Olympiad problems

2024 LMT Fall, 29

Tags: guts

Let $P(x)$ be a quartic polynomial with integer coefficients and leading coefficient $1$ such that $P(\sqrt 2+\sqrt 3+\sqrt 6)=0$. Find $P(1)$.

2024 LMT Fall, 15

Tags: guts

Regular hexagon $ABCDEF$ with side length $2$ is inscribed within a sphere of radius $4$. Let point $X$ be on the sphere. Find the maximum value of the volume of the pyramid $ABCDEFX$.

2023 Harvard-MIT Mathematics Tournament, 22

Tags: guts

Let $a_0, a_1, a_2, \ldots$ be an infinite sequence where each term is independently and uniformly at random in the set $\{1, 2, 3, 4\}.$ Define an infinite sequence $b_0, b_1, b_2, \ldots$ recursively by $b_0=1$ and $b_{i+1}=a_i^{b_i}.$ Compute the expected value of the smallest positive integer $k$ such that $b_k \equiv 1 \pmod{5}.$

2024 Harvard-MIT Mathematics Tournament, 10

Tags: guts

Alice, Bob, and Charlie are playing a game with $6$ cards numbered $1$ through $6.$ Each player is dealt $2$ cards uniformly at random. On each player’s turn, they play one of their cards, and the winner is the person who plays the median of the three cards played. Charlie goes last, so Alice and Bob decide to tell their cards to each other, trying to prevent him from winning whenever possible. Compute the probability that Charlie wins regardless.

2023 Harvard-MIT Mathematics Tournament, 12

Tags: guts

The number $770$ is written on a blackboard. Melody repeatedly performs moves, where a move consists of subtracting either $40$ or $41$ from the number on the board. She performs moves until the number is not positive, and then she stops. Let $N$ be the number of sequences of moves that Melody could perform. Suppose $N = a\cdot 2^b$ where $a$ is an odd positive integer and $b$ is a nonnegative integer. Compute $100a+b.$

2024 LMT Fall, 30

Tags: guts

Find \[\sum_{n=1}^{\infty} \frac{\varphi(n)}{(-4)^n-1},\]where $\varphi(n)$ is the number of positive integers $k \le n$ relatively prime to $n$. (Note $\varphi(1)=1$.)

2025 Harvard-MIT Mathematics Tournament, 7

Tags: guts

The number $$\frac{9^9-8^8}{1001}$$ is an integer. Compute the sum of its prime factors.

2024 Harvard-MIT Mathematics Tournament, 2

Tags: guts

Jerry and Neil have a $3$-sided die that rolls the numbers $1,2,$ and $3,$ each with probability $\tfrac{1}{3}.$ Jerry rolls first, then Neil rolls the die repeatedly until his number is at least as large as Jerry's. Compute the probability that Neil's final number is $3.$

2025 Harvard-MIT Mathematics Tournament, 5

Tags: guts

Compute the largest possible radius of a circle contained in the region defined by $|x+|y|| \le 1$ in the coordinate plane.

2023 Harvard-MIT Mathematics Tournament, 7

Tags: guts

Let $\Omega$ be a sphere of radius $4$ and $\Gamma$ be a sphere of radius $2.$ Suppose that the center of $\Gamma$ lies on the surface of $\Omega.$ The intersection of the surfaces of $\Omega$ and $\Gamma$ is a circle. Compute this circle's circumfrence.

2025 Harvard-MIT Mathematics Tournament, 3

Tags: guts

Jacob rolls two fair six-sided dice. If the outcomes of these dice rolls are the same, he rolls a third fair six-sided die. Compute the probability that the sum of the outcomes of all the dice he rolls is even.

2024 Harvard-MIT Mathematics Tournament, 3

Tags: guts

Compute the number of even positive integers $n \le 2024$ such that $1, 2, \ldots, n$ can be split into $\tfrac{n}{2}$ pairs, and the sum of the numbers in each pair is a multiple of $3.$

2025 Harvard-MIT Mathematics Tournament, 18

Tags: guts

Let $f: \{1, 2, 3, \ldots, 9\} \to \{1, 2, 3, \ldots, 9\}$ be a permutation chosen uniformly at random from the $9!$ possible permutations. Compute the expected value of $\underbrace{f(f(\cdots f(f(}_{2025 \ f\text{'s}}1))\cdots )).$

2024 HMNT, 13

Tags: guts

Let $f$ and $g$ be two quadratic polynomials with real coefficients such that the equation $f(g(x)) = 0$ has four distinct real solutions: $112, 131, 146,$ and $a.$ Compute the sum of all possible values of $a.$

2024 HMNT, 4

Tags: guts

The number $17^6$ when written out in base $10$ contains $8$ distinct digits from $1,2,\ldots,9,$ with no repeated digits or zeroes. Compute the missing nonzero digit.

2024 LMT Fall, 3

Tags: guts

Two distinct positive even integers sum to $8$. Find the larger of the two integers.

2024 HMNT, 21

Tags: guts

Two points are chosen independently and uniformly at random from the interior of the $X$-pentomino shown below. Compute the probability that the line segment between these two points lies entirely within the $X$-pentomino. [center] [img] https://cdn.artofproblemsolving.com/attachments/b/1/17565ba86dbc2358f546fa57145a7726d1b0a9.png [/img] [/center]

2025 Harvard-MIT Mathematics Tournament, 14

Tags: guts

A parallelogram $P$ can be folded over a straight line so that the resulting shape is a regular pentagon with side length $1.$ Compute the perimeter of $P.$

2024 HMNT, 16

Tags: guts

Compute $$\frac{2+3+\cdots+100}{1}+\frac{3+4+\cdots+100}{1+2}+\cdots+\frac{100}{1+2+\cdots+99}.$$

2024 Harvard-MIT Mathematics Tournament, 28

Tags: guts

Given that the $32$-digit integer $$64 \ 312 \ 311 \ 692 \ 944 \ 269 \ 609 \ 355 \ 712 \ 372 \ 657$$ is the product of $6$ consecutive primes, compute the sum of these $6$ primes.

2025 Harvard-MIT Mathematics Tournament, 21

Tags: guts

Compute the unique five-digit positive integer $\underline{abcde}$ such that $a \neq 0, c \neq 0,$ and $$\underline{abcde}=(\underline{ab}+\underline{cde})^2.$$

2023 Harvard-MIT Mathematics Tournament, 13

Tags: guts

Suppose $a, b, c,$ and $d$ are pairwise distinct positive perfect squares such that $a^b = c^d.$ Compute the smallest possible value of $a + b + c + d.$

2024 HMNT, 25

Tags: guts

Let $ABC$ be an equilateral triangle. A regular hexagon $PXQYRZ$ of side length $2$ is placed so that $P, Q,$ and $R$ lie on segments $\overline{BC}, \overline{CA},$ and $\overline{AB}$, respectively. If points $A, X,$ and $Y$ are collinear, compute $BC.$

2024 LMT Fall, 16

Tags: guts

A new meme is circling around social media known as the [i]DaDerek Convertible[/i]. The license plate number of the [i]DaDerek Convertible[/i] is such that the product of its nonzero digits times $5$ is equal to itself. Given that its license plate number has less than or equal to $3$ digits and that it has at least one nonzero digit, find the [i]DaDerek Convertible[/i]'s license plate number.

2024 LMT Fall, 6

Tags: guts

Let $P$ be a point in rectangle $ABCD$ such that the area of $PAB$ is $20$ and the area of $PCD$ is $24$. Find the area of $ABCD$.