prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
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Let $A$ be a subset of $\{1, 2, \dots , 1000000\}$ such that for any $x, y \in A$ with $x \neq y$, we have $xy \notin A$. Determine the maximum possible size of $A$. | 999001 |
What is the remainder when $7^{8^9}$ is divided by $1000$? | 801 |
Let $X$ be the set consisting of twenty positive integers $n, n+2, \ldots, n+38$. Determine the smallest value of $n$ such that any three numbers $a, b, c \in X$, not necessarily distinct, can form the sides of an acute-angled triangle. | 92 |
Let $a$, $b$, and $c$ be positive integers satisfying the following equations: $$a^4 + a^2b^2 + b^4 = 9633$$ $$2a^2 + a^2b^2 + 2b^2 + c^5 = 3605$$ What is the sum of all distinct values of $a + b + c$? | 18 |
Find the number of 12-digit "words" that can be formed from the alphabet \( \{0,1,2,3,4,5,6\} \) if neighboring digits must differ by exactly 2. | 882 |
For each integer $k \ge 2$, the decimal expansions of the numbers $1024, 1024^2, \dots, 1024^k$ are concatenated, in that order, to obtain a number $X_k$. For example, $X_2 = 10241048576$. If \[ \frac{X_n}{1024^n} \] is an odd integer, find the smallest possible value of $n$, where $n \ge 2$ is an integer. | 5 |
In base $-2$ notation, digits are $0$ and $1$ only, and the places increase in powers of $-2$. For example, $11011$ stands for $(-2)^4 + (-2)^3 + (-2)^1 + (-2)^0$ and equals the number $7$ in base $10$. If the decimal number $2019$ is expressed in base $-2$, how many non-zero digits does it contain? | 6 |
If you multiply all positive integer factors of $24$, you get $24^x$. Find $x$. | 4 |
The continent of Trianglandia is an equilateral triangle with a side length of $9$. It is divided into $81$ smaller equilateral triangles, each with a side length of $1$. Each smaller triangle (country) can choose to build a wall along at most one of its $3$ sides. Due to ongoing conflicts, no two adjacent countries ca... | 45 |
Points $A$, $B$, $C$, $D$, and $E$ are on the same plane such that $A, E, C$ lie on a line in that order, $B, E, D$ lie on a line in that order, $AE = 1$, $BE = 4$, $CE = 3$, $DE = 2$, and $\angle AEB = 60^\circ$. Let $AB$ and $CD$ intersect at $P$. The square of the area of quadrilateral $PAED$ can be expressed as $\f... | 967 |
Let $f(x) = x^2 + 4x + 2$. Let $r$ be the difference between the largest and smallest real solutions of the equation $f(f(f(f(x)))) = 0$. Then $r = a^{\frac{p}{q}}$ for some positive integers $a$, $p$, $q$ such that $a$ is square-free and $p,q$ are relatively prime positive integers. Compute $a + p + q$. | 35 |
Navi and Ozna are playing a game where Ozna starts first, and the two take turns making moves. A positive integer is written on the board. A move consists of either: 1. Subtracting any positive integer at most 2015 from it, or 2. If the integer on the board is divisible by $2014$, dividing it by $2014$. The first perso... | 2013 |
Find the least prime number greater than $1000$ that divides $2^{1010} \cdot 23^{2020} + 1$. | 1013 |
Find the sum of all perfect squares that divide $2016$. | 210 |
Pete chooses $1004$ monic quadratic polynomials $f_1, \cdots, f_{1004}$ such that each integer from $0$ to $2007$ is a root of at least one of them. Vasya considers all equations of the form $f_i = f_j$ for $i \neq j$ and computes their roots. For each such root, Pete has to pay Vasya $1$ ruble. Find the least possible... | 0 |
Let $n = (10^{2020} + 2020)^2$. Find the sum of all the digits of $n$. | 25 |
Joshua rolls two dice and records the product of the numbers face up. The probability that this product is composite can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. | 65 |
