Found problems: 233
2011 Harvard-MIT Mathematics Tournament, 6
Let $ABCD$ be a cyclic quadrilateral, and suppose that $BC = CD = 2$. Let $I$ be the incenter of triangle $ABD$. If $AI = 2$ as well, find the minimum value of the length of diagonal $BD$.
2013 Harvard-MIT Mathematics Tournament, 11
Compute the prime factorization of $1007021035035021007001$. (You should write your answer in the form $p_1^{e_1}p_2^{e_2}\ldots p_k^{e_k}$ where $p_1,\ldots,p_k$ are distinct prime numbers and $e_1,\ldots,e_k$ are positive integers.)
2016 HMNT, 2
Point $P_1$ is located $600$ miles West of point $P_2$. At $7:00\text{AM}$ a car departs from $P_1$ and drives East at a speed of $50$mph. At $8:00\text{AM}$ another car departs from $P_2$ and drives West at a constant speed of $x$ miles per hour. If the cars meet each other exactly halfway between $P_1$ and $P_2$, what is the value of $x$?
2014 Contests, 2
Find the integer closest to
\[\frac{1}{\sqrt[4]{5^4+1}-\sqrt[4]{5^4-1}}\]
2013 Harvard-MIT Mathematics Tournament, 27
Let $W$ be the hypercube $\{(x_1,x_2,x_3,x_4)\,|\,0\leq x_1,x_2,x_3,x_4\leq 1\}$. The intersection of $W$ and a hyperplane parallel to $x_1+x_2+x_3+x_4=0$ is a non-degenerate $3$-dimensional polyhedron. What is the maximum number of faces of this polyhedron?
2019 Harvard-MIT Mathematics Tournament, 5
Isosceles triangle $ABC$ with $AB = AC$ is inscibed is a unit circle $\Omega$ with center $O$. Point $D$ is the reflection of $C$ across $AB$. Given that $DO = \sqrt{3}$, find the area of triangle $ABC$.
2019 Harvard-MIT Mathematics Tournament, 9
How many ways can you fill a $3 \times 3$ square grid with nonnegative integers such that no [i]nonzero[/i] integer appears more than once in the same row or column and the sum of the numbers in every row and column equals 7?
2016 Harvard-MIT Mathematics Tournament, 8
Let $P_1P_2 \ldots P_8$ be a convex octagon. An integer $i$ is chosen uniformly at random from $1$ to $7$, inclusive. For each vertex of the octagon, the line between that vertex and the vertex $i$ vertices to the right is painted red. What is the expected number times two red lines intersect at a point that is not one of the vertices, given that no three diagonals are concurrent?
2013 Harvard-MIT Mathematics Tournament, 34
For how many unordered sets $\{a,b,c,d\}$ of positive integers, none of which exceed $168$, do there exist integers $w,x,y,z$ such that $(-1)^wa+(-1)^xb+(-1)^yc+(-1)^zd=168$? If your answer is $A$ and the correct answer is $C$, then your score on this problem will be $\left\lfloor25e^{-3\frac{|C-A|}C}\right\rfloor$.
2016 Harvard-MIT Mathematics Tournament, 2
I have five different pairs of socks. Every day for five days, I pick two socks at random without replacement to wear for the day. Find the probability that I wear matching socks on both the third day and the fifth day.
1999 Harvard-MIT Mathematics Tournament, 7
Carl and Bob can demolish a building in 6 days, Anne and Bob can do it in $3$, Anne and Carl in $5$. How many days does it take all of them working together if Carl gets injured at the end of the first day and can't come back?
2016 HMNT, 19-21
19. Let $S$ be the set of all positive integers whose prime factorizations only contain powers of the primes $2$ and $2017$ ($1$, powers of $2$, and powers of $2017$ are thus contained in $S$). Compute $\sum_{s\in S}\frac1s$.
20. Let $\mathcal{V}$ be the volume enclosed by the graph
$$x^ {2016} + y^{2016} + z^2 = 2016$$
Find $\mathcal{V}$ rounded to the nearest multiple of ten.
21. Zlatan has $2017$ socks of various colours. He wants to proudly display one sock of each of the
colours, and he counts that there are $N$ ways to select socks from his collection for display. Given this
information, what is the maximum value of $N$?
2020 Harvest Math Invitational Team Round Problems, HMI Team #2
2. Let $A$ be a set of $2020$ distinct real numbers. Call a number [i]scarily epic[/i] if it can be expressed as the product of two (not necessarily distinct) numbers from $A$. What is the minimum possible number of distinct [i]scarily epic[/i] numbers?
[i]Proposed by Monkey_king1[/i]
2011 Harvard-MIT Mathematics Tournament, 5
Let $ABCDEF$ be a convex equilateral hexagon such that lines $BC$, $AD$, and $EF$ are parallel. Let $H$ be the orthocenter of triangle $ABD$. If the smallest interior angle of the hexagon is $4$ degrees, determine the smallest angle of the triangle $HAD$ in degrees.
