prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
|---|---|
Find a set of positive integers with the greatest possible number of elements such that the least common multiple of all of them is less than $2011$. | 40 |
Let $ABCD$ be a regular tetrahedron with side length $1$. Let $EFGH$ be another regular tetrahedron such that the volume of $EFGH$ is $\frac{1}{8}$-th the volume of $ABCD$. The height of $EFGH$ (the minimum distance from any of the vertices to its opposing face) can be written as $\sqrt{\frac{a}{b}}$, where $a$ and $b$... | 7 |
Nair has puzzle pieces shaped like an equilateral triangle. She has pieces of two sizes: large and small. Nair builds triangular figures by following these rules: - Figure $1$ is made up of $4$ small pieces. - Figure $2$ is made up of $2$ large pieces and $8$ small pieces. - Figure $3$ is made up of $6$ large pieces an... | 460 |
An equilateral triangle $T$ with side length $2022$ is divided into equilateral unit triangles with lines parallel to its sides to obtain a triangular grid. The grid is covered with figures shown in the image below, which consist of $4$ equilateral unit triangles and can be rotated by any angle $k \cdot 60^{\circ}$ for... | 1011 |
Let $\triangle ABC$ be an equilateral triangle with side length $1$ and $M$ be the midpoint of side $AC$. Let $N$ be the foot of the perpendicular from $M$ to side $AB$. The angle bisector of angle $\angle BAC$ intersects $MB$ and $MN$ at $X$ and $Y$, respectively. If the area of triangle $\triangle MXY$ is $\mathcal{A... | 769 |
The straight line $y = ax + 16$ intersects the graph of $y = x^3$ at 2 distinct points. What is the value of $a$? | 12 |
Let $a_n$ be a sequence with $a_0 = 1$ and defined recursively by \[ a_{n+1} = \begin{cases} a_n + 2 & \text{if } n \text{ is even},\\ 2a_n & \text{if } n \text{ is odd.} \end{cases} \] What are the last two digits of $a_{2015}$? | 38 |
Let $AB$ be a diameter of a circle and let $C$ be a point on the segment $AB$ such that $AC : CB = 6 : 7$. Let $D$ be a point on the circle such that $DC$ is perpendicular to $AB$. Let $DE$ be the diameter through $D$. If $[XYZ]$ denotes the area of the triangle $XYZ$, find $\frac{[ABD]}{[CDE]}$ to the nearest integer. | 13 |
The real numbers $x$ and $y$ satisfy the following system of equations: \[ \begin{cases} \sqrt{3x}\left(1+\dfrac{1}{x+y}\right)=2 \\ \sqrt{7y}\left(1-\dfrac{1}{x+y}\right)=4\sqrt{2} \end{cases} \] Find the numerical value of the ratio $\frac{y}{x}$. | 6 |
Let $P(x)$ be a polynomial with real coefficients such that $P(12)=20$ and \[ (x-1) \cdot P(16x) = (8x-1) \cdot P(8x) \] holds for all real numbers $x$. Compute the remainder when $P(2014)$ is divided by $1000$. | 545 |
Consider all ordered pairs \((a, b)\) of positive integers such that \(\frac{a^2 + b^2 + 2}{ab}\) is an integer and \(a \leq b\). We label all such pairs in increasing order by their distance from the origin. (It is guaranteed that no ties exist.) Thus \(P_1 = (1, 1)\), \(P_2 = (1, 3)\), and so on. If \(P_{2020} = (x, ... | 52 |
Steve needed to address a letter to $2743$ Becker Road. He remembered the digits of the address but forgot the correct order of the digits, so he wrote them down in random order. The probability that Steve got exactly two of the four digits in their correct positions is $\frac{m}{n}$, where $m$ and $n$ are relatively p... | 5 |
Determine the natural number $a = \frac{p+q}{r} + \frac{q+r}{p} + \frac{r+p}{q}$ where $p$, $q$, and $r$ are prime positive numbers. | 6 |
Let $x$ and $y$ be integers satisfying both $x^2 - 16x + 3y = 20$ and $y^2 + 4y - x = -12$. Find $x + y$. | 18 |
Points $A_1, A_2, \ldots, A_{2022}$ are chosen on a plane such that no three of them are collinear. Consider all angles $A_iA_jA_k$ for distinct points $A_i, A_j, A_k$. What is the largest possible number of these angles that can be equal to $90^\circ$? | 2042220 |
For a positive integer $n$, let $S(n)$ denote the sum of its digits. Find the largest possible value of the expression $\frac{S(n)}{S(16n)}$. | 13 |
