prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
|---|---|
For $n$ a positive integer, denote by $P(n)$ the product of all positive integer divisors of $n$. Find the smallest $n$ for which \[ P(P(P(n))) > 10^{12} \] | 6 |
A contest consists of six problems, each worth 7 points. For any given problem, a contestant can score either $0$, $1$, or $7$ points. Determine the number of possible total scores a contestant can achieve across all six problems. | 28 |
Let $P$ and $Q$ be points on $AC$ and $AB$, respectively, of triangle $\triangle ABC$ such that $PB=PC$ and $PQ \perp AB$. Suppose $\frac{AQ}{QB}=\frac{AP}{PB}$. Find $\angle CBA$, in degrees. | 90 |
Let $a_1 < a_2 < \cdots < a_k$ denote the sequence of all positive integers between $1$ and $91$ which are relatively prime to $91$, and set $\omega = e^{2\pi i/91}$. Define \[S = \prod_{1\leq q < p\leq k}(\omega^{a_p} - \omega^{a_q}).\] Given that $S$ is a positive integer, compute the number of positive divisors of $... | 1054 |
The side lengths of a triangle are natural number multiples of $7$, and each side is smaller than $40$. How many triangles satisfy these conditions? | 22 |
For an olympiad geometry problem, Tina wants to draw an acute triangle whose angles each measure a multiple of $10^\circ$. She doesn't want her triangle to have any special properties, so none of the angles can measure $30^\circ$ or $60^\circ$, and the triangle should definitely not be isosceles. How many different tri... | 0 |
A positive integer is considered **happy** if: 1. All its digits are different and not $0$. 2. One of its digits is equal to the sum of the other digits. For example, 253 is a **happy** number. How many **happy** numbers are there? | 264 |
A number is called \textit{6-composite} if it has exactly 6 composite factors. What is the 6th smallest 6-composite number? (A number is \textit{composite} if it has a factor not equal to 1 or itself. In particular, 1 is not composite.) | 441 |
Suppose that $20^{21} = 2^a 5^b = 4^c 5^d = 8^e 5^f$ for positive integers $a, b, c, d, e,$ and $f$. Find $\frac{100bdf}{ace}$. | 75 |
Define the two sequences $a_0, a_1, a_2, \ldots$ and $b_0, b_1, b_2, \ldots$ by $a_0 = 3$ and $b_0 = 1$ with the recurrence relations: - $a_{n+1} = 3a_n + b_n$ - $b_{n+1} = 3b_n - a_n$ for all nonnegative integers $n$. Let $r$ and $s$ be the remainders when $a_{32}$ and $b_{32}$ are divided by $31$, respectively. Compu... | 3010 |
The polynomial $P$ is a quadratic with integer coefficients. For every positive integer $n$, the integers $P(n)$ and $P(P(n))$ are relatively prime to $n$. Given that $P(3) = 89$, determine the value of $P(10)$. | 859 |
Let $f$ be a quadratic function that satisfies the following condition. Find the value of $\frac{f(8)-f(2)}{f(2)-f(1)}$. For two distinct real numbers $a$ and $b$, if $f(a) = f(b)$, then $f(a^2 - 6b - 1) = f(b^2 + 8)$. | 13 |
Determine the sum of the positive integers $n$ such that there exist primes $p, q, r$ satisfying $p^n + q^2 = r^2$. | 5 |
Let $M$ be the smallest positive multiple of $2012$ that has $2012$ divisors. Suppose $M$ can be written as $\prod_{k=1}^{n}p_k^{a_k}$, where the $p_k$'s are distinct primes and the $a_k$'s are positive integers. Find $\sum_{k=1}^{n}(p_k + a_k)$. | 1012 |
The four sets $A$, $B$, $C$, and $D$ each have $400$ elements. The intersection of any two of the sets has $115$ elements. The intersection of any three of the sets has $53$ elements. The intersection of all four sets has $28$ elements. How many elements are there in the union of the four sets? | 1094 |
For a natural number $n$, a string $s$ of $n$ binary digits, and a natural number $k \leq n$, define an $n,s,k$-block as a string of $k$ consecutive elements from $s$. We say that two $n,s,k$-blocks, namely, $a_1a_2\ldots a_k$ and $b_1b_2\ldots b_k$, are incompatible if there exists an $i \in \{1,2,\ldots ,k\}$ such th... | 128 |
