Found problems: 233
2014 HMNT, 5
Let $A,B,C,D,E$ be five points on a circle; some segments are drawn between the points so that each of the $5C2 = 10$ pairs of points is connected by either zero or one segments. Determine the number of sets of segments that can be drawn such that:
• It is possible to travel from any of the five points to any other of the five points along drawn
segments.
• It is possible to divide the five points into two nonempty sets $S$ and $T$ such that each segment
has one endpoint in $S$ and the other endpoint in $T$.
2016 Harvard-MIT Mathematics Tournament, 3
The three points $A, B, C$ form a triangle. $AB=4, BC=5, AC=6$. Let the angle bisector of $\angle A$ intersect side $BC$ at $D$. Let the foot of the perpendicular from $B$ to the angle bisector of $\angle A$ be $E$. Let the line through $E$ parallel to $AC$ meet $BC$ at $F$. Compute $DF$.
2013 Harvard-MIT Mathematics Tournament, 9
Let $z$ be a non-real complex number with $z^{23}=1$. Compute \[\sum_{k=0}^{22}\dfrac{1}{1+z^k+z^{2k}}.\]
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?
2013 Harvard-MIT Mathematics Tournament, 7
Compute \[\sum_{a_1=0}^\infty\sum_{a_2=0}^\infty\cdots\sum_{a_7=0}^\infty\dfrac{a_1+a_2+\cdots+a_7}{3^{a_1+a_2+\cdots+a_7}}.\]
2019 Harvard-MIT Mathematics Tournament, 1
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
\[\frac{AB}{AX} + \frac{AD}{AY} = \frac{AC}{AZ}.\]
2016 HMIC, 5
Let $S = \{a_1, \ldots, a_n \}$ be a finite set of positive integers of size $n \ge 1$, and let $T$ be the set of all positive integers that can be expressed as sums of perfect powers (including $1$) of distinct numbers in $S$, meaning
\[ T = \left\{ \sum_{i=1}^n a_i^{e_i} \mid e_1, e_2, \dots, e_n \ge 0 \right\}. \]
Show that there is a positive integer $N$ (only depending on $n$) such that $T$ contains no arithmetic progression of length $N$.
[i]Yang Liu[/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 $ABCD$.
2014 HMNT, 4
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.
2019 Harvard-MIT Mathematics Tournament, 2
Your math friend Steven rolls five fair icosahedral dice (each of which is labelled $1,2, \dots,20$ on its sides). He conceals the results but tells you that at least half the rolls are $20$. Suspicious, you examine the first two dice and find that they show $20$ and $19$ in that order. Assuming that Steven is truthful, what is the probability that all three remaining concealed dice show $20$?
2016 HMIC, 3
Denote by $\mathbb{N}$ the positive integers. Let $f:\mathbb{N} \rightarrow \mathbb{N}$ be a function such that, for any $w,x,y,z \in \mathbb{N}$, \[ f(f(f(z)))f(wxf(yf(z)))=z^{2}f(xf(y))f(w). \] Show that $f(n!) \ge n!$ for every positive integer $n$.
[i]Pakawut Jiradilok[/i]
2019 HMNT, 2
$2019$ students are voting on the distribution of $N$ items. For each item, each student submits a vote on who should receive that item, and the person with the most votes receives the item (in case of a tie, no one gets the item). Suppose that no student votes for the same person twice. Compute the maximum possible number of items one student can receive, over all possible values of $N$ and all possible ways of voting.
2013 Harvard-MIT Mathematics Tournament, 3
Let $S$ be the set of integers of the form $2^x+2^y+2^z$, where $x,y,z$ are pairwise distinct non-negative integers. Determine the $100$th smallest element of $S$.
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 $x>1$.
2011 Harvard-MIT Mathematics Tournament, 3
Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a
running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either player wins, or else inde nitely. If Nathaniel goes fi rst, determine the probability that he ends up winning.
