABC$ has $AB = 3$, $BC = 4$, and $CA = 5$. Points $D$, $E$, $F$, $G$, $H$, and $I$ are the reflections of $A$ over $B$, $B$ over $A$, $B$ over $C$, $C$ over $B$, $C$ over $A$, and $A$ over $C$, respectively. Find the area of hexagon $EFIDGH$. | 78 |
How many values of $x$, where $-19 < x < 98$, satisfy the equation $\cos^2 x + 2\sin^2 x = 1$? | 38 |
A four-digit number has the following properties: 1. The units digit equals the tens digit increased by $1$. 2. The hundreds digit equals twice the tens digit. 3. The thousands digit is at least twice the units digit. 4. The number is twice a prime number. Determine this four-digit number. | 9634 |
Kelvin the Frog likes numbers whose digits strictly decrease, but numbers that violate this condition in at most one place are good enough. In other words, if $d_i$ denotes the $i$th digit, then $d_i \le d_{i+1}$ for at most one value of $i$. For example, Kelvin likes the numbers $43210$, $132$, and $3$, but not the nu... | 14034 |
At Stanford in 1988, human calculator Shakuntala Devi was asked to compute $m = \sqrt[3]{61,629,875}$ and $n = \sqrt[7]{170,859,375}$. Given that $m$ and $n$ are both integers, compute $100m+n$. | 39515 |
We record one number every workday: Monday, Tuesday, Wednesday, Thursday, and Friday. On the first Monday, the number we record is $10.5$. On every Tuesday and every Thursday, the number we record is one third of what it was on the previous workday. On every Monday, Wednesday, and Friday, the number we record is double... | 252 |
Let $\alpha = \cos^{-1} \left( \frac{3}{5} \right)$ and $\beta = \sin^{-1} \left( \frac{3}{5} \right)$. \[\sum_{n=0}^{\infty}\sum_{m=0}^{\infty} \frac{\cos(\alpha n +\beta m)}{2^n3^m}\] can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A +B$. | 15013 |
How many ways are there to write $657$ as a sum of powers of two, where each power of two is used at most twice in the sum? For example, $256 + 256 + 128 + 16 + 1$ is a valid sum. | 41 |
In $\triangle ABC$, $AB = 2019$, $BC = 2020$, and $CA = 2021$. Yannick draws three regular $n$-gons in the plane of $\triangle ABC$ so that each $n$-gon shares a side with a distinct side of $\triangle ABC$ and no two of the $n$-gons overlap. What is the maximum possible value of $n$? | 11 |
$A, B, C$ are points in the plane such that $\angle ABC = 90^\circ$. Circles with diameters $BA$ and $BC$ meet at $D$. If $BA = 20$ and $BC = 21$, then the length of segment $BD$ can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$? | 449 |
Let $P$ be a polynomial satisfying $P(x + 1) + P(x - 1) = x^3$ for all real numbers $x$. Find the value of $P(12)$. | 846 |
Parallelogram $ABCD$ is given such that $\angle ABC = 30^\circ$. Let $X$ be the foot of the perpendicular from $A$ onto $BC$, and $Y$ the foot of the perpendicular from $C$ onto $AB$. If $AX = 20$ and $CY = 22$, find the area of the parallelogram. | 880 |
A $3 \times 3$ determinant has three entries equal to $2$, three entries equal to $5$, and three entries equal to $8$. Find the maximum possible value of the determinant. | 405 |
For each integer $1 \le j \le 2017$, let $S_j$ denote the set of integers $0 \le i \le 2^{2017} - 1$ such that $\left\lfloor \frac{i}{2^{j-1}} \right\rfloor$ is an odd integer. Let $P$ be a polynomial such that \[P\left(x_0, x_1, \ldots, x_{2^{2017} - 1}\right) = \prod_{1 \le j \le 2017} \left(1 - \prod_{i \in S_j} x_i... | 1840 |
Find the 2000th positive integer that is not the difference between any two integer squares. | 7998 |
After moving his sticky toy spider one morning, Tony heads outside to play "pirates" with his pal Nick, who lives a few doors down the street from the Kubiks. Tony and Nick imagine themselves as pirates in a rough skirmish over a chest of gold. Victorious over their foes, Tony and Nick claim the prize. However, they mu... | 982 |
Five bricklayers working together finish a job in $3$ hours. Working alone, each bricklayer takes at most $36$ hours to finish the job. What is the smallest number of minutes it could take the fastest bricklayer to complete the job alone? | 270 |