Initially, five variables are defined: $a_1 = 1, a_2 = 0, a_3 = 0, a_4 = 0, a_5 = 0$. On a turn, Evan can choose an integer $2 \le i \le 5$. Then, the integer $a_{i-1}$ will be added to $a_i$. For example, if Evan initially chooses $i = 2$, then now $a_1 = 1, a_2 = 0 + 1 = 1, a_3 = 0, a_4 = 0, a_5 = 0$. Find the minimu... | 127 |
Let $P(A)$ be the arithmetic mean of all elements of the set $A = \{ a_1, a_2, \ldots, a_n \}$, defined as $P(A) = \frac{1}{n} \sum^{n}_{i=1} a_i$. A subset $B$ of $A$ is called a "balanced subset" if $B$ is non-empty and $P(B) = P(A)$. Given the set $M = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \}$, find the number of all "balanc... | 51 |
Find the number of perfect squares of five digits whose last two digits are equal. | 31 |
There is an isosceles triangle $\triangle ABC$ with $\overline{AB} = \overline{AC} = 5$ and $\overline{BC} = 6$. Points $D$ and $E$ are on $\overline{AC}$ such that $\overline{AD} = 1$ and $\overline{EC} = 2$. If the area of $\triangle BDE$ is $S$, find $15S$. | 72 |
Find the number of ordered pairs $(x, y)$, where $x$ and $y$ are both integers between $1$ and $9$, inclusive, such that the product $x \times y$ ends in the digit $5$. | 9 |
Find the sum of all possible $n$ such that $n$ is a positive integer and there exist real numbers $a, b, c$ such that for every integer $m$, the quantity \( \frac{2013m^3 + am^2 + bm + c}{n} \) is an integer. | 2976 |
The following diagram shows an equilateral triangle $\triangle ABC$ and three other triangles congruent to it. The other three triangles are obtained by sliding copies of $\triangle ABC$ a distance $\frac{1}{8} AB$ along a side of $\triangle ABC$ in the directions from $A$ to $B$, from $B$ to $C$, and from $C$ to $A$. ... | 768 |
Rectangle $R_0$ has sides of lengths $3$ and $4$. Rectangles $R_1$, $R_2$, and $R_3$ are formed such that: - All four rectangles share a common vertex $P$. - For each $n = 1, 2, 3$, one side of $R_n$ is a diagonal of $R_{n-1}$. - For each $n = 1, 2, 3$, the opposite side of $R_n$ passes through a vertex of $R_{n-1}$ su... | 30 |
Across the face of a rectangular post-it note, you idly draw lines that are parallel to its edges. Each time you draw a line, there is a $50\%$ chance it'll be in each direction and you never draw over an existing line or the edge of the post-it note. After a few minutes, you notice that you've drawn 20 lines. What is ... | 116 |
Let $P(x) = a_0 + a_1 x + a_2 x^2 + \cdots + a_n x^n$ be a polynomial where $a_i$ is a non-negative integer for each $i \in \{0, 1, 2, 3, \ldots, n\}$. If $P(1) = 4$ and $P(5) = 136$, what is the value of $P(3)$? | 34 |
A permutation $(a_1, a_2, a_3, \dots, a_{2012})$ of $(1, 2, 3, \dots, 2012)$ is selected at random. If $S$ is the expected value of \[ \sum_{i = 1}^{2012} | a_i - i |, \] then compute the sum of the prime factors of $S$. | 2083 |
What is the sum of all two-digit odd numbers whose digits are all greater than $6$? | 528 |
Find the sum of all values of $a + b$, where $(a, b)$ is an ordered pair of positive integers such that $a^2 + \sqrt{2017 - b^2}$ is a perfect square. | 67 |
Three vertices of a parallelogram are \((2,-4), (-2,8),\) and \((12,7)\). Determine the sum of the three possible x-coordinates of the fourth vertex. | 12 |
Let $N$ be the set of natural numbers. Suppose $f: N \to N$ is a function satisfying the following conditions: 1. $f(mn) = f(m)f(n)$ 2. $f(m) < f(n)$ if $m < n$ 3. $f(2) = 2$ What is the sum of $\sum_{k=1}^{20}f(k)$? | 210 |
Integers $a$, $b$, and $c$ satisfy the equations $a + b - c = 1$ and $a^2 + b^2 - c^2 = -1$. Determine the sum of all possible values of $a^2 + b^2 + c^2$. | 18 |
Non-degenerate quadrilateral $ABCD$ has integer side lengths with $AB = AD$ and $BC = CD$. The angles satisfy $\angle ABC = \angle BCD = \angle CDA$. Given that $AB = 3$ and $B \neq D$, determine the number of possible lengths for $BC$. | 8 |
Call an ordered triple $(a, b, c)$ of integers feral if $b - a$, $c - a$, and $c - b$ are all prime numbers. Find the number of feral triples where $1 \leq a < b < c \leq 20$. | 72 |