Find the maximum possible value of $9\sqrt{x} + 8\sqrt{y} + 5\sqrt{z}$ where $x, y,$ and $z$ are positive real numbers satisfying $9x + 4y + z = 128$. | 80 |
Determine all even numbers $n$, $n \in \mathbb{N}$ such that: \[ \frac{1}{d_{1}} + \frac{1}{d_{2}} + \cdots + \frac{1}{d_{k}} = \frac{1620}{1003}, \] where $d_1, d_2, \ldots, d_k$ are all different divisors of $n$. | 2006 |
Find all prime numbers of the form $\frac{1}{11} \cdot \underbrace{11\ldots 1}_{2n \text{ ones}}$, where $n$ is a natural number. | 101 |
Find the sum of all the positive integers that are divisors of either $96$ or $180$. | 770 |
Find the largest positive integer $n$ ($n \ge 3$), such that there exists a convex $n$-gon where the tangent of each interior angle is an integer. | 8 |
Find the smallest positive integer $n$ such that $\sqrt{\frac{1^2 + 2^2 + \ldots + n^2}{n}}$ is an integer. | 337 |
Hong and Song each have a shuffled deck of eight cards, consisting of four red cards and four black cards. On each turn, both players reveal the top two cards of their decks. The possible pairs that can be played are: - Two black cards - Two red cards - One black and one red card Determine the probability that Hong and... | 25150 |
Suppose that $n$ is a positive integer and that $a$ is the integer equal to \( \frac{10^{2n}-1}{3(10^n+1)} \). If the sum of the digits of $a$ is 567, what is the value of $n$? | 189 |
Compute the number of digits in $2015!$. | 5786 |
A set $D$ of positive integers is called indifferent if there are at least two integers in the set, and for any two distinct elements $x, y \in D$, their positive difference $|x-y|$ is also in $D$. Let $M(x)$ be the smallest size of an indifferent set whose largest element is $x$. Compute the sum $M(2) + M(3) + \dots +... | 1257 |
Let $\zeta = \cos \frac{2\pi}{13} + i \sin \frac{2\pi}{13}$. Suppose $a > b > c > d$ are positive integers satisfying: $$|\zeta^a + \zeta^b + \zeta^c + \zeta^d| = \sqrt{3}.$$ Compute the smallest possible value of $1000a + 100b + 10c + d$. | 7521 |
How many positive integers $n < 2549$ are there such that $x^2 + x - n$ has an integer root? | 49 |
Let $\mathbb{N}_0$ be the set of non-negative integers. Consider a triple $(f, a, b)$, where $f$ is a function from $\mathbb{N}_0$ to $\mathbb{N}_0$, and $a, b \in \mathbb{N}_0$ that satisfies the following conditions: - $f(1) = 2$ - $f(a) + f(b) \leq 2\sqrt{f(a)}$ - For all $n > 0$, $f(n) = f(n-1)f(b) + 2n - f(b)$ Fin... | 10201 |
There are $360$ permutations of the letters in $MMATHS$. When ordered alphabetically, starting from $AHMMST$, $MMATHS$ is in the $n$th permutation. What is $n$? | 173 |
Suppose that $P(x)$ is a monic quadratic polynomial satisfying $aP(a) = 20P(20) = 22P(22)$ for some integer $a \neq 20, 22$. Find the minimum possible positive value of $P(0)$. | 20 |
In the triangle $ABC$, $\angle ABC = \angle ACB = 78^\circ$. On the sides $AB$ and $AC$, respectively, the points $D$ and $E$ are chosen such that $\angle BCD = 24^\circ$, $\angle CBE = 51^\circ$. Find the measure of angle $\angle BED$. | 12 |
Let $ABC$ be a triangle such that $AB=\sqrt{10}$, $BC=4$, and $CA=3\sqrt{2}$. Circle $\omega$ has diameter $BC$, with center at $O$. Extend the altitude from $A$ to $BC$ to hit $\omega$ at $P$ and $P'$, where $AP < AP'$. Suppose line $P'O$ intersects $AC$ at $X$. Given that $PX$ can be expressed as $m\sqrt{n}-\sqrt{p}$... | 11 |
Let $a$, $b$, $c$, and $d$ be positive real numbers such that $abcd=17$. Let $m$ be the minimum possible value of \[a^2 + b^2 + c^2 + a(b+c+d) + b(c+d) + cd.\] Compute $\lfloor 17m \rfloor$. | 594 |
What are the last two digits of the number $$(11 + 12 + 13 + \ldots + 2006)^2$$? | 56 |
In $\triangle ABC$, $AB = 13$, $BC = 14$, and $AC = 15$. Draw the circumcircle of $\triangle ABC$, and suppose that the circumcircle has center $O$. Extend $AO$ past $O$ to a point $D$, $BO$ past $O$ to a point $E$, and $CO$ past $O$ to a point $F$ such that $D, E, F$ also lie on the circumcircle. Compute the area of t... | 168 |