Compute the number of three-digit numbers such that all three digits are distinct, in descending order, and one of the digits is a $5$. | 36 |
Henry starts with a list of the first 1000 positive integers and performs a series of steps on the list. At each step, he erases any non-positive integers or any integers that have a repeated digit, and then decreases everything in the list by 1. How many steps does it take for Henry's list to be empty? | 11 |
Consider two parallel lines $a$ and $b$. The circles $C, C_1$ are tangent to each other and to the line $a$. The circles $C, C_2$ are tangent to each other and to the line $b$. The circles $C_1, C_2$ are tangent to each other, with radii $R_1 = 9$ and $R_2 = 16$. What is the radius $R$ of the circle $C$? | 24 |
The circle centered at point $A$ with radius $19$ and the circle centered at point $B$ with radius $32$ are both internally tangent to a circle centered at point $C$ with radius $100$, such that point $C$ lies on segment $\overline{AB}$. Point $M$ is on the circle centered at $A$, and point $N$ is on the circle centere... | 140 |
Find a sequence using all the digits from $1$ to $9$ without repetition, such that any two consecutive digits form a number divisible by $7$ or $13$. | 784913526 |
Let $ABCD$ be a cyclic quadrilateral with circumcircle $\omega$ and let $AC$ and $BD$ intersect at $X$. Let the line through $A$ parallel to $BD$ intersect line $CD$ at $E$ and $\omega$ at $Y \neq A$. If $AB = 10$, $AD = 24$, $XA = 17$, and $XB = 21$, then the area of $\triangle DEY$ can be written in simplest form as ... | 421 |
Let $f(x) = x^2 + 4x + 2$. Determine the difference $r$ between the largest and smallest real solutions of the equation $f(f(f(f(x)))) = 0$. Express $r$ in the form $a^{\frac{p}{q}}$, where $a$ is square-free, and $p, q$ are relatively prime positive integers. Compute $a + p + q$. | 35 |
We define a number as "funny" if it is divisible by the sum of its digits plus one. For example, since $1 + 2 + 1 \mid 12$, the number $12$ is considered funny. What is the maximum number of consecutive funny numbers? | 2 |
Let the three sides of a triangle be $\ell, m, n$, respectively, satisfying $\ell > m > n$ and $\{\frac{3^\ell}{10^4}\} = \{\frac{3^m}{10^4}\} = \{\frac{3^n}{10^4}\}$, where $\{x\} = x - \lfloor{x}\rfloor$ and $\lfloor{x}\rfloor$ denotes the integral part of the number $x$. Find the minimum perimeter of such a triangle... | 3003 |
Archimedes drew a square with side length $36$ cm into the sand and he also drew a circle of radius $36$ cm around each vertex of the square. If the total area of the grey parts is $n \cdot \pi$ cm$^2$, what is the value of $n$? | 3888 |
Find the least natural number $n$, which has at least 6 different divisors $1 = d_1 < d_2 < d_3 < d_4 < d_5 < d_6 < \ldots$, such that: 1. $d_3 + d_4 = d_5 + 6$ 2. $d_4 + d_5 = d_6 + 7$ | 494 |
Alice is counting up by fives, starting with the number $3$. Meanwhile, Bob is counting down by fours, starting with the number $2021$. How many numbers between $3$ and $2021$, inclusive, are counted by both Alice and Bob? | 101 |
Let the sequence $\{x_n\}$ be defined by $x_1 \in \{5, 7\}$ and, for $k \ge 1$, $x_{k+1} \in \{5^{x_k} , 7^{x_k} \}$. For example, the possible values of $x_3$ are $5^{5^5}, 5^{5^7}, 5^{7^5}, 5^{7^7}, 7^{5^5}, 7^{5^7}, 7^{7^5}$, and $7^{7^7}$. Determine the sum of all possible values for the last two digits of $x_{2012... | 75 |
A quadrilateral is given, where two of its sides have lengths of $1$ and $4$. Additionally, a diagonal of length $2$ divides the quadrilateral into two isosceles triangles. Determine the perimeter of this quadrilateral. | 11 |
Gabriel plays a game of drawing triangles using the vertices of a regular polygon with $2019$ sides, following these rules: 1. The vertices used by each triangle must not have been previously used. 2. The sides of the triangle to be drawn must not intersect with the sides of the triangles previously drawn. If Gabriel c... | 673 |