2023 Harvard-MIT Mathematics Tournament, 6
Each cell of a $3 $ × $3$ grid is labeled with a digit in the set {$1, 2, 3, 4, 5$} Then, the maximum entry in
each row and each column is recorded. Compute the number of labelings for which every digit from $1$
to $5$ is recorded at least once.
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?
2016 HMIC, 1
Theseus starts at the point $(0, 0)$ in the plane. If Theseus is standing at the point $(x, y)$ in the plane, he can step one unit to the north to point $(x, y+1)$, one unit to the west to point $(x-1, y)$, one unit to the south to point $(x, y-1)$, or one unit to the east to point $(x+1, y)$. After a sequence of more than two such moves, starting with a step one unit to the south (to point $(0, -1)$), Theseus finds himself back at the point $(0, 0)$. He never visited any point other than $(0, 0)$ more than once, and never visited the point $(0, 0)$ except at the start and end of this sequence of moves.
Let $X$ be the number of times that Theseus took a step one unit to the north, and then a step one unit to the west immediately afterward. Let $Y$ be the number of times that Theseus took a step one unit to the west, and then a step one unit to the north immediately afterward. Prove that $|X - Y| = 1$.
[i]Mitchell Lee[/i]
2016 HMNT, 7-9
7. What is the minimum value of the product $$\prod_{i=1}^6\frac{a_i-a_{i+1}}{a_{i+2}-a_{i+3}}$$ given that $(a_1, a_2, a_3, a_4, a_5, a_6)$ is a permutation of $(1, 2, 3, 4, 5, 6)$? (note $a_7 = a_1, a_8 = a_2 \ldots$)
8. Danielle picks a positive integer $1 \le n \le 2016$ uniformly at random. What is the probability that $\text{gcd}(n, 2015) = 1$?
9. How many $3$-element subsets of the set $\{1, 2, 3, . . . , 19\}$ have sum of elements divisible by $4$?
2019 Harvard-MIT Mathematics Tournament, 1
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, 6
For positive reals $p$ and $q$, define the [i]remainder[/i] when $p$ and $q$ as the smallest nonnegative real $r$ such that $\tfrac{p-r}{q}$ is an integer. For an ordered pair $(a, b)$ of positive integers, let $r_1$ and $r_2$ be the remainder when $a\sqrt{2} + b\sqrt{3}$ is divided by $\sqrt{2}$ and $\sqrt{3}$ respectively. Find the number of pairs $(a, b)$ such that $a, b \le 20$ and $r_1 + r_2 = \sqrt{2}$.
2014 Harvard-MIT Mathematics Tournament, 3
Let \[ A = \frac{1}{6}((\log_2(3))^3-(\log_2(6))^3-(\log_2(12))^3+(\log_2(24))^3) \].
Compute $2^A$.
2016 HMNT, 3
The three points $A, B, C$ form a triangle. $AB=4, BC=5, AC=6$. Let the angle bisector of $\angle A$ intersect side $BC$ at $D$. Let the foot of the perpendicular from $B$ to the angle bisector of $\angle A$ be $E$. Let the line through $E$ parallel to $AC$ meet $BC$ at $F$. Compute $DF$.
2011 Harvard-MIT Mathematics Tournament, 7
Let $A = \{1,2,\ldots,2011\}$. Find the number of functions $f$ from $A$ to $A$ that satisfy $f(n) \le n$ for all $n$ in $A$ and attain exactly $2010$ distinct values.
2013 Harvard-MIT Mathematics Tournament, 25
The sequence $(z_n)$ of complex numbers satisfies the following properties:
[list]
[*]$z_1$ and $z_2$ are not real.
[*]$z_{n+2}=z_{n+1}^2z_n$ for all integers $n\geq 1$.
[*]$\dfrac{z_{n+3}}{z_n^2}$ is real for all integers $n\geq 1$.
[*]$\left|\dfrac{z_3}{z_4}\right|=\left|\dfrac{z_4}{z_5}\right|=2$. [/list]
Find the product of all possible values of $z_1$.