How many (possibly empty) sets of lattice points $\{P_1, P_2, \ldots, P_M\}$, where each point $P_i = (x_i, y_i)$ for $x_i, y_i \in \{0, 1, 2, 3, 4, 5, 6\}$, satisfy that the slope of the line $P_iP_j$ is positive for each $1 \le i < j \le M$? An infinite slope, e.g., $P_i$ is vertically above $P_j$, does not count as ... | 3432 |
Let \( f(x) = x^4 - 6x^3 + 26x^2 - 46x + 65 \). Let the roots of \( f(x) \) be \( a_k + ib_k \) for \( k = 1, 2, 3, 4 \). Given that the \( a_k \), \( b_k \) are all integers, find \( |b_1| + |b_2| + |b_3| + |b_4| \). | 10 |
For a rational number $r$, its *period* is the length of the smallest repeating block in its decimal expansion. For example, the number $r = 0.123123123\ldots$ has period $3$. If $S$ denotes the set of all rational numbers of the form $r = \overline{abcdefgh}$ having period $8$, find the sum of all elements in $S$. | 49995000 |
How many real numbers are roots of the polynomial \[x^9 - 37x^8 - 2x^7 + 74x^6 + x^4 - 37x^3 - 2x^2 + 74x\]? | 5 |
There are $N$ students in a class. Each possible nonempty group of students selected a positive integer. All of these integers are distinct and add up to 2014. Compute the greatest possible value of $N$. | 5 |
Let $ABCD$ be an isosceles trapezoid with $AD \parallel BC$. Points $P$ and $Q$ are placed on segments $\overline{CD}$ and $\overline{DA}$ respectively such that $AP \perp CD$ and $BQ \perp DA$, and point $X$ is the intersection of these two altitudes. Suppose that $BX = 3$ and $XQ = 1$. Compute the largest possible ar... | 32 |
Find all natural numbers $n$ for which $2^8 + 2^{11} + 2^n$ is a perfect square. | 12 |
How many non-empty subsets of $\{1,2,\dots,11\}$ are there such that the product of its elements is the cube of an integer? | 19 |
A company of \( n \) soldiers is such that: 1. \( n \) is a palindrome number (reads the same forwards and backwards). 2. When the soldiers arrange in rows of 3, 4, or 5 soldiers, the last row contains 2, 3, and 5 soldiers, respectively. Find the smallest \( n \) satisfying these conditions and prove that there are inf... | 515 |
How many factors of $$(20^{12})^2$$ less than $$20^{12}$$ are not factors of $$20^{12}$$? | 288 |
Let $S=\{1,2,3,\ldots,12\}$. How many subsets of $S$, excluding the empty set, have an even sum but not an even product? | 31 |
For positive integers $i$ and $j$, define $d(i,j)$ as follows: $d(1,j) = 1$, $d(i,1) = 1$ for all $i$ and $j$, and for $i, j > 1$, $d(i,j) = d(i-1,j) + d(i,j-1) + d(i-1,j-1)$. Compute the remainder when $d(3,2016)$ is divided by $1000$. | 481 |
The sequence $a_n$ satisfies the equation $a_{m+n} + a_{m-n} = \frac{1}{2}(a_{2m} + a_{2n})$ for all $m \geq n \geq 0$. Given that $a_1 = 1$, find $a_{1995}$. | 3980025 |
Segment $AB$ of length $13$ is the diameter of a semicircle. Points $C$ and $D$ are located on the semicircle but not on segment $AB$. Segments $AC$ and $BD$ both have length $5$. Given that the length of $CD$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers, find $a + b$. | 132 |
Let $N$ be the number of distinct rearrangements of the 34 letters in "SUPERCALIFRAGILISTICEXPIALIDOCIOUS". How many positive factors does $N$ have? | 3225600 |
Starting with a list of three numbers, the "Make-My-Day" procedure creates a new list by replacing each number with the sum of the other two. For example, from \( \{1, 3, 8\} \) "Make-My-Day" gives \( \{11, 9, 4\} \) and a new "Make-My-Day" leads to \( \{13, 15, 20\} \). If we begin with \( \{20, 1, 8\} \), what is the... | 19 |
Alice starts with the number $0$. She can apply $100$ operations on her number. In each operation, she can either add $1$ to her number, or square her number. After applying all operations, her score is the minimum distance from her number to any perfect square. What is the maximum score she can attain? | 94 |
