Olympiad problems

2019 Harvard-MIT Mathematics Tournament, 3

For any angle $0 < \theta < \pi/2$, show that \[0 < \sin \theta + \cos \theta + \tan \theta + \cot \theta - \sec \theta - \csc \theta < 1.\]

2013 Harvard-MIT Mathematics Tournament, 6

Tags: geometry , incenter , geometric transformation , hmmt , reflection , analytic geometry , circumcircle

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 $AD$.

2019 Harvard-MIT Mathematics Tournament, 1

Tags: hmmt , algebra

Let $d$ be a real number such that every non-degenerate quadrilateral has at least two interior angles with measure less than $d$ degrees. What is the minimum possible value for $d$?

2019 Harvard-MIT Mathematics Tournament, 5

Tags: hmmt , algebra

Let $a_1, a_2, \dots$ be an arithmetic sequence and $b_1, b_2, \dots$ be a geometric sequence. Suppose that $a_1 b_1 = 20$, $a_2 b_2 = 19$, and $a_3 b_3 = 14$. Find the greatest possible value of $a_4 b_4$.

2020 Harvest Math Invitational Team Round Problems, HMI Team #1

Tags: hmmt

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]

2019 Harvard-MIT Mathematics Tournament, 9

Tags: hmmt , number theory

Let $p > 2$ be a prime number. $\mathbb{F}_p[x]$ is defined as the set of polynomials in $x$ with coefficients in $\mathbb{F}_p$ (the integers modulo $p$ with usual addition and subtraction), so that two polynomials are equal if and only if the coefficients of $x^k$ are equal in $\mathbb{F}_p$ for each nonnegative integer $k$. For example, $(x+2)(2x+3) = 2x^2 + 2x + 1$ in $\mathbb{F}_5[x]$ because the corresponding coefficients are equal modulo 5. Let $f, g \in \mathbb{F}_p[x]$. The pair $(f, g)$ is called [i]compositional[/i] if \[f(g(x)) \equiv x^{p^2} - x\] in $\mathbb{F}_p[x]$. Find, with proof, the number of compositional pairs.

2013 Harvard-MIT Mathematics Tournament, 15

Tags: hmmt , expected value , probability

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?

2014 AMC 10, 19

Tags: probability , hmmt , geometric probability , geometry

Two concentric circles have radii $1$ and $2$. Two points on the outer circle are chosen independently and uniformly at random. What is the probability that the chord joining the two points intersects the inner circle? $\textbf{(A) }\frac{1}{6}\qquad\textbf{(B) }\frac{1}{4}\qquad\textbf{(C) }\frac{2-\sqrt{2}}{2}\qquad\textbf{(D) }\frac{1}{3}\qquad\textbf{(E) }\frac{1}{2}\qquad$

2014 HMNT, 4

Tags: rotation , hmmt , reflection , geometric transformation , geometry

How many ways are there to color the vertices of a triangle red, green, blue, or yellow such that no two vertices have the same color? Rotations and reflections are considered distinct.

2013 Harvard-MIT Mathematics Tournament, 8

Tags: hmmt , distinguishability , counting , probability

In a game, there are three indistinguishable boxes; one box contains two red balls, one contains two blue balls, and the last contains one ball of each color. To play, Raj first predicts whether he will draw two balls of the same color or two of different colors. Then, he picks a box, draws a ball at random, looks at the color, and replaces the ball in the same box. Finally, he repeats this; however, the boxes are not shuffled between draws, so he can determine whether he wants to draw again from the same box. Raj wins if he predicts correctly; if he plays optimally, what is the probability that he will win?

2014 PUMaC Individual Finals A, 2

Tags: algebra , function , hmmt , inequalities , polynomial

Given $a,b,c \in\mathbb{R}^+$, and that $a^2+b^2+c^2=3$. Prove that \[ \frac{1}{a^3+2}+\frac{1}{b^3+2}+\frac{1}{c^3+2}\ge 1 \]

2013 Harvard-MIT Mathematics Tournament, 10

Tags: hmmt , geometry

Wesyu is a farmer, and she's building a cao (a relative of the cow) pasture. Shw starts with a triangle $A_0A_1A_2$ where angle $A_0$ is $90^\circ$, angle $A_1$ is $60^\circ$, and $A_0A_1$ is $1$. She then extends the pasture. FIrst, she extends $A_2A_0$ to $A_3$ such that $A_3A_0=\dfrac12A_2A_0$ and the new pasture is triangle $A_1A_2A_3$. Next, she extends $A_3A_1$ to $A_4$ such that $A_4A_1=\dfrac16A_3A_1$. She continues, each time extending $A_nA_{n-2}$ to $A_{n+1}$ such that $A_{n+1}A_{n-2}=\dfrac1{2^n-2}A_nA_{n-2}$. What is the smallest $K$ such that her pasture never exceeds an area of $K$?

