Found problems: 161
2024 HMNT, 12
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 $\mathcal{P}.$
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2023 Harvard-MIT Mathematics Tournament, 19
Compute the number of ways to select $99$ cells in a $19 \times 19$ square grid such that no two selected cells share an edge or a vertex.
2024 Harvard-MIT Mathematics Tournament, 3
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 $3.$
2024 LMT Fall, 29
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 $P(1)$.
2025 Harvard-MIT Mathematics Tournament, 29
Points $A$ and $B$ lie on circle $\omega$ with center $O.$ Let $X$ be a point inside $\omega.$ Suppose that $XO=2\sqrt{2}, XA=1, XB=3,$ and $\angle{AXB}=90^\circ.$ Points $Y$ and $Z$ are on $\omega$ such that $Y \neq A$ and triangles $\triangle{AXB}$ and $\triangle{YXZ}$ are similar with the same orientation. Compute $XY.$
2023 Harvard-MIT Mathematics Tournament, 15
Let $A$ and $B$ be points in space such that $AB=1.$ Let $\mathcal{R}$ be the region of points $P$ for which $AP \le 1$ and $BP \le 1.$ Compute the largest possible side length of a cube contained in $\mathcal{R}.$
2024 LMT Fall, 14
Find the number of trailing $0$s in the base $12$ expression of $99!$ (Note: $99$ is in base $10$).
2025 Harvard-MIT Mathematics Tournament, 11
Let $f(n)=n^2+100.$ Compute the remainder when $\underbrace{f(f(\cdots f(f(}_{2025 \ f\text{'s}}1))\cdots ))$ is divided by $10^4.$
2024 LMT Fall, 24
Let $ABC$ be a triangle with $AB=13, BC=15, AC=14$. Let $P$ be the point such that $AP$ $=$ $CP$ $=$ $\tfrac12 BP$. Find $AP^2$.
2024 HMNT, 11
A four-digit integer in base $10$ is [i]friendly[/i] if its digits are four consecutive digits in any order. A four-digit integer is [i]shy[/i] if there exist two adjacent digits in its representation that differ by $1.$ Compute the number of four-digit integers that are both friendly and shy.
2023 Harvard-MIT Mathematics Tournament, 11
The Fibonacci numbers are defined recursively by $F_0=0, F_1=1,$ and $F_i=F_{i-1}+F_{i-2}$ for $i \ge 2.$ Given $15$ wooden blocks of weights $F_2, F_3, \ldots, F_{16},$ compute the number of ways to paint each block either red or blue such that the total weight of the red blocks equals the total weight of the blue blocks.
2025 Harvard-MIT Mathematics Tournament, 22
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 $abc.$
2023 Harvard-MIT Mathematics Tournament, 5
If $a$ and $b$ are positive real numbers such that $a \cdot 2^b=8$ and $a^b=2,$ compute $a^{\log_2 a} 2^{b^2}.$
2024 Harvard-MIT Mathematics Tournament, 22
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, 20
Compute $\sqrt[4]{5508^3+5625^3+5742^3},$ given that it is an integer.
2025 Harvard-MIT Mathematics Tournament, 23
Regular hexagon $ABCDEF$ has side length $2.$ Circle $\omega$ lies inside the hexagon and is tangent to segments $\overline{AB}$ and $\overline{AF}.$ There exist two perpendicular lines tangent to $\omega$ that pass through $C$ and $E,$ respectively. Given that these two lines do not intersect on line $AD,$ compute the radius of $\omega.$
2024 LMT Fall, 1
Find the least prime factor of $2024^{2024}-1$.
2024 Harvard-MIT Mathematics Tournament, 30
Let $ABC$ be an equilateral triangle with side length $1.$ Points $D, E,$ and $F$ lie inside triangle $ABC$ such that $A, E, F$ are collinear, $B, F, D$ are collinear, $C, D, E$ are collinear, and triangle $DEF$ is equilateral. Suppose that there exists a unique equilateral triangle $XYZ$ with $X$ on side $\overline{BC},$ $Y$ is on side $\overline{AB},$ and $Z$ is on side $\overline{AC}$ such that $D$ lies on side $\overline{XZ},$ $E$ lies on side $\overline{YZ},$ and $F$ lies on side $\overline{XY}.$ Compute $AZ.$
2024 Harvard-MIT Mathematics Tournament, 18
An ordered pair $(a,b)$ of positive integers is called [i]spicy[/i] if $\gcd(a+b, ab+1)=1.$ Compute the probability that both $(99, n)$ and $(101,n)$ are spicy when $n$ is chosen from $\{1, 2, \ldots, 2024!\}$ uniformly at random.
2025 Harvard-MIT Mathematics Tournament, 27
Compute the number of ordered pairs $(m,n)$ of [i]odd[/i] positive integers both less than $80$ such that $$\gcd(4^m+2^m+1, 4^n+2^n+1)>1.$$
2024 LMT Fall, 20
A base $9$ number [i]probably places[/i] if it has a $7$ as one of its digits. Find the number of base $9$ numbers less than or equal to $100$ in base $10$ that probably place.
2025 Harvard-MIT Mathematics Tournament, 18
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 $\underbrace{f(f(\cdots f(f(}_{2025 \ f\text{'s}}1))\cdots )).$
2025 Harvard-MIT Mathematics Tournament, 2
Compute $$\frac{20+\frac{1}{25-\frac{1}{20}}}{25+\frac{1}{20-\frac{1}{25}}}.$$
2024 Harvard-MIT Mathematics Tournament, 24
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, 17
Let $f$ be a quadratic polynomial with real coefficients, and let $g_1, g_2, g_3, \ldots$ be a geometric progression of real numbers. Define $a_n=f(n)+g_n.$ Given that $a_1, a_2, a_3, a_4,$ and $a_5$ are equal to $1, 2, 3, 14,$ and $16,$ respectively, compute $\tfrac{g_2}{g_1}.$