Olympiad problems

2023 CMIMC Team, 5

Tags: team

$1296$ CMU Students sit in a circle. Every pair of adjacent students rolls a standard six-sided die, and the `score' of any individual student is the sum of their two dice rolls. A 'matched pair' of students is an (unordered) pair of distinct students with the same score. What is the expected value of the number of matched pairs of students? [i]Proposed by Dilhan Salgado[/i]

2019 MOAA, 8

Tags: team , algebra

Suppose that $$\frac{(\sqrt2)^5 + 1}{\sqrt2 + 1} \times \frac{2^5 + 1}{2 + 1} \times \frac{4^5 + 1}{4 + 1} \times \frac{16^5 + 1}{16 + 1} =\frac{m}{7 + 3\sqrt2}$$ for some integer $m$. How many $0$’s are in the binary representation of $m$? (For example, the number $20 = 10100_2$ has three $0$’s in its binary representation.)

2020 CMIMC Team, 1

Tags: team

In a game of ping-pong, the score is $4-10$. Six points later, the score is $10-10$. You remark that it was impressive that I won the previous $6$ points in a row, but I remark back that you have won $n$ points in a row. What the largest value of $n$ such that this statement is true regardless of the order in which the points were distributed?

MOAA Team Rounds, 2021.18

Tags: team

Let $\triangle ABC$ be a triangle with side length $BC= 4\sqrt{6}$. Denote $\omega$ as the circumcircle of $\triangle{ABC}$. Point $D$ lies on $\omega$ such that $AD$ is the diameter of $\omega$. Let $N$ be the midpoint of arc $BC$ that contains $A$. $H$ is the intersection of the altitudes in $\triangle{ABC}$ and it is given that $HN = HD= 6$. If the area of $\triangle{ABC}$ can be expressed as $\frac{a\sqrt{b}}{c}$, where $a,b,c$ are positive integers with $a$ and $c$ relatively prime and $b$ not divisible by the square of any prime, compute $a+b+c$. [i]Proposed by Andy Xu[/i]

2018 MOAA, 4

Tags: combinatorics , team

Michael and Andrew are playing the game Bust, which is played as follows: Michael chooses a positive integer less than or equal to $99$, and writes it on the board. Andrew then makes a move, which consists of him choosing a positive integer less than or equal to $ 8$ and increasing the integer on the board by the integer he chose. Play then alternates in this manner, with each person making exactly one move, until the integer on the board becomes greater than or equal to $100$. The person who made the last move loses. Let S be the sum of all numbers for which Michael could choose initially and win with both people playing optimally. Find S.

MOAA Team Rounds, 2021.11

Tags: team

Find the product of all possible real values for $k$ such that the system of equations $$x^2+y^2= 80$$ $$x^2+y^2= k+2x-8y$$ has exactly one real solution $(x,y)$. [i]Proposed by Nathan Xiong[/i]

2025 CMIMC Team, 2

Tags: team

We are searching for the number $7$ in the following binary tree: [center] [img] https://cdn.artofproblemsolving.com/attachments/8/c/70ad159d239e9fd8dd9775e6391965e1016f03.png [/img] [/center] We use the following algorithm (which terminates with probability $1$): [list=1] [*] Write down the number currently at the root node. [*] If we wrote down $7,$ terminate. [*] Else, pick a random edge, and swap the two numbers at the endpoints of that edge [*] Go back to step $1.$ [/list] Let $p(a)$ be the probability that we ever write down the number $a$ after running the algorithm once. Find $$p(1)+p(2)+p(3)+p(5)+p(6).$$

2024 LMT Fall, 2

Tags: team

Currently, Selena’s analog clock says $4{:}00$. Suddenly her clock breaks, so the hour hand moves $12$ times as fast as it normally does, but the minute hand stays the same speed. Find the degree measure of the smaller angle formed by the minute and the hour hand $2024$ minutes from now.

MOAA Team Rounds, 2021.1

Tags: team

The value of \[\frac{1}{20}-\frac{1}{21}+\frac{1}{20\times 21}\] can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$. [i]Proposed by Nathan Xiong[/i]

2022 CMIMC, 5

Tags: team

For any integer $a$, let $f(a) = |a^4 - 36a^2 + 96a - 64|$. What is the sum of all values of $f(a)$ that are prime? [i]Proposed by Alexander Wang[/i]

2024 HMNT, 8

Tags: team

Compute the unique real numbers $x<3$ such that $$\sqrt{(3-x)(4-x)}+\sqrt{(4-x)(6-x)}+\sqrt{(6-x)(3-x)}=x.$$

2024 HMNT, 3

Tags: team

Rectangle $R$ with area $20$ and diagonal of length $7$ is translated $2$ units in some direction to form a new rectangle $R'.$ The vertices of $R$ and $R'$ that are not contained in the other rectangle form a convex hexagon. Compute the maximum possible area of this hexagon.

