Olympiad problems

2024 Harvard-MIT Mathematics Tournament, 16

Tags: guts

Let $ABC$ be an isosceles triangle with orthocenter $H.$ Let $M$ and $N$ be the midpoints of sides $\overline{AB}$ and $\overline{AC},$ respectively. The circumcircle of triangle $MHN$ intersects line $BC$ at two points $X$ and $Y.$ Given $XY=AB=AC=2,$ compute $BC^2.$

2025 Harvard-MIT Mathematics Tournament, 22

Tags: guts

Let $a,b,$ and $c$ be real numbers such that $a^2(b+1)=1, b^2(c+a)=2,$ and $c^2(a+b)=5.$ Given that there are three possible values for $abc,$ compute the minimum possible value of $abc.$

2024 HMNT, 5

Tags: guts

Let $ABCD$ be a trapezoid with $AB \parallel CD, AB=20, CD=24,$ and area $880.$ Compute the area of the triangle formed by the midpoints of $AB, AC,$ and $BD.$

2024 LMT Fall, 19

Tags: guts

Given $\sum_{n=1}^{\infty} \tfrac{1}{n^2}=\tfrac{\pi^2}{6}$, find$$\sum_{j=1}^{\infty} \sum_{i=1}^j \frac{1}{ij(i+1)(j+1)}.$$

2023 Harvard-MIT Mathematics Tournament, 10

Tags: guts

The number $$316990099009901=\frac{32016000000000001}{101}$$ is the product of two distinct prime numbers. Compute the smaller of these two primes.

2024 Harvard-MIT Mathematics Tournament, 24

Tags: guts

A circle is tangent to both branches of the hyperbola $x^2 - 20y^2 = 24$ as well as the $x$-axis. Compute the area of this circle.

2025 Harvard-MIT Mathematics Tournament, 9

Tags: guts

Let $P$ and $Q$ be points selected uniformly and independently at random inside a regular hexagon $ABCDEF.$ Compute the probability that segment $\overline{PQ}$ is entirely contained in at least one of the quadrilaterals $ABCD,$ $BCDE,$ $CDEF,$ $DEFA,$ $EFAB,$ or $FABC.$

2023 Harvard-MIT Mathematics Tournament, 4

Tags: guts

A [i]standard $n$-sided die[/i] has $n$ sides labeled $1$ to $n.$ Luis, Luke, and Sean play a game in which they roll a fair standard $4$-sided die, a fair standard $6$-sided die, and a fair standard $8$-sided die, respectively. They lose the game if Luis's roll is less than Luke's roll, and Luke's roll is less than Sean's roll. Compute the probability that they lose the game.

2024 LMT Fall, 23

Tags: guts

Define $\overline{a}$ of a positive integer $a$ to be the number $a$ with its digits reversed. For example, $\overline{31564} = 46513.$ Find the sum of all positive integers $n \leq 100$ such that $(\overline{n})^2=\overline{n^2}.$ (Note: For a number that ends with a zero, like 450, the reverse would exclude the zero, so $\overline{450}=54$).

2023 Harvard-MIT Mathematics Tournament, 25

Tags: guts

The [i]spikiness[/i] of a sequence $a_1, a_2, \ldots, a_n$ of at least two real numbers is the sum $\textstyle\sum_{i=1}^{n-1} |a_{i+1}-a_i|.$ Suppose $x_1, x_2, \ldots, x_9$ are chosen uniformly at random from the set $[0, 1].$ Let $M$ be the largest possible value of the spikiness of a permutation of $x_1, x_2, \ldots, x_9.$ Compute the expected value of $M.$

2024 LMT Fall, 26

Tags: guts

Let $P$ be a point in the interior of square $ABCD$ such that $\angle APB+\angle CPD=180^\circ$ and $\angle APB$ $ <$ $\angle CPD$. If $PC=7$ and $PD=5$, find $\tfrac{PA}{PB}$.

2023 Harvard-MIT Mathematics Tournament, 32

Tags: guts

Let $ABC$ be a triangle with $\angle{BAC}>90^\circ.$ Let $D$ be the foot of the perpendicular from $A$ to side $BC.$ Let $M$ and $N$ be the midpoints of segments $BC$ and $BD,$ respectively. Suppose that $AC=2, \angle{BAN}=\angle{MAC},$ and $AB \cdot BC = AM.$ Compute the distance from $B$ to line $AM.$

2025 Harvard-MIT Mathematics Tournament, 26

Tags: guts

Isabella has a bag with $20$ blue diamonds and $25$ purple diamonds. She repeats the following process $44$ times: she removes a diamond from the bag uniformly at random, then puts one blue diamond and one purple diamond into the bag. Compute the expected number of blue diamonds in the bag after all $44$ repetitions.

2024 HMNT, 12

Tags: guts

A dodecahedron is a polyhedron shown on the left below. One of its nets is shown on the right. Compute the label of the face opposite to $\mathcal{P}.$ [center] [img] https://cdn.artofproblemsolving.com/attachments/a/8/7607ee5d199471fd13b09a41a473c71d5d935b.png [/img] [/center]

2025 Harvard-MIT Mathematics Tournament, 20

Tags: guts

Compute the $100$th smallest positive multiple of $7$ whose digits in base $10$ are all strictly less than $3.$

2024 HMNT, 9

Tags: guts

Compute the remainder when $$1002003004005006007008009$$ is divided by $13.$

2024 Harvard-MIT Mathematics Tournament, 22

Tags: guts

Let $x<y$ be positive real numbers such that $$\sqrt{x}+\sqrt{y}=4 \quad \text{and} \quad \sqrt{x+2}+\sqrt{y+2}=5.$$ Compute $x.$

2024 Harvard-MIT Mathematics Tournament, 5

Tags: guts

Let $a,b,$ and $c$ be real numbers such that \begin{align*} a+b+c &= 100 \\ ab+bc+ca &= 20, \text{ and} \\ (a+b)(a+c) &=24. \end{align*} Compute all possible values of $bc.$

2024 HMNT, 14

Tags: guts

Let $ABCD$ be a trapezoid with $AB \parallel CD.$ Point $X$ is placed on segment $BC$ such that $\angle{BAX} = \angle{XDC}.$ Given that $AB = 5, BX =3, CX =4,$ and $CD =12,$ compute $AX.$

2024 HMNT, 30

Tags: guts

Compute the number of ways to shade exactly $4$ distinct cells of a $4\times4$ grid such that no two shaded cells share one or more vertices.

2024 HMNT, 19

Tags: guts

An equilateral triangle is inscribed in a circle $\omega.$ A chord of $\omega$ is cut by the perimeter of the triangle into three segments of lengths $55, 121,$ and $55,$ in that order. Compute the sum of all possible side lengths of the triangle.

2024 Harvard-MIT Mathematics Tournament, 14

Tags: guts

Compute the smallest positive integer such that, no matter how you rearrange its digits (in base ten), the resulting number is a multiple of $63.$

2024 LMT Fall, 28

Tags: guts

Find the number of ways to tile a $2 \times 2 \times 2 \times 2$ four dimensional hypercube with $2 \times 1 \times 1 \times 1$ blocks, with reflections and rotations of the large hypercube distinct.

2025 Harvard-MIT Mathematics Tournament, 24

Tags: guts

For any integer $x,$ let $$f(x)=100!\left(1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots+\frac{x^{100}}{100!}\right).$$ A positive integer $a$ is chosen such that $f(a)-20$ is divisible by $101^2.$ Compute the remainder when $f(x+101)$ is divided by $101^2.$

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.