2019 Harvard-MIT Mathematics Tournament, 2
Let $N = 2^{\left(2^2\right)}$ and $x$ be a real number such that $N^{\left(N^N\right)} = 2^{(2^x)}$. Find $x$.
2013 Harvard-MIT Mathematics Tournament, 18
Define the sequence of positive integers $\{a_n\}$ as follows. Let $a_1=1$, $a_2=3$, and for each $n>2$, let $a_n$ be the result of expressing $a_{n-1}$ in base $n-1$, then reading the resulting numeral in base $n$, then adding $2$ (in base $n$). For example, $a_2=3_{10}=11_2$, so $a_3=11_3+2_3=6_{10}$. Express $a_{2013}$ in base $10$.
2015 HMIC, 2
Let $m,n$ be positive integers with $m \ge n$. Let $S$ be the set of pairs $(a,b)$ of relatively prime positive integers such that $a,b \le m$ and $a+b > m$.
For each pair $(a,b)\in S$, consider the nonnegative integer solution $(u,v)$ to the equation $au - bv = n$ chosen with $v \ge 0$ minimal, and let $I(a,b)$ denote the (open) interval $(v/a, u/b)$.
Prove that $I(a,b) \subseteq (0,1)$ for every $(a,b)\in S$, and that any fixed irrational number $\alpha\in(0,1)$ lies in $I(a,b)$ for exactly $n$ distinct pairs $(a,b)\in S$.
[i]Victor Wang, inspired by 2013 ISL N7[/i]
2016 HMIC, 4
Let $P$ be an odd-degree integer-coefficient polynomial. Suppose that $xP(x)=yP(y)$ for infinitely many pairs $x,y$ of integers with $x\ne y$. Prove that the equation $P(x)=0$ has an integer root.
[i]Victor Wang[/i]
2012 Harvard-MIT Mathematics Tournament, 8
Let $x_1=y_1=x_2=y_2=1$, then for $n\geq 3$ let $x_n=x_{n-1}y_{n-2}+x_{n-2}y_{n-1}$ and $y_n=y_{n-1}y_{n-2}-x_{n-1}x_{n-2}$. What are the last two digits of $|x_{2012}|?$
2024 Harvard-MIT Mathematics Tournament, 9
Compute the number of triples $(f,g,h)$ of permutations on $\{1,2,3,4,5\}$ such that \begin{align*}
& f(g(h(x))) = h(g(f(x))) = g(x) \\
& g(h(f(x))) = f(h(g(x))) = h(x), \text{ and } \\
& h(f(g(x))) = g(f(h(x))) = f(x), \\
\end{align*} for all $x\in \{1,2,3,4,5\}$.
2016 HMNT, 1-3
1. If five fair coins are flipped simultaneously, what is the probability that at least three of them show heads?
2. How many perfect squares divide $10^{10}$?
3. Evaluate $\frac{2016!^2}{2015!2017!}$ . Here $n!$ denotes $1 \times 2 \times \ldots \times n$.
2019 Harvard-MIT Mathematics Tournament, 10
In triangle $ABC$, $AB = 13$, $BC = 14$, $CA = 15$. Squares $ABB_1A_2$, $BCC_1B_2$, $CAA_1B_2$ are constructed outside the triangle. Squares $A_1A_2A_3A_4$, $B_1B_2B_3B_4$ are constructed outside the hexagon $A_1A_2B_1B_2C_1C_2$. Squares $A_3B_4B_5A_6$, $B_3C_4C_5B_6$, $C_3A_4A_5C_6$ are constructed outside the hexagon $A_4A_3B_4B_3C_4C_3$. Find the area of the hexagon $A_5A_6B_5B_6C_5C_6$.
2023 Harvard-MIT Mathematics Tournament, 5
Let $ABC$ be a triangle with $AB = 13, BC = 14, $and$ CA = 15$. Suppose $PQRS$ is a square such that $P$ and $R$ lie on line $BC, Q$ lies on line $CA$, and $S$ lies on line $AB$. Compute the side length of this square.
2014 Harvard-MIT Mathematics Tournament, 4
Let $b$ and $c$ be real numbers and define the polynomial $P(x)=x^2+bx+c$. Suppose that $P(P(1))=P(P(2))=0$, and that $P(1) \neq P(2)$. Find $P(0)$.
2012 AIME Problems, 14
In a group of nine people each person shakes hands with exactly two of the other people from the group. Let N be the number of ways this handshaking can occur. Consider two handshaking arrangements different if and only if at least two people who shake hands under one arrangement do not shake hands under the other arrangement. Find the remainder when N is divided by 1000.