Determine the last (rightmost) three decimal digits of $n$ where: \[ n=1 \times 3 \times 5 \times 7 \times \ldots \times 2019. \] | 875 |
Let $f(x, y) = \frac{x + y}{2}$, $g(x, y) = \sqrt{xy}$, $h(x, y) = \frac{2xy}{x + y}$, and let $$ S = \{ (a, b) \in \mathbb{N} \times \mathbb{N} \mid a \neq b \text{ and } f(a, b), g(a, b), h(a, b) \in \mathbb{N} \} $$ where $\mathbb{N}$ denotes the positive integers. Find the minimum of $f$ over $S$. | 25 |
Ralph is standing along a road which heads straight east. If you go nine miles east, make a left turn, and travel seven miles north, you will find Pamela with her mountain bike. At exactly the same time that Ralph begins running eastward along the road at 6 miles per hour, Pamela begins biking in a straight line at 10 ... | 29 |
$N$ students are seated at desks in an $m \times n$ array, where $m, n \ge 3$. Each student shakes hands with the students who are adjacent horizontally, vertically, or diagonally. If there are $1020$ handshakes, what is $N$? | 280 |
A permutation of a finite set is a one-to-one function from the set onto itself. A cycle in a permutation $P$ is a nonempty sequence of distinct items $x_1, \ldots, x_n$ such that $P(x_1) = x_2$, $P(x_2) = x_3$, $\ldots$, $P(x_n) = x_1$. Note that we allow the 1-cycle $x_1$ where $P(x_1) = x_1$ and the 2-cycle $x_1, x_... | 1764 |
If $x = \sqrt{2} + \sqrt{3} + \sqrt{6}$ is a root of the polynomial $x^4 + ax^3 + bx^2 + cx + d = 0$, where $a, b, c, d$ are integers, what is the value of $|a + b + c + d|$? | 93 |
One rainy afternoon, you write the number $1$ once, the number $2$ twice, the number $3$ three times, and so forth until you have written the number $99$ ninety-nine times. What is the $2005^{th}$ digit that you write? | 5 |
Let $N \ge 5$ be given. Consider all sequences $(e_1,e_2,...,e_N)$ with each $e_i$ equal to $1$ or $-1$. Per move, one can choose any five consecutive terms and change their signs. Two sequences are said to be similar if one of them can be transformed into the other in finitely many moves. Find the maximum number of pa... | 16 |
Let $x$, $y$, and $z$ be positive integers satisfying the following system of equations: \[ \begin{align*} &x^2 + \frac{2023}{x} = 2y^2, \\ &y + \frac{2028}{y^2} = z^2, \\ &2z + \frac{2025}{z^2} = xy. \end{align*} \] Find $x + y + z$. | 25 |
Find the sum of all positive integers $n$ such that exactly $2\%$ of the numbers in the set $\{1, 2, \ldots, n\}$ are perfect squares. | 7350 |
Find the largest positive integer value of $n < 1000$ such that $\phi(36n) = \phi(25n) + \phi(16n)$, where $\phi(n)$ denotes the number of positive integers less than $n$ that are relatively prime to $n$. | 996 |
Let $\triangle ABC$ be a triangle with $\angle ABC = 76^\circ$ and $\angle ACB = 72^\circ$. Points $P$ and $Q$ lie on the sides $AB$ and $AC$, respectively, such that $\angle ABQ = 22^\circ$ and $\angle ACP = 44^\circ$. Find the measure of $\angle APQ$. | 36 |
A $9 \times 9$ board has a number written on each square: all squares in the first row have $1$, all squares in the second row have $2$, $\ldots$, all squares in the ninth row have $9$. We will call a "special" rectangle any rectangle of $2 \times 3$, $3 \times 2$, $4 \times 5$, or $5 \times 4$ on the board. The permit... | 1 |
What is the remainder when $128^{2023}$ is divided by $126$? | 2 |
There is a real number $x$ between $0$ and $\frac{\pi}{2}$ such that: $$\frac{\sin^3 x + \cos^3 x}{\sin^5 x + \cos^5 x} = \frac{12}{11}$$ and: $$\sin x + \cos x = \frac{\sqrt{m}}{n}$$ where $m$ and $n$ are positive integers, and $m$ is not divisible by the square of any prime. Find $m + n$. | 8 |
Let $0 < \theta < 2\pi$ be a real number for which $\cos (\theta) + \cos (2\theta) + \cos (3\theta) + \cdots + \cos (2020\theta) = 0$ and $\theta =\frac{\pi}{n}$ for some positive integer $n$. Compute the sum of the possible values of $n \leq 2020$. | 1926 |
A positive integer $N$ has 20 digits when written in base 9 and 13 digits when written in base 27. How many digits does $N$ have when written in base 3? | 39 |
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