Found problems: 233
2016 Harvard-MIT Mathematics Tournament, 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 $a_{1000} - a_1$.
2013 Harvard-MIT Mathematics Tournament, 10
Let $N$ be a positive integer whose decimal representation contains $11235$ as a contiguous substring, and let $k$ be a positive integer such that $10^k>N$. Find the minimum possible value of \[\dfrac{10^k-1}{\gcd(N,10^k-1)}.\]
2016 HMNT, 7
Seven lattice points form a convex heptagon with all sides having distinct lengths. Find the minimum possible value of the sum of the squares of the sides of the heptagon.
2019 Harvard-MIT Mathematics Tournament, 7
Let $ABC$ be a triangle with $AB = 13$, $BC = 14$, $CA = 15$. Let $H$ be the orthocenter of $ABC$. Find the radius of the circle with nonzero radius tangent to the circumcircles of $AHB$, $BHC$, $CHA$.
2013 Harvard-MIT Mathematics Tournament, 33
Compute the value of $1^{25}+2^{24}+3^{23}+\ldots+24^2+25^1$. If your answer is $A$ and the correct answer is $C$, then your score on this problem will be $\left\lfloor25\min\left(\left(\frac AC\right)^2,\left(\frac CA\right)^2\right)\right\rfloor$.
2019 Harvard-MIT Mathematics Tournament, 10
Prove that for all positive integers $n$, all complex roots $r$ of the polynomial
\[P(x) = (2n)x^{2n} + (2n-1)x^{2n-1} + \dots + (n+1)x^{n+1} + nx^n + (n+1)x^{n-1} + \dots + (2n-1)x + 2n\]
lie on the unit circle (i.e. $|r| = 1$).
2016 Harvard-MIT Mathematics Tournament, 1
Let $a$ and $b$ be integers (not necessarily positive). Prove that $a^3+5b^3 \neq 2016$.
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?
2016 HMNT, 4
A rectangular pool table has vertices at $(0, 0) (12, 0) (0, 10),$ and $(12, 10)$. There are pockets only in the four corners. A ball is hit from $(0, 0)$ along the line $y = x$ and bounces off several walls before eventually entering a pocket. Find the number of walls that the ball bounces off of before entering a pocket.
2023 Harvard-MIT Mathematics Tournament, 1
Let $ABCDEF$ be a regular hexagon, and let $P$ be a point inside quadrilateral $ABCD$. If the area of triangle $PBC$ is $20$, and the area of triangle $PAD$ is $23$, compute the area of hexagon $ABCDEF$.
2012 AMC 10, 23
A solid tetrahedron is sliced off a solid wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face down. What is the height of this object?
$ \textbf{(A)}\ \dfrac{\sqrt{3}}{3}\qquad\textbf{(B)}\ \dfrac{2\sqrt{2}}{3}\qquad\textbf{(C)}\ 1\qquad\textbf{(D)}\ \dfrac{2\sqrt{3}}{3}\qquad\textbf{(E)}\ \sqrt{2} $
2011 Harvard-MIT Mathematics Tournament, 3
Evaluate $\displaystyle \int_1^\infty \left(\frac{\ln x}{x}\right)^{2011} dx$.
2012 Purple Comet Problems, 27
You have some white one-by-one tiles and some black and white two-bye-one tiles as shown below. There are four different color patterns that can be generated when using these tiles to cover a three-by-one rectangoe by laying these tiles side by side (WWW, BWW, WBW, WWB). How many different color patterns can be generated when using these tiles to cover a ten-by-one rectangle?
[asy]
import graph; size(5cm);
real labelscalefactor = 0.5;
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
draw((12,0)--(12,1)--(11,1)--(11,0)--cycle);
fill((13.49,0)--(13.49,1)--(12.49,1)--(12.49,0)--cycle, black);
draw((13.49,0)--(13.49,1)--(14.49,1)--(14.49,0)--cycle);
draw((15,0)--(15,1)--(16,1)--(16,0)--cycle);
fill((17,0)--(17,1)--(16,1)--(16,0)--cycle, black);
[/asy]
2019 Harvard-MIT Mathematics Tournament, 4
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.
2012 Harvard-MIT Mathematics Tournament, 8
Hexagon $ABCDEF$ has a circumscribed circle and an inscribed circle. If $AB = 9$, $BC = 6$, $CD = 2$, and $EF = 4$. Find $\{DE, FA\}$.
2019 Harvard-MIT Mathematics Tournament, 3
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?
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.
2016 HMNT, 28-30
28. The numbers $1-10$ are written in a circle randomly. Find the expected number of numbers which are at least $2$ larger than an adjacent number.
29. We want to design a new chess piece, the American, with the property that (i) the American can never attack itself, and (ii) if an American $A_1$ attacks another American $A_2$, then $A_2$ also attacks $A_1$. Let $m$ be the number of squares that an American attacks when placed in the top left corner of an 8 by 8 chessboard. Let $n$ be the maximal number of Americans that can be placed on the $8$ by $8$ chessboard such that no Americans attack each other, if one American must be in the top left corner. Find the largest possible value of $mn$.
30. On the blackboard, Amy writes $2017$ in base-$a$ to get $133201_a$. Betsy notices she can erase a digit from Amy’s number and change the base to base-$b$ such that the value of the the number remains the same. Catherine then notices she can erase a digit from Betsy’s number and change the base to base-$c$ such that the value still remains the same. Compute, in decimal, $a + b + c$.
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 $N$?
2013 Harvard-MIT Mathematics Tournament, 9
I have $8$ unit cubes of different colors, which I want to glue together into a $2\times 2\times 2$ cube. How many distinct $2\times 2\times 2$ cubes can I make? Rotations of the same cube are not considered distinct, but reflections are.
2011 Harvard-MIT Mathematics Tournament, 6
How many polynomials $P$ with integer coefficients and degree at most $5$ satisfy $0 \le P(x) < 120$ for all $x \in \{0,1,2,3,4,5\}$?
2019 Harvard-MIT Mathematics Tournament, 1
How many distinct permutations of the letters in the word REDDER are there that do not contain a palindromic substring of length at least two? (A [i]substring[/i] is a continuous block of letters that is part of the string. A string is [i]palindromic[/i] if it is the same when read backwards.)
2012 Harvard-MIT Mathematics Tournament, 10
Suppose that there are $16$ variables $\{a_{i,j}\}_{0\leq i,j\leq 3}$, each of which may be $0$ or $1$. For how many settings of the variables $a_{i,j}$ do there exist positive reals $c_{i,j}$ such that the polynomial \[f(x,y)=\sum_{0\leq i,j\leq 3}a_{i,j}c_{i,j}x^iy^j\] $(x,y\in\mathbb{R})$ is bounded below?
2013 Harvard-MIT Mathematics Tournament, 16
The walls of a room are in the shape of a triangle $ABC$ with $\angle ABC = 90^\circ$, $\angle BAC = 60^\circ$, and $AB=6$. Chong stands at the midpoint of $BC$ and rolls a ball toward $AB$. Suppose that the ball bounces off $AB$, then $AC$, then returns exactly to Chong. Find the length of the path of the ball.