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problem_idx
int64 | problem
string | answer
string | id
string |
|---|---|---|---|
1 | One hundred concentric circles are labelled $C_{1}, C_{2}, C_{3}, \ldots, C_{100}$. Each circle $C_{n}$ is inscribed within an equilateral triangle whose vertices are points on $C_{n+1}$. Given $C_{1}$ has a radius of $1$, what is the radius of $C_{100}$ ?
| 2^{99} | brumo_1 |
2 | An infinite geometric sequence with common ratio $r$ sums to $91$. A new sequence starting with the same term has common ratio $r^{3}$. The sum of the new sequence produced is $81$. What was the common ratio of the original sequence?
| \frac{1}{9} | brumo_2 |
3 | Let $A, B, C, D$, and $E$ be five equally spaced points on a line in that order. Let $F, G, H$, and $I$ all be on the same side of line $A E$ such that triangles $A F B, B G C, C H D$, and $D I E$ are equilateral with side length $1$. Let $S$ be the region consisting of the interiors of all four triangles. Compute the length of segment $A I$ that is contained in $S$. | \frac{\sqrt{13}}{2} | brumo_3 |
4 | If $5 f(x)-x f\left(\frac{1}{x}\right)=\frac{1}{17} x^{2}$, determine $f(3)$. | \frac{1}{9} | brumo_4 |
5 | How many ways are there to arrange $1,2,3,4,5,6$ such that no two consecutive numbers have the same remainder when divided by $3$ ? | 240 | brumo_5 |
6 | Joshua is playing with his number cards. He has $9$ cards of $9$ lined up in a row. He puts a multiplication sign between two of the $9 \mathrm{~s}$ and calculates the product of the two strings of $9 \mathrm{~s}$. For example, one possible result is $999 \times 999999=998999001$. Let $S$ be the sum of all possible distinct results (note that $999 \times 999999$ yields the same result as $999999 \times 999$ ). What is the sum of digits of $S$ ? | 72 | brumo_6 |
7 | Bruno the Bear is tasked to organize $16$ identical brown balls into $7$ bins labeled 1-7. He must distribute the balls among the bins so that each odd-labeled bin contains an odd number of balls, and each even-labeled bin contains an even number of balls (with $0$ considered even). In how many ways can Bruno do this? | 924 | brumo_7 |
8 | Let $f(n)$ be the number obtained by increasing every prime factor in $f$ by one. For instance, $f(12)=(2+1)^{2}(3+1)=36$. What is the lowest $n$ such that $6^{2025}$ divides $f^{(n)}(2025)$, where $f^{(n)}$ denotes the $n$th iteration of $f$ ? | 20 | brumo_8 |
9 | How many positive integer divisors of $63^{10}$ do not end in a $1$ ? | 173 | brumo_9 |
10 | Bruno is throwing a party and invites $n$ guests. Each pair of party guests are either friends or enemies. Each guest has exactly $12$ enemies. All guests believe the following: the friend of an enemy is an enemy. Calculate the sum of all possible values of $n$. (Please note: Bruno is not a guest at his own party) | 100 | brumo_10 |
11 | In acute $\triangle A B C$, let $D$ be the foot of the altitude from $A$ to $B C$ and $O$ be the circumcenter. Suppose that the area of $\triangle A B D$ is equal to the area of $\triangle A O C$. Given that $O D=2$ and $B D=3$, compute $A D$. | 3+2\sqrt{2} | brumo_11 |
12 | Alice has $10$ gifts $g_{1}, g_{2}, \ldots, g_{10}$ and $10$ friends $f_{1}, f_{2}, \ldots, f_{10}$. Gift $g_{i}$ can be given to friend $f_{j}$ if
$$
i-j=-1,0, \text { or } 1 \quad(\bmod 10)
$$
How many ways are there for Alice to pair the $10$ gifts with the $10$ friends such that each friend receives one gift? | 125 | brumo_12 |
13 | Let $\triangle A B C$ be an equilateral triangle with side length $1$. A real number $d$ is selected uniformly at random from the open interval $(0,0.5)$. Points $E$ and $F$ lie on sides $A C$ and $A B$, respectively, such that $A E=d$ and $A F=1-d$. Let $D$ be the intersection of lines $B E$ and $C F$.