Compute the number of sequences of integers $(a_1, \ldots, a_{200})$ such that the following conditions hold: 1. $0 \leq a_1 < a_2 < \cdots < a_{200} \leq 202$. 2. There exists a positive integer $N$ with the following property: for every index $i \in \{1, \ldots, 200\}$, there exists an index $j \in \{1, \ldots, 200\}... | 20503 |
The repeating decimal $2.0151515\ldots$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$. | 199 |
What is the minimal number of operations needed to repaint an entirely white grid $100 \times 100$ to be entirely black, if in one move we can choose $99$ cells from any row or column and change their color? | 200 |
Compute the number of integer solutions $(x, y)$ to the equation $xy - 18x - 35y = 1890$. | 96 |
Two positive integers $m$ and $n$ are both less than $500$ and $\text{lcm}(m,n) = (m-n)^2$. What is the maximum possible value of $m+n$? | 840 |
If $m$, $n$, and $p$ are three different natural numbers, each between $2$ and $9$, what are all the possible integer values of the expression $\frac{m+n+p}{m+n}$? | 2 |
Pick out three numbers from $0, 1, \ldots, 9$, such that their sum is an even number and not less than $10$. We have ________ different ways to pick these numbers. | 51 |
What is the smallest positive integer that cannot be written as the sum of two nonnegative palindromic integers? (An integer is palindromic if the sequence of decimal digits is the same when read backwards.) | 21 |
For each $n \in \mathbb{N}$, let $d_n$ denote the gcd of $n$ and $(2019-n)$. Find the value of $d_1 + d_2 + \cdots + d_{2018} + d_{2019}$. | 6725 |
Several players try out for the USAMTS basketball team, and they all have integer heights and weights when measured in centimeters and pounds, respectively. In addition, they all weigh less in pounds than they are tall in centimeters. All of the players weigh at least $190$ pounds and are at most $197$ centimeters tall... | 128 |
The intersection of two squares with a perimeter of $8$ is a rectangle with a diagonal length of $1$. Given that the distance between the centers of the two squares is $2$, the perimeter of the rectangle can be expressed as $P$. Find $10P$. | 25 |
A board measuring $64$ inches in length and $4$ inches in height is inclined such that the longer side of the board forms a $30^\circ$ angle with the ground. The distance from the ground to the highest point on the board can be expressed in the form $a + b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers, and $c... | 37 |
In the figure below, each side of the rhombus has a length of 5 centimeters. \[ \text{[asy]} \text{import graph;} \text{unitsize}(2.0\text{cm}); \text{real } w = \sqrt{3}; \text{draw}((w, 0) -- (0, 1) -- (-w, 0) -- (0, -1) -- \text{cycle}); \text{filldraw}( \text{Circle}((-0.5, 0), \frac{0.8}{\sqrt{\pi}}), \text{gray} ... | 14 |
Find the sum of all real numbers $x$ such that \( x^2 = 5x + 6\sqrt{x} - 3 \). | 7 |
What is the largest positive integer $n$ such that $10 \times 11 \times 12 \times \ldots \times 50$ is divisible by $10^n$? | 11 |
We call $\triangle ABC$ with centroid $G$ "balanced" on side $AB$ if the foot of the altitude from $G$ onto line $\overline{AB}$ lies between $A$ and $B$. $\triangle XYZ$, with $XY=2023$ and $\angle ZXY=120^\circ$, is balanced on $XY$. What is the maximum value of $XZ$? | 4046 |
Let $n$ be the answer to this problem. Suppose square $ABCD$ has side-length $3$. Then, congruent non-overlapping squares $EHGF$ and $IHJK$ of side-length $\frac{n}{6}$ are drawn such that $A$, $C$, and $H$ are collinear, $E$ lies on $BC$ and $I$ lies on $CD$. Given that $\triangle AJG$ is an equilateral triangle, the ... | 48 |
A positive integer $a > 1$ is given in decimal notation. We copy it twice to obtain a number $b = \overline{aa}$, which is a multiple of $a^2$. Find all possible values of $\frac{b}{a^2}$. | 7 |
