Found problems: 233
2014 Contests, 2
Find the integer closest to
\[\frac{1}{\sqrt[4]{5^4+1}-\sqrt[4]{5^4-1}}\]
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.
2016 Harvard-MIT Mathematics Tournament, 1
If $a$ and $b$ satisfy the equations $a +\frac1b=4$ and $\frac1a+b=\frac{16}{15}$, determine the product of all possible values of $ab$.
2019 Harvard-MIT Mathematics Tournament, 4
Yannick is playing a game with $100$ rounds, starting with $1$ coin. During each round, there is an $n\%$ chance that he gains an extra coin, where $n$ is the number of coins he has at the beginning of the round. What is the expected number of coins he will have at the end of the game?
2012 Harvard-MIT Mathematics Tournament, 7
Let $\otimes$ be a binary operation that takes two positive real numbers and returns a positive real number. Suppose further that $\otimes$ is continuous, commutative $(a\otimes b=b\otimes a)$, distributive across multiplication $(a\otimes(bc)=(a\otimes b)(a\otimes c))$, and that $2\otimes 2=4$. Solve the equation $x\otimes y=x$ for $y$ in terms of $x$ for $x>1$.
2019 Harvard-MIT Mathematics Tournament, 6
Six unit disks $C_1$, $C_2$, $C_3$, $C_4$, $C_5$, $C_6$ are in the plane such that they don't intersect each other and $C_i$ is tangent to $C_{i+1}$ for $1 \le i \le 6$ (where $C_7 = C_1$). Let $C$ be the smallest circle that contains all six disks. Let $r$ be the smallest possible radius of $C$, and $R$ the largest possible radius. Find $R - r$.
2019 Harvard-MIT Mathematics Tournament, 6
A point $P$ lies at the center of square $ABCD$. A sequence of points $\{P_n\}$ is determined by $P_0 = P$, and given point $P_i$, point $P_{i+1}$ is obtained by reflecting $P_i$ over one of the four lines $AB$, $BC$, $CD$, $DA$, chosen uniformly at random and independently for each $i$. What is the probability that $P_8 = P$?
2013 Harvard-MIT Mathematics Tournament, 15
Tim and Allen are playing a match of [i]tenus[/i]. In a match of [i]tenus[/i], the two players play a series of games, each of which is won by one of the two players. The match ends when one player has won exactly two more games than the other player, at which point the player who has won more games wins the match. In odd-numbered games, Tim wins with probability $3/4$, and in the even-numbered games, Allen wins with probability $3/4$. What is the expected number of games in a match?
2012 Harvard-MIT Mathematics Tournament, 1
Let $f$ be the function such that
\[f(x)=\begin{cases}2x & \text{if }x\leq \frac{1}{2}\\2-2x & \text{if }x>\frac{1}{2}\end{cases}\]
What is the total length of the graph of $\underbrace{f(f(\ldots f}_{2012\text{ }f's}(x)\ldots))$ from $x=0$ to $x=1?$
2013 Harvard-MIT Mathematics Tournament, 1
Let $x$ and $y$ be real numbers with $x>y$ such that $x^2y^2+x^2+y^2+2xy=40$ and $xy+x+y=8$. Find the value of $x$.
2020 Harvest Math Invitational Team Round Problems, HMI Team #1
1. Let $f(n) = n^2+6n+11$ be a function defined on positive integers. Find the sum of the first three prime values $f(n)$ takes on.
[i]Proposed by winnertakeover[/i]
2016 Harvard-MIT Mathematics Tournament, 10
Quadrilateral $ABCD$ satisfies $AB = 8, BC = 5, CD = 17, DA = 10$. Let $E$ be the intersection of $AC$ and $BD$. Suppose $BE : ED = 1 : 2$. Find the area of $ABCD$.
2013 Harvard-MIT Mathematics Tournament, 6
Let triangle $ABC$ satisfy $2BC = AB+AC$ and have incenter $I$ and circumcircle $\omega$. Let $D$ be the intersection of $AI$ and $\omega$ (with $A, D$ distinct). Prove that $I$ is the midpoint of $AD$.
