Olympiad problems

2019 Harvard-MIT Mathematics Tournament, 3

Tags: hmmt , probability

Reimu and Sanae play a game using $4$ fair coins. Initially both sides of each coin are white. Starting with Reimu, they take turns to color one of the white sides either red or green. After all sides are colored, the four coins are tossed. If there are more red sides showing up, then Reimu wins, and if there are more green sides showing up, then Sanae wins. However, if there is an equal number of red sides and green sides, then [i]neither[/i] of them wins. Given that both of them play optimally to maximize the probability of winning, what is the probability that Reimu wins?

2014 Harvard-MIT Mathematics Tournament, 2

Tags: hmmt

Find the integer closest to \[\frac{1}{\sqrt[4]{5^4+1}-\sqrt[4]{5^4-1}}\]

2013 Harvard-MIT Mathematics Tournament, 11

Tags: hmmt , number theory , prime factorization , prime number , binomial coefficient , pascal triangle

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.)

2013 Harvard-MIT Mathematics Tournament, 13

Tags: hmmt , inequalities

Find the smallest positive integer $n$ such that $\dfrac{5^{n+1}+2^{n+1}}{5^n+2^n}>4.99$.

2019 Harvard-MIT Mathematics Tournament, 4

Tags: hmmt , algebra , function

Let $\mathbb{N}$ be the set of positive integers, and let $f: \mathbb{N} \to \mathbb{N}$ be a function satisfying [list] [*] $f(1) = 1$, [*] for $n \in \mathbb{N}$, $f(2n) = 2f(n)$ and $f(2n+1) = 2f(n) - 1$. [/list] Determine the sum of all positive integer solutions to $f(x) = 19$ that do not exceed 2019.

2013 Harvard-MIT Mathematics Tournament, 12

Tags: hmmt , inequalities

For how many integers $1\leq k\leq 2013$ does the decimal representation of $k^k$ end with a $1$?

2009 Harvard-MIT Mathematics Tournament, 3

Tags: hmmt

How many rearrangements of the letters of "$HMMTHMMT$" do not contain the substring "$HMMT$"? (For instance, one such arrangement is $HMMHMTMT$.)

2016 HMNT, 2

Tags: hmmt

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 Harvard-MIT Mathematics Tournament, 4

Tags: hmmt , counting , distinguishability

Find the number of triples of sets $(A, B, C)$ such that: (a) $A, B, C \subseteq \{1, 2, 3, \dots , 8 \}$. (b) $|A \cap B| = |B \cap C| = |C \cap A| = 2$. (c) $|A| = |B| = |C| = 4$. Here, $|S|$ denotes the number of elements in the set $S$.

2013 Harvard-MIT Mathematics Tournament, 18

Tags: hmmt

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 $10$.

2013 Harvard-MIT Mathematics Tournament, 26

Tags: hmmt , geometry , perimeter

Triangle $ABC$ has perimeter $1$. Its three altitudes form the side lengths of a triangle. Find the set of all possible values of $\min(AB,BC,CA)$.

2018 HMNT, 6

Tags: hmmt , combinatorics , geometry

Call a polygon [i]normal[/i] if it can be inscribed in a unit circle. How many non-congruent normal polygons are there such that the square of each side length is a positive integer?

2019 Harvard-MIT Mathematics Tournament, 10

Tags: hmmt , geometry

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 $A_5A_6B_5B_6C_5C_6$.

2019 Harvard-MIT Mathematics Tournament, 1

Tags: hmmt , geometry , analytic geometry

Let $ABCD$ be a parallelogram. Points $X$ and $Y$ lie on segments $AB$ and $AD$ respectively, and $AC$ intersects $XY$ at point $Z$. Prove that \[\frac{AB}{AX} + \frac{AD}{AY} = \frac{AC}{AZ}.\]

2011 Harvard-MIT Mathematics Tournament, 4

Tags: hmmt , geometry , ratio

Let $ABCD$ be a square of side length $13$. Let $E$ and $F$ be points on rays $AB$ and $AD$ respectively, so that the area of square $ABCD$ equals the area of triangle $AEF$. If $EF$ intersects $BC$ at $X$ and $BX = 6$, determine $DF$.

2013 Harvard-MIT Mathematics Tournament, 7

Tags: hmmt , number theory , prime factorization

Find the number of positive divisors $d$ of $15!=15\cdot 14\cdot\cdots\cdot 2\cdot 1$ such that $\gcd(d,60)=5$.

2011 Harvard-MIT Mathematics Tournament, 8

Tags: hmmt , trigonometry , complex number

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 $P(z)P(z^2)P(z^3) \cdots P(z^{2010})$.

2019 Harvard-MIT Mathematics Tournament, 8

Tags: hmmt , combinatorics

For a positive integer $N$, we color the positive divisors of $N$ (including 1 and $N$) with four colors. A coloring is called [i]multichromatic[/i] if whenever $a$, $b$ and $\gcd(a, b)$ are pairwise distinct divisors of $N$, then they have pairwise distinct colors. What is the maximum possible number of multichromatic colorings a positive integer can have if it is not the power of any prime?

2024 Harvard-MIT Mathematics Tournament, 9

Tags: hmmt , combinatorics

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 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 $$\prod_{i=0}^\infty a_i=a_0a_1a_2\ldots$$

2013 Harvard-MIT Mathematics Tournament, 35

Tags: hmmt , floor function , function , geometry , geometric transformation , reflection , logarithm

Let $P$ be the number of ways to partition $2013$ into an ordered tuple of prime numbers. What is $\log_2 (P)$? If your answer is $A$ and the correct answer is $C$, then your score on this problem will be $\left\lfloor\frac{125}2\left(\min\left(\frac CA,\frac AC\right)-\frac35\right)\right\rfloor$ or zero, whichever is larger.

2011 Harvard-MIT Mathematics Tournament, 3

Tags: hmmt , calculus , integration , logarithm

Evaluate $\displaystyle \int_1^\infty \left(\frac{\ln x}{x}\right)^{2011} dx$.

2011 Harvard-MIT Mathematics Tournament, 2

Tags: hmmt

Let $H$ be a regular hexagon of side length $x$. Call a hexagon in the same plane a "distortion" of $H$ if and only if it can be obtained from $H$ by translating each vertex of $H$ by a distance strictly less than $1$. Determine the smallest value of $x$ for which every distortion of $H$ is necessarily convex.

2019 Harvard-MIT Mathematics Tournament, 7

Tags: hmmt , combinatorics , algebra

In an election for the Peer Pressure High School student council president, there are 2019 voters and two candidates Alice and Celia (who are voters themselves). At the beginning, Alice and Celia both vote for themselves, and Alice's boyfriend Bob votes for Alice as well. Then one by one, each of the remaining 2016 voters votes for a candidate randomly, with probabilities proportional to the current number of the respective candidate's votes. For example, the first undecided voter David has a $\tfrac{2}{3}$ probability of voting for Alice and a $\tfrac{1}{3}$ probability of voting for Celia. What is the probability that Alice wins the election (by having more votes than Celia)?

2016 HMIC, 2

Tags: geometry , hmmt

Let $ABC$ be an acute triangle with circumcenter $O$, orthocenter $H$, and circumcircle $\Omega$. Let $M$ be the midpoint of $AH$ and $N$ the midpoint of $BH$. Assume the points $M$, $N$, $O$, $H$ are distinct and lie on a circle $\omega$. Prove that the circles $\omega$ and $\Omega$ are internally tangent to each other. [i]Dhroova Aiylam and Evan Chen[/i]