2019 Harvard-MIT Mathematics Tournament, 10

Tags: combinatorics , hmmt

Fred the Four-Dimensional Fluffy Sheep is walking in 4-dimensional space. He starts at the origin. Each minute, he walks from his current position $(a_1, a_2, a_3, a_4)$ to some position $(x_1, x_2, x_3, x_4)$ with integer coordinates satisfying \[(x_1-a_1)^2 + (x_2-a_2)^2 + (x_3-a_3)^2 + (x_4-a_4)^2 = 4 \quad \text{and} \quad |(x_1 + x_2 + x_3 + x_4) - (a_1 + a_2 + a_3 + a_4)| = 2.\] In how many ways can Fred reach $(10, 10, 10, 10)$ after exactly 40 minutes, if he is allowed to pass through this point during his walk?

2016 HMNT, 6

Tags: hmmt

The numbers $1, 2\ldots11$ are arranged in a line from left to right in a random order. It is observed that the middle number is larger than exactly one number to its left. Find the probability that it is larger than exactly one number to its right.

2013 Harvard-MIT Mathematics Tournament, 5

Tags: hmmt

Rahul has ten cards face-down, which consist of five distinct pairs of matching cards. During each move of his game, Rahul chooses one card to turn face-up, looks at it, and then chooses another to turn face-up and looks at it. If the two face-up cards match, the game ends. If not, Rahul flips both cards face-down and keeps repeating this process. Initially, Rahul doesn't know which cards are which. Assuming that he has perfect memory, find the smallest number of moves after which he can guarantee that the game has ended.

2013 Harvard-MIT Mathematics Tournament, 6

Tags: analytic geometry , hmmt , graphing lines , slope , ceiling function , floor function

Find the number of integers $n$ such that \[1+\left\lfloor\dfrac{100n}{101}\right\rfloor=\left\lceil\dfrac{99n}{100}\right\rceil.\]

2020 Harvest Math Invitational Team Round Problems, HMI Team #2

Tags: hmmt , set theory

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, 8

Tags: hmmt , complex number , trigonometry

Let $z = \cos \frac{2\pi}{2011} + i\sin \frac{2\pi}{2011}$, and let \[ P(x) = x^{2008} + 3x^{2007} + 6x^{2006} + \cdots + \frac{2008 \cdot 2009}{2} x + \frac{2009 \cdot 2010}{2} \] for all complex numbers $x$. Evaluate $P(z)P(z^2)P(z^3) \cdots P(z^{2010})$.

2016 Harvard-MIT Mathematics Tournament, 5

Tags: hmmt

Steph Curry is playing the following game and he wins if he has exactly $5$ points at some time. Flip a fair coin. If heads, shoot a $3$-point shot which is worth $3$ points. If tails, shoot a free throw which is worth $1$ point. He makes $\frac12$ of his $3$-point shots and all of his free throws. Find the probability he will win the game. (Note he keeps ﬂipping the coin until he has exactly $5$ or goes over $5$ points)

2016 HMNT, 1-3

Tags: hmmt

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 $1 \times 2 \times \ldots \times n$.

2016 Harvard-MIT Mathematics Tournament, 3

Tags: hmmt , 3d geometry , prism , geometry

Let $V$ be a rectangular prism with integer side lengths. The largest face has area $240$ and the smallest face has area $48$. A third face has area $x$, where $x$ is not equal to $48$ or $240$. What is the sum of all possible values of $x$?

2016 Harvard-MIT Mathematics Tournament, 5

Tags: hmmt

Let the sequence $\{a_i\}^\infty_{i=0}$ be defined by $a_0 =\frac12$ and $a_n = 1 + (a_{n-1} - 1)^2$. Find the product $$\prod_{i=0}^\infty a_i=a_0a_1a_2\ldots$$

2014 NIMO Problems, 8

Tags: analytic geometry , trigonometry , slope , derivative , hmmt , graphing lines , calculus

Triangle $ABC$ lies entirely in the first quadrant of the Cartesian plane, and its sides have slopes $63$, $73$, $97$. Suppose the curve $\mathcal V$ with equation $y=(x+3)(x^2+3)$ passes through the vertices of $ABC$. Find the sum of the slopes of the three tangents to $\mathcal V$ at each of $A$, $B$, $C$. [i]Proposed by Akshaj[/i]

2014 Contests, 3

Tags: hmmt , logarithm

Let \[ A = \frac{1}{6}((\log_2(3))^3-(\log_2(6))^3-(\log_2(12))^3+(\log_2(24))^3) \]. Compute $2^A$.

2013 Harvard-MIT Mathematics Tournament, 31

Tags: geometric transformation , hmmt , geometry , reflection

Let $ABCD$ be a quadrilateral inscribed in a unit circle with center $O$. Suppose that $\angle AOB = \angle COD = 135^\circ$, $BC=1$. Let $B^\prime$ and $C^\prime$ be the reflections of $A$ across $BO$ and $CO$ respectively. Let $H_1$ and $H_2$ be the orthocenters of $AB^\prime C^\prime$ and $BCD$, respectively. If $M$ is the midpoint of $OH_1$, and $O^\prime$ is the reflection of $O$ about the midpoint of $MH_2$, compute $OO^\prime$.