Let $a$, $b$, $c$ be positive integers such that $abc + bc + c = 2014$. Find the minimum possible value of $a + b + c$. | 40 |
The set of positive real numbers $x$ that satisfy $2 |x^2 - 9| \le 9 |x|$ is an interval $[m, M]$. Find $10m + M$. | 21 |
There are 8 distinct points on a plane, where no three are collinear. An ant starts at one of the points, then walks in a straight line to each one of the other points, visiting each point exactly once and stopping at the final point. This creates a trail of 7 line segments. What is the maximum number of times the ant ... | 15 |
Let $a$, $b$, $c$, and $d$ be positive real numbers such that \[ a^2 + b^2 - c^2 - d^2 = 0 \] and \[ a^2 - b^2 - c^2 + d^2 = \frac{56}{53}(bc + ad). \] Let $M$ be the maximum possible value of $\frac{ab+cd}{bc+ad}$. If $M$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, find... | 4553 |
Let $ABC$ be a triangle such that $AB = 7$, $BC = 8$, and $CA = 9$. There exists a unique point $X$ such that $XB = XC$ and $XA$ is tangent to the circumcircle of $\triangle ABC$. If $XA = \frac{a}{b}$, where $a$ and $b$ are coprime positive integers, find $a + b$. | 61 |
What is the smallest positive integer $n$ such that $n!$ is divisible by $2012^{2012}$? | 1010527 |
Let $f(x) = x^3 + 3x^2 + 1$. There is a unique line of the form $y = mx + b$ such that $m > 0$ and this line intersects $f(x)$ at three points, $A, B, C$ such that $AB = BC = 2$. Find $\lfloor 100m \rfloor$. | 41 |
Let $ABCD$ be a quadrilateral with $AD = 20$ and $BC = 13$. The area of $\triangle ABC$ is $338$ and the area of $\triangle DBC$ is $212$. Compute the smallest possible perimeter of $ABCD$. | 118 |
Let $O$ be the center of a circle with radius $26$, and let $A$ and $B$ be two distinct points on the circle. Let $M$ be the midpoint of segment $AB$. Consider a point $C$ such that $CO = 34$ and $\angle COM = 15^\circ$. Let $N$ be the midpoint of segment $CO$. Suppose that $\angle ACB = 90^\circ$. Find the length of $... | 7 |
The degree-$6$ polynomial $f$ satisfies the following conditions: - $f(7) - f(1) = 1$, - $f(8) - f(2) = 16$, - $f(9) - f(3) = 81$, - $f(10) - f(4) = 256$, - $f(11) - f(5) = 625$. Compute $f(15) - f(-3)$. | 6723 |
Let $ABCD$ be an isosceles trapezoid with $AD = BC = 15$ such that the distance between its bases $AB$ and $CD$ is $7$. Suppose further that the circles with diameters $\overline{AD}$ and $\overline{BC}$ are tangent to each other. What is the area of the trapezoid? | 105 |
How many numbers $\overline{abcd}$ with different digits satisfy the following property: if we replace the largest digit with the digit $1$, the resulting number is a multiple of $30$? | 162 |
Integers $1, 2, 3, \ldots, n$, where $n > 2$, are written on a board. Two numbers $m, k$ such that $1 < m < n, 1 < k < n$ are removed, and the average of the remaining numbers is found to be $17$. What is the maximum sum of the two removed numbers? | 51 |
Determine the value of \((101 \times 99) - (102 \times 98) + (103 \times 97) - (104 \times 96) + \ldots + (149 \times 51) - (150 \times 50)\). | 1275 |
Right triangle $ABC$ has a right angle at $C$. Point $D$ on side $\overline{AB}$ is the base of the altitude of $\triangle ABC$ from $C$. Point $E$ on side $\overline{BC}$ is the base of the altitude of $\triangle CBD$ from $D$. Given that $\triangle ACD$ has area $48$ and $\triangle CDE$ has area $40$, find the area o... | 200 |
Find the positive integer $n$ such that the least common multiple of $n$ and $n - 30$ is $n + 1320$. | 165 |
Solve in positive real numbers: $n + \lfloor \sqrt{n} \rfloor + \lfloor \sqrt[3]{n} \rfloor = 2014$ | 1958 |
Compute the number of ways to rearrange nine white cubes and eighteen black cubes into a $3 \times 3 \times 3$ cube such that each $1 \times 1 \times 3$ row or column contains exactly one white cube. Note that rotations are considered \textit{distinct}. | 12 |