Calculate the least integer greater than $5^{(-6)(-5)(-4)...(2)(3)(4)}$. | 2 |
Let $OABC$ be a tetrahedron such that $\angle AOB = \angle BOC = \angle COA = 90^\circ$ and its faces have integral surface areas. If $[OAB] = 20$ and $[OBC] = 14$, find the sum of all possible values of $[OCA][ABC]$. (Here $[\triangle]$ denotes the area of $\triangle$.) | 22200 |
Ryan is altering Brice’s coin so that one side appears twice as often as the other. He randomly decides to weight either heads or tails more heavily, with equal probability. Brice flips this modified coin twice, and it lands on heads both times. We need to find the probability that the coin lands on heads in the next f... | 8 |
Find the largest positive integer $n$ such that $n$ is divisible by all the positive integers less than $\sqrt[3]{n}$. | 420 |
Let $a$ and $b$ be integer solutions to $17a + 6b = 13$. What is the smallest possible positive value for $a - b$? | 17 |
You have a $10 \times 10$ grid of squares. You write a number in each square as follows: you write $1$, $2$, $3$, $\ldots$, $10$ from left to right across the top row, then $11$, $12$, $\ldots$, $20$ across the second row, and so on, ending with a $100$ in the bottom right square. You then write a second number in each... | 10 |
Compute the number of nonempty subsets $S \subseteq \{-10, -9, -8, \ldots, 8, 9, 10\}$ that satisfy $|S| + \min(S) \cdot \max(S) = 0$. | 335 |
There are $1988$ birds in $994$ cages, two in each cage. Every day we change the arrangement of the birds so that no cage contains the same two birds as ever before. What is the greatest possible number of days we can keep doing so? | 1987 |
Let $N$ be the number of ordered pairs of integers $(x, y)$ such that \[ 4x^2 + 9y^2 \le 1000000000. \] Let $a$ be the first digit of $N$ (from the left) and let $b$ be the second digit of $N$. What is the value of $10a + b$? | 52 |
On a 30-question test, Question 1 is worth 1 point, Question 2 is worth 2 points, and so on up to Question 30. David takes the test and afterward finds out he answered nine of the questions incorrectly. However, he was not told which nine were incorrect. What is the highest possible score he could have attained? | 420 |
Isosceles trapezoid $ABCD$ has side lengths $AB = 6$ and $CD = 12$, with $AD = BC$. It is given that $O$, the circumcenter of $ABCD$, lies in the interior of the trapezoid. The extensions of lines $AD$ and $BC$ intersect at $T$. Given that $OT = 18$, the area of $ABCD$ can be expressed as $a + b\sqrt{c}$, where $a$, $b... | 84 |
Let $ABC$ be a triangle with $AB=7$, $BC=9$, and $CA=4$. Let $D$ be the point such that $AB \parallel CD$ and $CA \parallel BD$. Let $R$ be a point within triangle $BCD$. Lines $\ell$ and $m$ going through $R$ are parallel to $CA$ and $AB$ respectively. Line $\ell$ meets $AB$ and $BC$ at $P$ and $P^\prime$ respectively... | 180 |
Given the integers $a, b, c, d, e$ are equal to $1, 2, 3, 4, 5$ in some order, with the condition that no two of $a, b, c, d, e$ are equal to the same integer. Additionally, the following conditions are satisfied: - $b \leq d$ - $c \geq a$ - $a \leq e$ - $b \geq e$ - $d \neq 5$ Determine the value of $a^b + c^d + e$. | 628 |
A village is surrounded by a circular wall with four gates oriented towards the north, south, east, and west. A tree is located outside the village, precisely $16 \, \mathrm{m}$ north of the north gate. This tree is visible on the horizon from a point $48 \, \mathrm{m}$ east of the south gate. Determine the diameter, i... | 48 |
A store offers packages of $12$ pens for $\$10$ and packages of $20$ pens for $\$15$. Using only these two types of packages of pens, find the greatest number of pens $\$173$ can buy at this store. | 224 |