The integers $1, 2, \dots, 20$ are written on the blackboard. Consider the following operation as one step: choose two integers $a$ and $b$ such that $a - b \ge 2$ and replace them with $a - 1$ and $b + 1$. Determine the maximum number of steps that can be done. | 330 |
A \textit{divisibility chain} is a sequence of positive integers $(a_1, a_2, \ldots, a_n)$ such that $a_k$ divides $a_{k+1}$ for all $1 \le k < n$. Compute the number of divisibility chains of the form $(a, b, a^2, c, a^3, 360^9)$. | 1617000 |
The diagram below shows a large square with each of its sides divided into four equal segments. The shaded square, whose sides are diagonals drawn to these division points, has an area of $13$. Find the area of the large square. | 104 |
Let $r_1$, $r_2$, $r_3$ be the (possibly complex) roots of the polynomial $x^3 + ax^2 + bx + \frac{4}{3}$. How many pairs of integers $a$, $b$ exist such that $r_1^3 + r_2^3 + r_3^3 = 0$? | 3 |
Find the least real number $m$ such that with all $5$ equilateral triangles with a total area of $m$, we can cover an equilateral triangle with side length $1$. | 2 |
The sum of the decimal digits of a natural number $n$ equals 100, and the sum of the digits of $44n$ equals 800. Determine the sum of the digits of $3n$. | 300 |
The altitudes in triangle $ABC$ have lengths 10, 12, and 15. The area of $\triangle ABC$ can be expressed as $\frac{m\sqrt{n}}{p}$, where $m$ and $p$ are relatively prime positive integers and $n$ is a positive integer not divisible by the square of any prime. Find $m + n + p$. | 254 |
A cuboid has an integer volume. Three of the faces have different areas, namely $7$, $27$, and $L$. What is the smallest possible integer value for $L$? | 21 |
Principal Skinner is thinking of two integers $m$ and $n$ and bets Superintendent Chalmers that he will not be able to determine these integers with a single piece of information. Chalmers asks Skinner the numerical value of $mn + 13m + 13n - m^2 - n^2$. From the value of this expression alone, he miraculously determin... | 169 |
We define $\lfloor x \rfloor$ as the largest integer less than or equal to $x$. What is \[ \left\lfloor \frac{5^{2017015}}{5^{2015}+7} \right\rfloor \mod 1000?\] | 0 |
A point $P$ is given in the square $ABCD$ such that $PA=3$, $PB=7$, and $PD=5$. Find the area of the square. | 58 |
Alice has three daughters, each of whom has two daughters. Each of Alice's six grand-daughters has one daughter. How many sets of women from the family of $16$ can be chosen such that no woman and her daughter are both in the set? (Include the empty set as a possible set.) | 2926 |
Triangle $ABC$ has side lengths $AB=65$, $BC=33$, and $AC=56$. Find the radius of the circle that is tangent to sides $AC$ and $BC$, and also tangent to the circumcircle of triangle $ABC$. | 24 |
Evin's calculator is broken and can only perform three operations: 1. Given a number $x$, output $2x$. 2. Given a number $x$, output $4x + 1$. 3. Given a number $x$, output $8x + 3$. Initially starting with the number $0$, determine how many distinct numbers at most $128$ can be generated using these operations. | 82 |
Determine the number of positive integers $n$ satisfying the following conditions: 1. $n < 10^6$ 2. $n$ is divisible by 7 3. $n$ does not contain any of the digits 2, 3, 4, 5, 6, 7, 8. | 104 |
The triangle $ABC$ satisfies $AB=10$ and has angles $\angle A=75^{\circ}$, $\angle B=60^{\circ}$, and $\angle C = 45^{\circ}$. Let $I_A$ be the center of the excircle opposite $A$, and let $D$, $E$ be the circumcenters of triangle $BCI_A$ and $ACI_A$ respectively. If $O$ is the circumcenter of triangle $ABC$, then the ... | 29 |
Suppose that $g$ and $h$ are polynomials of degree $10$ with integer coefficients such that $g(2) < h(2)$ and \[ g(x) h(x) = \sum_{k=0}^{10} \left( \binom{k+11}{k} x^{20-k} - \binom{21-k}{11} x^{k-1} + \binom{21}{11}x^{k-1} \right) \] holds for all nonzero real numbers $x$. Find $g(2)$. | 2047 |