You are playing a game in which you have $3$ envelopes, each containing a uniformly random amount of money between $0$ and $1000$ dollars. That is, for any real $0 \leq a < b \leq 1000$, the probability that the amount of money in a given envelope is between $a$ and $b$ is $\frac{b-a}{1000}$. At any step, you take an e... | 695 |
Determine the sum of absolute values for the complex roots of \( 20x^8 + 7ix^7 - 7ix + 20 \). | 8 |
In the figure below, how many ways are there to select two squares which do not share an edge? \[ \begin{asy} \text{size}(3\text{cm}); \text{for} \left( \text{int} \ t = -2; \ t \leq 2; \ t=t+1 \right) \{ \text{draw}( \text{shift}((t,0))*\text{unitsquare} ) ; \ \text{if} (t \neq 0) \text{draw}( \text{shift}((0,t))*\tex... | 28 |
Using squares of side 1, a stair-like figure is formed in stages following a specific pattern. For example, the first stage uses 1 square, the second stage uses 5 squares, and so on. Determine the last stage for which the corresponding figure uses less than 2014 squares. | 32 |
Find the number of three-digit positive integers with three distinct digits such that the sum of the digits is an even number. For example, 925 and 824. | 328 |
The equation $166 \times 56 = 8590$ is valid in some base $b \ge 10$ (where $1, 6, 5, 8, 9, 0$ are digits in base $b$). Find the sum of all possible values of $b \ge 10$ that satisfy the equation. | 12 |
Let $\Omega_1$ and $\Omega_2$ be two circles in the plane. Suppose the common external tangent to $\Omega_1$ and $\Omega_2$ has length $2017$, while their common internal tangent has length $2009$. Find the product of the radii of $\Omega_1$ and $\Omega_2$. | 8052 |
Suppose you have $27$ identical unit cubes colored such that $3$ faces adjacent to a vertex are red and the other $3$ are colored blue. Suppose further that you assemble these $27$ cubes randomly into a larger cube with $3$ cubes to an edge (in particular, the orientation of each cube is random). The probability that t... | 53 |
Suppose $x$, $y$, and $z$ are nonzero complex numbers such that $(x+y+z)(x^2+y^2+z^2)=x^3+y^3+z^3$. Compute $$(x+y+z)\left(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\right).$$ | 3 |
Given the sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, \ldots$, find $n$ such that the sum of the first $n$ terms is $2008$ or $2009$. | 1026 |
Let $x=\sqrt{37-20\sqrt{3}}$. Find the value of \(\frac{x^4-9x^3+5x^2-7x+68}{x^2-10x+19}\). | 7 |
Compute the only element of the set \[ \{1, 2, 3, 4, \ldots\} \cap \left\{\frac{404}{r^2-4} \;\bigg| \; r \in \mathbb{Q} \setminus \{-2, 2\}\right\}. \] | 2500 |
In $\triangle ABC$, three lines are drawn parallel to side $BC$, dividing the altitude of the triangle into four equal parts. If the area of the second largest part is $35$, what is the area of the whole $\triangle ABC$? | 112 |
One test is a multiple choice test with $5$ questions, each having $4$ options. There are $2000$ candidates, each choosing only one answer for each question. Find the smallest possible integer $n$ such that a student's answer sheet, labeled $n$, has the following property: The answer sheet contains four sheets, and any... | 5 |
How many possible values are there for the sum $a + b + c + d$ if $a$, $b$, $c$, and $d$ are positive integers and $abcd = 2007$? | 4 |
Twelve friends participate in a tennis tournament, where each friend plays exactly one game against each of the other eleven friends. The winner of each game receives one point, while the loser receives zero points. There are no draws. The final points for the participants are denoted as $B_1, B_2, \ldots, B_{12}$. Det... | 4356 |
Find the greatest value of the positive integer $x$, such that the number \( A = 2^{182} + 4^x + 8^{700} \) is a perfect square. | 2008 |