Let $\zeta = e^{2\pi i/99}$ and $\omega = e^{2\pi i/101}$. The polynomial \[ x^{9999} + a_{9998}x^{9998} + \cdots + a_1x + a_0 \] has roots $\zeta^m + \omega^n$ for all pairs of integers $(m, n)$ with $0 \le m < 99$ and $0 \le n < 101$. Compute $a_{9799} + a_{9800} + \cdots + a_{9998}$. | 0 |
Equilateral triangle $\triangle ABC$ has side length $6$. Points $D$ and $E$ lie on $\overline{BC}$ such that $BD = CE$ and $B$, $D$, $E$, $C$ are collinear in that order. Points $F$ and $G$ lie on $\overline{AB}$ such that $\overline{FD} \perp \overline{BC}$, and $GF = GA$. If the minimum possible value of the sum of ... | 14 |
Find the number of positive integers $k \leq 2018$ for which there exist integers $m$ and $n$ such that $k = 2^m + 2^n$. For example, $64 = 2^5 + 2^5$, $65 = 2^0 + 2^6$, and $66 = 2^1 + 2^6$. | 65 |
Let $f(x) = x^3 - 3x + b$ and $g(x) = x^2 + bx - 3$, where $b$ is a real number. What is the sum of all possible values of $b$ for which the equations $f(x) = 0$ and $g(x) = 0$ have a common root? | 0 |
Three distinct real numbers form (in some order) a 3-term arithmetic sequence, and also form (in possibly a different order) a 3-term geometric sequence. Compute the greatest possible value of the common ratio of this geometric sequence. | -2 |
Lazy Linus wants to minimize his amount of laundry over the course of a week (seven days), so he decides to wear only three different T-shirts and three different pairs of pants for the week. However, he doesn't want to look dirty or boring, so he decides to wear each piece of clothing for either two or three (possibly... | 90720 |
Of the thirteen members of the volunteer group, Hannah selects herself, Tom Morris, Jerry Hsu, Thelma Paterson, and Louise Bueller to teach the September classes. When she is done, she decides that it's not necessary to balance the number of female and male teachers with the proportions of girls and boys at the hospita... | 1261 |
In the Bank of Shower, a bored customer lays $n$ coins in a row. Each second, the customer performs "The Process." In The Process, all coins with exactly one neighboring coin heads-up before The Process are placed heads-up (in its initial location), and all other coins are placed tails-up. The customer stops once all c... | 1040 |
Let $LMT$ represent a 3-digit positive integer where $L$ and $M$ are nonzero digits. Suppose that the 2-digit number $MT$ divides $LMT$. Compute the difference between the maximum and minimum possible values of $LMT$. | 880 |
In $\triangle ABC$, $\angle A = 30^{\circ}$ and $AB = AC = 16$ inches. Let $D$ lie on segment $BC$ such that $\frac{DB}{DC} = \frac{2}{3}$. Let $E$ and $F$ be the orthogonal projections of $D$ onto $AB$ and $AC$, respectively. Find $DE + DF$ in inches. | 8 |
Recall that the Fibonacci numbers $F(n)$ are defined by: - $F(0) = 0$ - $F(1) = 1$ - $F(n) = F(n - 1) + F(n - 2)$ for $n \geq 2$. Determine the last digit of $F(2006)$. | 3 |
During a recent campaign for office, a candidate made a tour of a country which we assume lies in a plane. On the first day of the tour, he went east, on the second day he went north, on the third day west, on the fourth day south, on the fifth day east, etc. If the candidate went $\frac{n^2}{2}$ miles on the $n^{\text... | 580 |
Let $S$ be the set of all positive integers between 1 and 2017, inclusive. Suppose that the least common multiple of all elements in $S$ is $L$. Find the number of elements in $S$ that do not divide \( \frac{L}{2016} \). | 44 |
Determine the number of real solutions $a$ to the equation: \[ \left\lfloor \frac{1}{2}a \right\rfloor + \left\lfloor \frac{1}{3}a \right\rfloor + \left\lfloor \frac{1}{5}a \right\rfloor = a. \] Here, if $x$ is a real number, then $\left\lfloor x \right\rfloor$ denotes the greatest integer that is less than or equal to... | 30 |
Find the largest integer that divides $p^4 - 1$ for all primes $p > 4$. | 48 |