Compute the smallest positive integer that gives a remainder of $1$ when divided by $11$, a remainder of $2$ when divided by $21$, and a remainder of $5$ when divided by $51$. | 1178 |
Consider a sequence $x_1, x_2, \cdots, x_{12}$ of real numbers such that $x_1 = 1$ and for $n = 1, 2, \dots, 10$, let \[ x_{n+2} = \frac{(x_{n+1} + 1)(x_{n+1} - 1)}{x_n}. \] Suppose $x_n > 0$ for $n = 1, 2, \dots, 11$ and $x_{12} = 0$. Then the value of $x_2$ can be written as $\frac{\sqrt{a} + \sqrt{b}}{c}$ for positi... | 622 |
Find the greatest integer $n$ such that $5^n$ divides $2019! - 2018! + 2017!$. | 504 |
Two perpendicular planes intersect a sphere in two circles. These circles intersect at two points, $A$ and $B$, such that $AB = 42$. If the radii of the two circles are $54$ and $66$, find $R^2$, where $R$ is the radius of the sphere. | 6831 |
A pharmaceutical company produces a disease test with the following characteristics: - **Accuracy on infected individuals**: $95\%$ - **Accuracy on non-infected individuals**: $90\%$ The test is applied to a population of mathletes, where $2\%$ of them actually have an infection. If a test result indicates that a mathl... | 136 |
In circle $\Omega$, let $\overline{AB}=65$ be the diameter and let points $C$ and $D$ lie on the same side of arc $\overarc{AB}$ such that $CD=16$, with $C$ closer to $B$ and $D$ closer to $A$. Moreover, let $AD, BC, AC,$ and $BD$ all have integer lengths. Two other circles, circles $\omega_1$ and $\omega_2$, have $\ov... | 961 |
Consider the grid of numbers shown below: \[ \begin{array}{ccccc} 20 & 01 & 96 & 56 & 16 \\ 37 & 48 & 38 & 64 & 60 \\ 96 & 97 & 42 & 20 & 98 \\ 35 & 64 & 96 & 40 & 71 \\ 50 & 58 & 90 & 16 & 89 \\ \end{array} \] Among all paths that start on the top row, move only left, right, and down, and end on the bottom row, what i... | 196 |
Compute the smallest positive integer $N$ for which $N \cdot 2^{2024}$ is a multiple of $2024$. | 253 |
Triangle $ABC$ has side lengths $AB=13$, $BC=14$, and $CA=15$. Let $\Gamma$ denote the circumcircle of $\triangle ABC$. Let $H$ be the orthocenter of $\triangle ABC$. Let $AH$ intersect $\Gamma$ at a point $D$ other than $A$. Let $BH$ intersect $AC$ at $F$ and $\Gamma$ at point $G$ other than $B$. Suppose $DG$ intersec... | 24 |
Points $A$, $B$, $C$, and $D$ lie on a circle $\Gamma$, in that order, with $AB=5$ and $AD=3$. The angle bisector of $\angle ABC$ intersects $\Gamma$ at point $E$ on the opposite side of $\overleftrightarrow{CD}$ as $A$ and $B$. Assume that $\overline{BE}$ is a diameter of $\Gamma$ and $AC=AE$. Compute $DE$. | 4 |
For a natural number $p$, one can move between two points with integer coordinates if the distance between them equals $p$. Find all prime numbers $p$ for which it is possible to reach the point $(2002,38)$ starting from the origin $(0,0)$. | 2 |
Matt has somewhere between $1000$ and $2000$ pieces of paper that he is trying to divide into piles of the same size. He attempts to divide them into $2$, $3$, $4$, $5$, $6$, $7$, and $8$ piles, but each time he ends up with one sheet left over. How many piles does he need? | 41 |
How many functions $f : \{1,2,3,4,5,6\} \to \{1,2,3,4,5,6\}$ have the property that $f(f(x)) + f(x) + x$ is divisible by $3$ for all $x \in \{1,2,3,4,5,6\}$? | 64 |
The figure below has only two sizes for its internal angles. The larger angles are three times the size of the smaller angles. Find the degree measure of one of the larger angles. Note: The figure is a nonagon (9-sided polygon). | 252 |
In $\triangle ABC$ with $AC>AB$, let $D$ be the foot of the altitude from $A$ to side $\overline{BC}$, and let $M$ be the midpoint of side $\overline{AC}$. Let lines $AB$ and $DM$ intersect at a point $E$. If $AC=8$, $AE=5$, and $EM=6$, find the square of the area of $\triangle ABC$. | 252 |
Two robots are programmed to communicate numbers using different bases. The first robot states: "I communicate in base 10, which interestingly is a perfect square. You communicate in base 16, which is not a perfect square." The second robot states: "I find it more interesting that the sum of our bases is the factorial ... | 4 |