Anthony denotes in sequence all positive integers divisible by $2$. Bertha denotes in sequence all positive integers divisible by $3$. Claire denotes in sequence all positive integers divisible by $4$. Orderly Dora denotes all numbers written by the other three. She puts them in order by size and does not repeat a numb... | 3026 |
Let $S_n$ be the sum of reciprocal values of non-zero digits of all positive integers up to (and including) $n$. For instance, $S_{13} = \frac{1}{1} + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \frac{1}{6} + \frac{1}{7} + \frac{1}{8} + \frac{1}{9} + \frac{1}{1} + \frac{1}{1} + \frac{1}{1} + \frac{1}{1} + \... | 7 |
Circles $P$, $Q$, and $R$ are externally tangent to one another. The external tangent of circles $P$ and $Q$ that does not intersect circle $R$ intersects circle $P$ at point $P_Q$ and circle $Q$ at point $Q_P$. Points $Q_R$, $R_Q$, $R_P$, and $P_R$ are defined similarly. If the radius of circle $Q$ is $4$ and the line... | 8 |
Joel is rolling a 6-sided die. After his first roll, he can choose to re-roll the die up to 2 more times. If he rerolls strategically to maximize the expected value of the final value the die lands on, the expected value of the final value the die lands on can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relati... | 17 |
On a $5 \times 5$ grid, we randomly place two \emph{cars}, each occupying a single cell and randomly facing one of the four cardinal directions. It is given that the two cars do not start in the same cell. In a \emph{move}, one chooses a car and shifts it one cell forward. The probability that there exists a sequence o... | 1148 |
David draws a $2 \times 2$ grid of squares in chalk on the sidewalk outside NIMO HQ. He then draws one arrow in each square, each pointing in one of the four cardinal directions (north, south, east, west) parallel to the sides of the grid. In how many ways can David draw his arrows such that no two of the arrows are po... | 194 |
Call a set of positive integers "conspiratorial" if no three of them are pairwise relatively prime. What is the largest number of elements in any "conspiratorial" subset of the integers $1$ to $16$? | 11 |
Nine distinct light bulbs are arranged in a circle, each initially turned off. Determine the number of ways to turn on some of these bulbs such that no four consecutive bulbs remain off at the same time. | 367 |
The positive integer $n$ has the following property: if the last three digits of $n$ are removed, the number becomes $\sqrt[3]{n}$. Find $n$. | 32768 |
In the diagram below, square $ABCD$ with side length 23 is cut into nine rectangles by two lines parallel to $\overline{AB}$ and two lines parallel to $\overline{BC}$. The areas of four of these rectangles are indicated in the diagram. Compute the largest possible value for the area of the central rectangle. [asy] size... | 180 |
Let $n = 2^{19} \times 3^{12}$. Let $M$ denote the number of positive divisors of $n^2$ which are less than $n$ but would not divide $n$. What is the number formed by taking the last two digits of $M$ (in the same order)? | 28 |
Consider non-negative real numbers $a, b, c$ satisfying the condition $a^2 + b^2 + c^2 = 2$. Find the maximum value of the following expression: \[ P = \frac{\sqrt{b^2 + c^2}}{3-a} + \frac{\sqrt{c^2 + a^2}}{3-b} + a + b - 2022c \] | 3 |
How many integer pairs $(a, b)$ with $1 < a, b \le 2015$ are there such that $\log_a b$ is an integer? | 2082 |
A T-tetromino is formed by adjoining three unit squares to form a $1 \times 3$ rectangle, and adjoining on top of the middle square a fourth unit square. Determine the least number of unit squares that must be removed from a $202 \times 202$ grid so that it can be tiled using T-tetrominoes. | 4 |
Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. Points $P$, $Q$, and $R$ are chosen on segments $BC$, $CA$, and $AB$, respectively, such that triangles $AQR$, $BPR$, and $CPQ$ have the same perimeter, which is $\frac{4}{5}$ of the perimeter of $\triangle PQR$. What is the perimeter of $\triangle PQR$? | 30 |
The parabolas $y = x^2 + 15x + 32$ and $x = y^2 + 49y + 593$ meet at one point $(x_0, y_0)$. Find $x_0 + y_0$. | -31 |
Let $A = (0,0)$, $B=(-1,-1)$, $C=(x,y)$, and $D=(x+1,y)$, where $x > y$ are positive integers. Suppose points $A$, $B$, $C$, and $D$ lie on a circle with radius $r$. Denote by $r_1$ and $r_2$ the smallest and second smallest possible values of $r$. Compute $r_1^2 + r_2^2$. | 2523 |
Let $A_1A_2A_3 \ldots A_{21}$ be a 21-sided regular polygon inscribed in a circle with center $O$. How many triangles $A_iA_jA_k$, $1 \leq i < j < k \leq 21$, contain the center point $O$ in their interior? | 385 |
For what largest $n$ are there $n$ seven-digit numbers that are successive members of one geometric progression? | 11 |
The sum of the following seven numbers is exactly 19: \[a_1=2.56,\ a_2=2.61,\ a_3=2.65,\ a_4=2.71,\ a_5=2.79,\ a_6=2.82,\ a_7=2.86.\] It is desired to replace each $a_i$ by an integer approximation $A_i$, where $1 \le i \le 7$, so that the sum of the $A_i$'s is also 19 and the maximum of the "errors" $|A_i - a_i|$, den... | 61 |
We say a positive integer $n$ is \emph{metallic} if there is no prime of the form $m^2-n$. What is the sum of the three smallest metallic integers? | 165 |
Let $ABC$ be a triangle. Let the $A$-altitude meet $BC$ at $D$, the $B$-altitude meet $AC$ at $E$, and let $T \neq A$ be the point on the circumcircle of $\triangle ABC$ such that $AT \parallel BC$. Given that $D, E, T$ are collinear, if $BD = 3$ and $AD = 4$, then the area of $\triangle ABC$ can be written as $a + \sq... | 112 |
Triangle $ABC$ is inscribed in a unit circle $\omega$. Let $H$ be its orthocenter and $D$ be the foot of the perpendicular from $A$ to $BC$. Let $\triangle XYZ$ be the triangle formed by drawing the tangents to $\omega$ at $A$, $B$, $C$. If $\overline{AH} = \overline{HD}$ and the side lengths of $\triangle XYZ$ form an... | 11 |
A trapezoid has side lengths $10, 10, 10$, and $22$. Each side of the trapezoid is the diameter of a semicircle. The two semicircles on the parallel sides of the trapezoid face outside the trapezoid, while the other two semicircles face inside, as shown in the figure. The region bounded by these four semicircles has an... | 176 |
Find the number of positive integers $n$ such that $\sqrt{n} + \sqrt{n + 1} < 11$. | 29 |
The coefficients of the polynomial $P(x)$ are nonnegative integers, each less than 100. Given that $P(10) = 331633$ and $P(-10) = 273373$, compute $P(1)$. | 100 |
A child lines up $2020^2$ pieces of bricks in a row and removes bricks whose positions are square numbers (i.e., the 1st, 4th, 9th, 16th, ... bricks). Then, he lines up the remaining bricks again and removes those that are in a 'square position'. This process is repeated until the number of bricks remaining drops below... | 240 |
Let $a_1, a_2, a_3, \ldots$ be an infinite sequence where for all positive integers $i$, $a_i$ is chosen to be a random positive integer between $1$ and $2016$, inclusive. Let $S$ be the set of all positive integers $k$ such that for all positive integers $j < k$, $a_j \neq a_k$. (So $1 \in S$; $2 \in S$ if and only if... | 4030 |
The number of proper subsets of the set $\left\{ x \mid -1 \leq \log_{\frac{1}{x}} 10 < -\frac{1}{2}, x \in \mathbb{Z}_{\geq 0} \right\}$ is ________. | 127 |
Aileen plays badminton where she and her opponent stand on opposite sides of a net and attempt to bat a birdie back and forth over the net. A player wins a point if their opponent fails to bat the birdie over the net. When Aileen is the server (the first player to try to hit the birdie over the net), she wins a point w... | 73 |
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