One thousand students participate in the $2011$ Canadian Closed Mathematics Challenge. Each student is assigned a unique three-digit identification number $abc$, where each of $a$, $b$, and $c$ is a digit between $0$ and $9$, inclusive. Later, when the contests are marked, a number of markers will be hired. Each of the... | 50 |
Let $f:\{1,2,\dots,2019\}\to\{-1,1\}$ be a function, such that for every $k\in\{1,2,\dots,2019\}$, there exists an $\ell\in\{1,2,\dots,2019\}$ such that $$ \sum_{i\in\mathbb{Z}:(\ell-i)(i-k)\geqslant 0} f(i)\leqslant 0. $$ Determine the maximum possible value of $$ \sum_{i\in\mathbb{Z}:1\leqslant i\leqslant 2019} f(i).... | 673 |
Five persons of different heights stand next to each other on numbered booths to take a picture. In how many ways can they be arranged such that the people in positions $1$ and $3$ are both taller than the person in position $2$? | 40 |
Bob has $30$ identical unit cubes. He can join two cubes together by gluing a face on one cube to a face on the other cube. He must join all the cubes together into one connected solid. Over all possible solids that Bob can build, what is the largest possible surface area of the solid? | 122 |
Let $ABC$ and $DEF$ be two triangles such that $AB = DE = 20$, $BC = EF = 13$, and $\angle A = \angle D$. If $AC - DF = 10$, determine the area of $\triangle ABC$. | 126 |
In $\triangle ABC$, let point $D$ be on $\overline{BC}$ such that the perimeters of $\triangle ADB$ and $\triangle ADC$ are equal. Let point $E$ be on $\overline{AC}$ such that the perimeters of $\triangle BEA$ and $\triangle BEC$ are equal. Let point $F$ be the intersection of $\overline{AB}$ with the line that passes... | 36 |
Find the remainder when $20^{20} + 21^{21} - 21^{20} - 20^{21}$ is divided by 100. | 20 |
On a blackboard, a stranger writes the values of $s_7(n)^2$ for $n = 0, 1, \ldots, 7^{20} - 1$, where $s_7(n)$ denotes the sum of digits of $n$ in base $7$. Compute the average value of all the numbers on the board. | 3680 |
For any nonnegative integer $n$, let $S(n)$ be the sum of the digits of $n$. Let $K$ be the number of nonnegative integers $n \leq 10^{10}$ that satisfy the equation \[ S(n) = (S(S(n)))^2. \] Find the remainder when $K$ is divided by $1000$. | 632 |
If $a, b, c$ are the roots of the polynomial $x^3 + 20x^2 + 1x + 5$, compute $(a^2 + 1)(b^2 + 1)(c^2 + 1)$. | 225 |
Rachel starts with the number $1000$ in her hands. Each minute, she flips a fair coin. If it lands heads, she adds 1 to the number by placing it in her left pocket, transforming it to $x+1$. If it lands tails, she inverts the number by placing it in her right pocket, transforming it to $x^{-1}$. After eight minutes, th... | 13 |
Suppose that $x$ and $y$ are complex numbers such that $x + y = 1$ and $x^{20} + y^{20} = 20$. Find the sum of all possible values of $x^2 + y^2$. | -90 |
We color each number in the set $S = \{1, 2, \ldots, 61\}$ with one of $25$ given colors. It is not necessary for every color to be used. Let $m$ be the number of non-empty subsets of $S$ such that every number in the subset has the same color. What is the minimum possible value of $m$? | 119 |
An equilateral triangle $ABC$ is divided by nine lines parallel to $BC$ into ten bands that are equally wide. The bands are colored alternately red and blue, with the smallest band colored red. The difference between the total area in red and the total area in blue is $20\, \text{cm}^2$. What is the area of triangle $A... | 200 |
Katherine makes Benj play a game called \(50\) Cent. Benj starts with \(\$0.50\), and every century thereafter has a \(50\%\) chance of doubling his money and a \(50\%\) chance of having his money reset to \(\$0.50\). What is the expected value of the amount of money Benj will have, in dollars, after \(50\) centuries? | 13 |
The roots of the equation $x^3 - x + 1 = 0$ are $a$, $b$, and $c$. Find $a^8 + b^8 + c^8$. | 10 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.