2013 Harvard-MIT Mathematics Tournament, 14
Consider triangle $ABC$ with $\angle A=2\angle B$. The angle bisectors from $A$ and $C$ intersect at $D$, and the angle bisector from $C$ intersects $\overline{AB}$ at $E$. If $\dfrac{DE}{DC}=\dfrac13$, compute $\dfrac{AB}{AC}$.
2016 HMNT, 8
Let $S = \{1, 2, \ldots, 2016\}$, and let $f$ be a randomly chosen bijection from $S$ to itself. Let $n$ be the smallest
positive integer such that $f^{(n)}(1) = 1$, where $f^{(i)}(x) = f(f^{(i-1)}(x))$. What is the expected value of $n$?
2018 PUMaC Combinatorics A, 6
Michael is trying to drive a bus from his home, $(0,0)$, to school, located at $(6,6)$. There are horizontal and vertical roads at every line $x=0,1,\ldots,6$ and $y=0,1,\ldots,6$. The city has placed $6$ roadblocks on lattice point intersections $(x,y)$ with $0\leq x,y \leq 6$. Michael notices that the only path he can take that only goes up and to the right is directly up from $(0,0)$ to $(0,6)$, and then right to $(6,6)$. How many sets of $6$ locations could the city have blocked?
2011 Harvard-MIT Mathematics Tournament, 4
For all real numbers $x$, let \[ f(x) = \frac{1}{\sqrt[2011]{1-x^{2011}}}. \] Evaluate $(f(f(\ldots(f(2011))\ldots)))^{2011}$, where $f$ is applied $2010$ times.
2000 Harvard-MIT Mathematics Tournament, 34
What is the largest $n$ such that $n! + 1$ is a square?
2019 Harvard-MIT Mathematics Tournament, 3
Let $x$ and $y$ be positive real numbers. Define $a = 1 + \tfrac{x}{y}$ and $b = 1 + \tfrac{y}{x}$. If $a^2 + b^2 = 15$, compute $a^3 + b^3$.
2019 Harvard-MIT Mathematics Tournament, 2
Let $\mathbb{N} = \{1, 2, 3, \dots\}$ be the set of all positive integers, and let $f$ be a bijection from $\mathbb{N}$ to $\mathbb{N}$. Must there exist some positive integer $n$ such that $(f(1), f(2), \dots, f(n))$ is a permutation of $(1, 2, \dots, n)$?
2016 HMNT, 22-24
22. Let the function $f : \mathbb{Z} \to \mathbb{Z}$ take only integer inputs and have integer outputs. For any integers $x$ and $y$, f satisfies
$$f(x) + f(y) = f(x + 1) + f(y - 1)$$
If $f(2016) = 6102$ and $f(6102) = 2016$, what is $f(1)$?
23. Let $d$ be a randomly chosen divisor of $2016$. Find the expected value of
$$\frac{d^2}{d^2 + 2016}$$
24. Consider an infinite grid of equilateral triangles. Each edge (that is, each side of a small triangle) is colored one of $N$ colors. The coloring is done in such a way that any path between any two nonadjecent vertices consists of edges with at least two different colors. What is the smallest possible value of $N$?
2016 Harvard-MIT Mathematics Tournament, 4
A positive integer is written on each corner of a square such that numbers on opposite vertices are relatively prime while numbers on adjacent vertices are not relatively prime. What is the smallest possible value of the sum of these $4$ numbers?
2013 Harvard-MIT Mathematics Tournament, 8
Let $x,y$ be complex numbers such that $\dfrac{x^2+y^2}{x+y}=4$ and $\dfrac{x^4+y^4}{x^3+y^3}=2$. Find all possible values of $\dfrac{x^6+y^6}{x^5+y^5}$.
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.
2011 Harvard-MIT Mathematics Tournament, 5
Let $f(x) = x^2 + 6x + c$ for all real number s$x$, where $c$ is some real number. For what values of $c$ does $f(f(x))$ have exactly $3$ distinct real roots?