Found problems: 233
2013 Harvard-MIT Mathematics Tournament, 26
Triangle $ABC$ has perimeter $1$. Its three altitudes form the side lengths of a triangle. Find the set of all possible values of $\min(AB,BC,CA)$.
2019 Harvard-MIT Mathematics Tournament, 9
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.
2019 Harvard-MIT Mathematics Tournament, 4
Convex hexagon $ABCDEF$ is drawn in the plane such that $ACDF$ and $ABDE$ are parallelograms with area 168. $AC$ and $BD$ intersect at $G$. Given that the area of $AGB$ is 10 more than the area of $CGB$, find the smallest possible area of hexagon $ABCDEF$.
2008 Harvard-MIT Mathematics Tournament, 9
([b]7[/b]) Evaluate the limit $ \lim_{n\rightarrow\infty}
n^{\minus{}\frac{1}{2}\left(1\plus{}\frac{1}{n}\right)}
\left(1^1\cdot2^2\cdot\cdots\cdot n^n\right)^{\frac{1}{n^2}}$.
2013 Harvard-MIT Mathematics Tournament, 4
Spencer is making burritos, each of which consists of one wrap and one filling. He has enough filling for up to four beef burritos and three chicken burritos. However, he only has five wraps for the burritos; in how many orders can he make exactly five burritos?
2016 HMNT, 10-12
10. Michael is playing basketball. He makes $10\%$ of his shots, and gets the ball back after $90\%$ of his missed shots. If he does not get the ball back he stops playing. What is the probability that Michael eventually makes a shot?
11. How many subsets $S$ of the set $\{1, 2, \ldots , 10\}$ satisfy the property that, for all $i \in [1, 9]$, either $i$ or $i + 1$ (or both) is in S?
12. A positive integer $\overline{ABC}$, where $A, B, C$ are digits, satisfies $$\overline{ABC} = B^C - A$$
Find $\overline{ABC}$.
2020 Harvard-MIT Mathematics Tournament, 9
Farmer James wishes to cover a circle with circumference $10\pi$ with six different types of colored arcs. Each type of arc has radius $5$, has length either $\pi$ or $2\pi$, and is colored either red, green, or blue. He has an unlimited number of each of the six arc types. He wishes to completely cover his circle without overlap, subject to the following conditions:
[list][*] Any two adjacent arcs are of different colors.
[*] Any three adjacent arcs where the middle arc has length $\pi$ are of three different colors. [/list]
Find the number of distinct ways Farmer James can cover his circle. Here, two coverings are equivalent if and only if they are rotations of one another. In particular, two colorings are considered distinct if they are reflections of one another, but not rotations of one another.
[i]Proposed by James Lin.[/i]
2011 Harvard-MIT Mathematics Tournament, 8
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 $P(z)P(z^2)P(z^3) \cdots P(z^{2010})$.
2013 Harvard-MIT Mathematics Tournament, 5
Thaddeus is given a $2013 \times 2013$ array of integers each between $1$ and $2013$, inclusive. He is allowed two operations:
1. Choose a row, and subtract $1$ from each entry.
2. Chooses a column, and add $1$ to each entry.
He would like to get an array where all integers are divisible by $2013$. On how many arrays is this possible?
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 $a^3 + b^3$.
2013 Harvard-MIT Mathematics Tournament, 32
For an even positive integer $n$ Kevin has a tape of length $4n$ with marks at $-2n,-2n+1,\ldots,2n-1,2n$. He then randomly picks $n$ points in the set $-n,-n+1,-n+2,\ldots,n-1,n$ and places a stone on each of these points. We call a stone 'stuck' if it is on $2n$ or $-2n$, or either all the points to the right, or all the points to the left, all contain stones. Then, every minute, Kevin shifts the unstruck stones in the following manner:
[list]
[*]He picks an unstuck stone uniformly at random and then flips a fair coin.
[*]If the coin came up heads, he then moves that stone and every stone in the largest contiguous set containing that stone one point to the left. If the coin came up tails, he moves every stone in that set one point right instead.
[*]He repeats until all the stones are stuck.[/list]
Let $p_n$ be the probability that at the end of the process there are exactly $k$ stones in the right half. Evaluate \[\dfrac{p_{n-1}-p_{n-2}+p_{n-3}+\ldots+p_3-p_2+p_1}{p_{n-1}+p_{n-2}+p_{n-3}+\ldots+p_3+p_2+p_1}\] in terms of $n$.
2013 Harvard-MIT Mathematics Tournament, 12
For how many integers $1\leq k\leq 2013$ does the decimal representation of $k^k$ end with a $1$?
2016 HMIC, 2
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]
2013 Harvard-MIT Mathematics Tournament, 24
Given a point $p$ and a line segment $l$, let $d(p,l)$ be the distance between them. Let $A$, $B$, and $C$ be points in the plane such that $AB=6$, $BC=8$, $AC=10$. What is the area of the region in the $(x,y)$-plane formed by the ordered pairs $(x,y)$ such that there exists a point $P$ inside triangle $ABC$ with $d(P,AB)+x=d(P,BC)+y=d(P,AC)?$
2019 Harvard-MIT Mathematics Tournament, 8
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?
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 $AD$.
2014 Harvard-MIT Mathematics Tournament, 4
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 $S$.
2011 Harvard-MIT Mathematics Tournament, 3
Let $ABCDEF$ be a regular hexagon of area $1$. Let $M$ be the midpoint of $DE$. Let $X$ be the intersection of $AC$ and $BM$, let $Y$ be the intersection of $BF$ and $AM$, and let $Z$ be the intersection of $AC$ and $BF$. If $[P]$ denotes the area of a polygon $P$ for any polygon $P$ in the plane, evaluate $[BXC] + [AYF] + [ABZ] - [MXZY]$.
2019 Harvard-MIT Mathematics Tournament, 1
What is the smallest positive integer that cannot be written as the sum of two nonnegative palindromic integers? (An integer is [i]palindromic[/i] if the sequence of decimal digits are the same when read backwards.)
2013 Harvard-MIT Mathematics Tournament, 8
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?
2013 Harvard-MIT Mathematics Tournament, 31
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 $OO^\prime$.
2016 Harvard-MIT Mathematics Tournament, 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 $x$?
2000 Stanford Mathematics Tournament, 9
Edward's formula for the stock market predicts correctly that the price of HMMT is directly proportional to a secret quantity $ x$ and inversely proportional to $ y$, the number of hours he slept the night before. If the price of HMMT is $ \$12$ when $ x\equal{}8$ and $ y\equal{}4$, how many dollars does it cost when $ x\equal{}4$ and $ y\equal{}8$?
2016 HMNT, 9
Let the sequence $a_i$ be defined as $a_{i+1} = 2^{a_i}$. Find the number of integers $1 \le n \le 1000$ such that if $a_0 = n$, then $100$ divides $a_{1000} - a_1$.
2016 HMNT, 6
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.