Consider line $\ell$ passing through both points of intersection of the circumcircles of triangles $\triangle D E F$ and $\triangle D B C . O$ is the circumcenter of $\triangle D E F$. Line $\ell$ intersects line $\overleftrightarrow{B C}$ at point $P$, and point $Q$ lies on $A P$ such that $\angle A Q B=120^{\circ}$. What is the probability that the line segment $\overline{Q O}$ has length less than $\frac{1}{3}$ ? | \frac{1}{3} | brumo_13 |
14 | Define sequence $\left\{a_{n}\right\}_{n=1}^{\infty}$ such that $a_{1}=\frac{\pi}{3}$ and $a_{n+1}=\cot ^{-1}\left(\csc \left(a_{n}\right)\right)$ for all positive integers $n$. Find the value of
$$
\frac{1}{\cos \left(a_{1}\right) \cos \left(a_{2}\right) \cos \left(a_{3}\right) \cdots \cos \left(a_{16}\right)}
$$ | 7 | brumo_14 |
15 | Define $\{x\}$ to be the fractional part of $x$. For example, $\{20.25\}=0.25$ and $\{\pi\}=\pi-3$. Let $A=\sum_{a=1}^{96} \sum_{n=1}^{96}\left\{\frac{a^{n}}{97}\right\}$, where $\{x\}$ denotes the fractional part of $x$. Compute $A$ rounded to the nearest integer. | 4529 | brumo_15 |
16 | Find the smallest positive integer $n$ such that $n$ is divisible by exactly $25$ different positive integers. | 1296 | brumo_16 |
17 | Two squares, $A B C D$ and $A E F G$, have equal side length $x$. They intersect at $A$ and $O$. Given that $C O=2$ and $O A=2 \sqrt{2}$, what is $x$ ? | 1+\sqrt{3} | brumo_17 |
18 | Bruno and Brutus are running on a circular track with a $20$ foot radius. Bruno completes $5$ laps every hour, while Brutus completes $7$ laps every hour. If they start at the same point but run in opposite directions, how far along the track's circumference (in feet) from the starting point are they when they meet for the sixth time? Note: Do not count the moment they start running as a meeting point. | 20\pi | brumo_18 |
19 | What is the smallest positive integer $n$ such that $z^{n}-1$ and $(z-\sqrt{3})^{n}-1$ share a common complex root? | 12 | brumo_19 |
20 | Consider a pond with lily pads numbered from $1$ to $12$ arranged in a circle. Bruno the frog starts on lily pad 1. Each turn, Bruno has an equal probability of making one of three moves: jumping $4$ lily pads clockwise, jumping $2$ lily pads clockwise, or jumping $1$ lily pad counterclockwise. What is the expected number of turns for Bruno to return to lily pad $1$ for the first time? | 12 | brumo_20 |
21 | $4$ bears - Aruno, Bruno, Cruno and Druno - are each given a card with a positive integer and are told that the sum of their $4$ numbers is $17$. They cannot show each other their cards, but discuss a series of observations in the following order:
Aruno: "I think it is possible that the other three bears all have the same card."
Bruno: "At first, I thought it was possible for the other three bears to have the same card. Now I know it is impossible for them to have the same card."
Cruno: "I think it is still possible that the other three bears have the same card."
Druno: "I now know what card everyone has."
What is the product of their four card values? | 160 | brumo_21 |
22 | Digits $1$ through $9$ are placed on a $3 x 3$ square such that all rows and columns sum to the same value. Please note that diagonals do not need to sum to the same value. How many ways can this be done? | 72 | brumo_22 |
23 | Define the operation $\oplus$ by
$$
x \oplus y=x y-2 x-2 y+6 .
$$
Compute all complex numbers $a$ such that
$$
a \oplus(a \oplus(a \oplus a))=a .
$$ | 2,3,\frac{3+i\sqrt{3}}{2},\frac{3-i\sqrt{3}}{2} | brumo_23 |
24 | Define the function $f$ on positive integers
$$
f(n)= \begin{cases}\frac{n}{2} & \text { if } n \text { is even } \\ n+1 & \text { if } n \text { is odd }\end{cases}
$$
Let $S(n)$ equal the smallest positive integer $k$ such that $f^{k}(n)=1$. How many positive integers satisfy $S(n)=11$ ? | 89 | brumo_24 |
25 | Let $A B C D E F$ be a convex cyclic hexagon. Suppose that $A B=D E=\sqrt{5}, B C=E F=3$, and $C D=F A=\sqrt{20}$. Compute the circumradius of $A B C D E F$. | \frac{1+\sqrt{31}}{2} | brumo_25 |
26 | A repetend is the infinitely repeated digit sequence of a repeating decimal. What are the last three digits of the repetend of the decimal representation of $\frac{1}{727}$, given that the repetend has a length of $726$ ? Express the answer as a three-digit number. Include preceding zeros if there are any. | 337 | brumo_26 |
27 | Consider a $54$-deck of cards, i.e. a standard $52$-card deck together with two jokers. Ada draws cards from the deck until Ada has drawn an ace, a king, and a queen. How many cards does Ada pick up on average? | \frac{737}{39} | brumo_27 |
28 | Let $\omega$ be a circle, and let a line $\ell$ intersect $\omega$ at two points, $P$ and $Q$. Circles $\omega_{1}$ and $\omega_{2}$ are internally tangent to $\omega$ at points $X$ and $Y$, respectively, and both are tangent to $\ell$ at a common point $D$. Similarly, circles $\omega_{3}$ and $\omega_{4}$ are externally tangent to $\omega$ at $X$ and $Y$, respectively, and are tangent to $\ell$ at points $E$ and $F$, respectively.
Given that the radius of $\omega$ is $13$, the segment $\overline{P Q}=24$, and $\overline{Y D}=\overline{Y E}$, find the length of segment $\overline{Y F}$. | 5\sqrt{2} | brumo_28 |
29 | Let $f$ be a degree $7$ polynomial satisfying
$$
f(k)=\frac{1}{k^{2}}
$$
for $k \in\{1 \cdot 2,2 \cdot 3, \ldots, 8 \cdot 9\}$. Find $f(90)-\frac{1}{90^{2}}$. | -\frac{2431}{50} | brumo_29 |
30 | Let $\triangle A B C$ be an isosceles triangle with $A B=A C$. Let $D$ be a point on the circumcircle of $\triangle A B C$ on minor arc $A B$. Let $\overline{A D}$ intersect the extension of $\overline{B C}$ at $E$. Let $F$ be the midpoint of segment $A C$, and let $G$ be the intersection of $\overline{E F}$ and $\overline{A B}$. Let the extension of $\overline{D G}$ intersect $\overline{A C}$ and the circumcircle of $\triangle A B C$ at $H$ and $I$, respectively. Given that $D G=3, G H=5$, and $H I=1$, compute the length of $A E$. | \frac{9\sqrt{30}}{4} | brumo_30 |
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