There are $22$ chairs around a round table. Find the minimum number $n$ such that for any group of $n$ people sitting at the table, there will always be two people with exactly $2$ or $8$ chairs between them. | 9 |
There are $63$ houses at distances of $1, 2, 3, \ldots, 63$ km from the North Pole, respectively. Santa Claus wants to distribute vaccines to each house. To do so, he will have his $63$ assistants, elves named $E_1, E_2, \ldots, E_{63}$, deliver the vaccine to each house; each elf will deliver the vaccine to exactly on... | 1024 |
Let the rest energy of a particle be $E$. Let the work done to increase the speed of this particle from rest to $v$ be $W$. If $W = \frac{13}{40} E$, then $v = kc$, where $k$ is a constant. Find $10000k$ and round to the nearest whole number. | 6561 |
Let $f: \mathbb{N} \to \mathbb{R}$ be the function \[ f(n) = \sum_{k=1}^{\infty} \frac{1}{\operatorname{lcm}(k,n)^2}. \] It is well-known that $f(1) = \frac{\pi^2}{6}$. What is the smallest positive integer $m$ such that $m \cdot f(10)$ is the square of a rational multiple of $\pi$? | 42 |
Determine the least possible value of $f(1998)$, where $f: \mathbb{N} \to \mathbb{N}$ is a function such that for all $m, n \in \mathbb{N}$, $$f\left( n^{2}f(m)\right) = m\left( f(n)\right)^{2}.$$ | 120 |
A positive integer $n$ is called bad if it cannot be expressed as the product of two distinct positive integers greater than $1$. Find the number of bad positive integers less than $100$. | 30 |
Consider the harmonic sequence $\frac{2017}{4},\frac{2017}{7},\frac{2017}{10},\ldots$, where the reciprocals of the terms of the sequence form an arithmetic sequence. How many terms of this sequence are integers? | 1 |
There are relatively prime positive integers $s$ and $t$ such that \[ \sum_{n=2}^{100}\left(\frac{n}{n^2-1}- \frac{1}{n}\right)=\frac{s}{t} \] Find $s + t$. | 25249 |
Let $p, q, r$ be distinct prime numbers and let \[ A = \{p^a q^b r^c \mid 0 \le a, b, c \le 5\} \] Find the least $n \in \mathbb{N}$ such that for any $B \subset A$ where $|B| = n$, there exist elements $x$ and $y$ such that $x$ divides $y$. | 28 |
Let $S$ be the set of ordered pairs $(x,y)$ of positive integers for which $x+y \le 20$. Evaluate \[ \sum_{(x, y) \in S} (-1)^{x+y}xy. \] | 715 |
An airline company is planning to introduce a network of connections between the ten different airports of Sawubonia. The airports are ranked by priority from first to last (with no ties). We call such a network *feasible* if it satisfies the following conditions: - All connections operate in both directions. - If ther... | 512 |
Find the first digit after the decimal point of the number $\displaystyle \frac{1}{1009} + \frac{1}{1010} + \cdots + \frac{1}{2016}$. | 6 |
An ellipse has foci at $(9, 20)$ and $(49, 55)$ in the $xy$-plane and is tangent to the $x$-axis. What is the length of its major axis? | 85 |
Starting with an empty string, we create a string by repeatedly appending one of the letters $H$, $M$, $T$ with probabilities $\frac{1}{4}$, $\frac{1}{2}$, $\frac{1}{4}$, respectively, until the letter $M$ appears twice consecutively. What is the expected value of the length of the resulting string? | 6 |
Let $ABCD$ be a convex quadrilateral with $AB=5$, $BC=6$, $CD=7$, and $DA=8$. Let $M$, $P$, $N$, $Q$ be the midpoints of sides $AB$, $BC$, $CD$, and $DA$ respectively. Compute $MN^2 - PQ^2$. | 13 |
The diagram below shows circles with radii $1$ and $2$ that are externally tangent to each other and internally tangent to a circle with radius $3$. There are relatively prime positive integers $m$ and $n$ such that a circle with radius $\frac{m}{n}$ is internally tangent to the circle with radius $3$ and externally ta... | 13 |
What is the largest prime factor of $4^9 + 9^4$? | 881 |