2020 CMIMC Team, 14

Tags: team

Let $a_0=1$ and for all $n\ge 1$ let $a_n$ be the smaller root of the equation $$4^{-n}x^2-x+a_{n-1} = 0.$$ Given that $a_n$ approaches a value $L$ as $n$ goes to infinity, what is the value of $L$?

2019 MOAA, 1

Tags: team , geometry

Jeffrey stands on a straight horizontal bridge that measures $20000$ meters across. He wishes to place a pole vertically at the center of the bridge so that the sum of the distances from the top of the pole to the two ends of the bridge is $20001$ meters. To the nearest meter, how long of a pole does Jeffrey need?

2018 CMIMC Team, 8-1/8-2

Tags: team

Let $\triangle ABC$ be a triangle with $AB=3$ and $AC=5$. Select points $D, E,$ and $F$ on $\overline{BC}$ in that order such that $\overline{AD}\perp \overline{BC}$, $\angle BAE=\angle CAE$, and $\overline{BF}=\overline{CF}$. If $E$ is the midpoint of segment $\overline{DF}$, what is $BC^2$? Let $T = TNYWR$, and let $T = 10X + Y$ for an integer $X$ and a digit $Y$. Suppose that $a$ and $b$ are real numbers satisfying $a+\frac1b=Y$ and $\frac{b}a=X$. Compute $(ab)^4+\frac1{(ab)^4}$.

2018 CMIMC Team, 1-1/1-2

Tags: team

Let $ABC$ be a triangle with $BC=30$, $AC=50$, and $AB=60$. Circle $\omega_B$ is the circle passing through $A$ and $B$ tangent to $BC$ at $B$; $\omega_C$ is defined similarly. Suppose the tangent to $\odot(ABC)$ at $A$ intersects $\omega_B$ and $\omega_C$ for the second time at $X$ and $Y$ respectively. Compute $XY$. Let $T = TNYWR$. For some positive integer $k$, a circle is drawn tangent to the coordinate axes such that the lines $x + y = k^2, x + y = (k+1)^2, \dots, x+y = (k+T)^2$ all pass through it. What is the minimum possible value of $k$?

2025 Harvard-MIT Mathematics Tournament, 2

Tags: team

A polyomino is a connected figure constructed by joining one or more unit squares edge-to-edge. Determine, with proof, the number of non-congruent polyominoes with no holes, perimeter $180,$ and area $2024.$

2025 Harvard-MIT Mathematics Tournament, 10

Tags: team

Determine, with proof, all possible values of $\gcd(a^2+b^2+c^2,abc)$ across all triples of positive integers $(a,b,c).$

2024 CMIMC Team, 6

Tags: team

Cyclic quadrilateral $ABCD$ has circumradius $3$. Additionally, $AC = 3\sqrt{2}$, $AB/CD = 2/3$, and $AD = BD$. Find $CD$. [i]Proposed by Justin Hsieh[/i]

2018 MOAA, 8

Tags: algebra , team

Suppose that k and x are positive integers such that $$\frac{k}{2}=\left( \sqrt{1 +\frac{\sqrt3}{2}}\right)^x+\left( \sqrt{1 -\frac{\sqrt3}{2}}\right)^x.$$ Find the sum of all possible values of $k$

2025 Harvard-MIT Mathematics Tournament, 9

Tags: team

Let $\mathbb{Z}$ be the set of integers. Determine, with proof, all primes $p$ for which there exists a function $f:\mathbb{Z}\to\mathbb{Z}$ such that for any integer $x,$ $\quad \bullet \ f(x+p)=f(x)\text{ and}$ $\quad \bullet \ p \text{ divides } f(x+f(x))-x.$

2023 CMIMC Team, 11

Tags: team

A positive integer is [i]detestable[/i] if the sum of its digits is a multiple of $11$. How many positive integers below $10000$ are detestable? [i]Proposed by Giacomo Rizzo[/i]

2017 CMIMC Team, 2

Tags: team

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 \[(x+y+z)\left(\dfrac1x+\dfrac1y+\dfrac1z\right).\]

2025 CMIMC Team, 5

Tags: team

Suppose we have a uniformly random function from $\{1, 2, 3, \ldots, 25\}$ to itself. Find the expected value of $$\sum_{x=1}^{25} (f(f(x))-x)^2.$$

2025 CMIMC Team, 7

Tags: team

The binomial coefficient $\tbinom{n}{k}$ can be defined as the coefficient of $x^k$ in the expansion of $(1+x)^n.$ Similarly, define the trinomial coefficient $\tbinom{n}{k}_3$ as the coefficient of $x^k$ in the expansion of $(1+x+x^2)^n.$ Determine the number of integers $k$ with $0 \le k \le 4048$ such that $\tbinom{2024}{k}_3 \equiv 1 \pmod{3}.$