The lengths of the three sides of a right triangle form a geometric sequence. The sine of the smallest of the angles in the triangle is $\frac{m+\sqrt{n}}{k}$ where $m$, $n$, and $k$ are integers, and $k$ is not divisible by the square of any prime. Find $m + n + k$. | 6 |
Given that $x$ is a real number, find the minimum value of $f(x) = |x + 1| + 3|x + 3| + 6|x + 6| + 10|x + 10|.$ | 54 |
A positive integer $n \geq 4$ is called *interesting* if there exists a complex number $z$ such that $|z|=1$ and \[1 + z + z^2 + z^{n-1} + z^n = 0.\] Find how many interesting numbers are smaller than $2022$. | 404 |
In the following sequence of numbers, each term, starting with the third, is obtained by adding three times the previous term plus twice the term before that to the previous one: $$a_1, a_2, 78, a_4, a_5, 3438, a_7, a_8, \ldots$$ As seen in the sequence, the third term is $78$ and the sixth term is $3438$. What is the ... | 12246 |
Let $x_0, x_1, \ldots$ be a sequence of real numbers such that $x_n = \frac{1 + x_{n -1}}{x_{n - 2}}$ for $n \geq 2$. Find the number of ordered pairs of positive integers $(x_0, x_1)$ such that the sequence gives $x_{2018} = \frac{1}{1000}$. | 128 |
If $x$ and $y$ are positive integers, and $x^4 + y^4 = 4721$, find all possible values of $x + y$. | 13 |
Let $f(x)$ be a quadratic function with a leading coefficient of $1$. The graph of $f(x)$ is tangent to the lines $y = -5x + 6$ and $y = x - 1$. The sum of the coefficients of $f(x)$ is $\frac{p}{q}$, where $p$ and $q$ are positive, relatively prime integers. Find $100p + q$. | 2509 |
Let $f(x)=\int_0^{x} e^{t} (\cos t+\sin t)\, dt$, $g(x)=\int_0^{x} e^{t} (\cos t-\sin t)\, dt$. For a real number $a$, find: \[ \sum_{n=1}^{\infty} \frac{e^{2a}}{\{f^{(n)}(a)\}^2+\{g^{(n)}(a)\}^2}. \] | 1 |
Two toads named Gamakichi and Gamatatsu are sitting at the points $(0,0)$ and $(2,0)$ respectively. Their goal is to reach $(5,5)$ and $(7,5)$ respectively by making one unit jumps in the positive $x$ or $y$ direction at a time. How many ways can they do this while ensuring that there is no point on the plane where bot... | 19152 |
On a plane, there is an invisible rabbit hiding on a lattice point. We want to place $n$ hunters on some lattice points to catch the rabbit. In each turn, the following steps occur in order: 1. The rabbit jumps to an adjacent lattice point in the top, bottom, left, or right direction. 2. Each hunter moves to an adjacen... | 4 |
In the universe of Pi Zone, points are labeled with $2 \times 2$ arrays of positive reals. One can teleport from point $M$ to point $M'$ if $M$ can be obtained from $M'$ by multiplying either a row or column by some positive real. For example, one can teleport from $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ to $\be... | 17 |
Suppose that $a$ and $b$ are natural numbers such that \[ p = \frac{b}{4} \sqrt{\frac{2a-b}{2a+b}} \] is a prime number. What is the maximum possible value of $p$? | 5 |
Consider a square $ABCD$ with side length $16$. Points $E$ and $F$ are on $CD$ such that $CE = EF = FD$. The lines $BF$ and $AE$ intersect at point $M$. Find the area of triangle $\triangle MAB$. | 96 |
Three lines are drawn parallel to each of the three sides of $\triangle ABC$ such that the three lines intersect in the interior of $\triangle ABC$. The resulting three smaller triangles have areas $1$, $4$, and $9$. Find the area of $\triangle ABC$. | 36 |
Let $P(x) = x^3 + x^2 - r^2 x - 2020$ be a polynomial with roots $r, s, t$. What is $P(1)$? | -4038 |
Find the value of $\lfloor 1 \rfloor + \lfloor 1.7 \rfloor + \lfloor 2.4 \rfloor + \lfloor 3.1 \rfloor + \cdots + \lfloor 99 \rfloor$. | 6987 |
Jeffrey writes the numbers $1$ and $100000000 = 10^8$ on the blackboard. Every minute, if $x, y$ are on the board, Jeffrey replaces them with \[ \frac{x + y}{2} \text{ and } 2 \left(\frac{1}{x} + \frac{1}{y}\right)^{-1}. \] After $2017$ minutes, the two numbers are $a$ and $b$. Find $\min(a, b)$ to the nearest integer. | 10000 |
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