Found problems: 233
2020 Harvest Math Invitational Team Round Problems, HMI Team #2
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]
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, 9
Tessa the hyper-ant has a 2019-dimensional hypercube. For a real number $k$, she calls a placement of nonzero real numbers on the $2^{2019}$ vertices of the hypercube [i]$k$-harmonic[/i] if for any vertex, the sum of all 2019 numbers that are edge-adjacent to this vertex is equal to $k$ times the number on this vertex. Let $S$ be the set of all possible values of $k$ such that there exists a $k$-harmonic placement. Find $\sum_{k \in S} |k|$.
2008 Harvard-MIT Mathematics Tournament, 6
A [i]root of unity[/i] is a complex number that is a solution to $ z^n \equal{} 1$ for some positive integer $ n$. Determine the number of roots of unity that are also roots of $ z^2 \plus{} az \plus{} b \equal{} 0$ for some integers $ a$ and $ b$.
2013 Harvard-MIT Mathematics Tournament, 23
Let $ABCD$ be a parallelogram with $AB=8$, $AD=11$, and $\angle BAD=60^\circ$. Let $X$ be on segment $CD$ with $CX/XD=1/3$ and $Y$ be on segment $AD$ with $AY/YD=1/2$. Let $Z$ be on segment $AB$ such that $AX$, $BY$, and $DZ$ are concurrent. Determine the area of triangle $XYZ$.
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 $AHB$, $BHC$, $CHA$.
2016 HMNT, 5
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 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 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.
2014 Harvard-MIT Mathematics Tournament, 9
Given $a$, $b$, and $c$ are complex numbers satisfying
\[ a^2+ab+b^2=1+i \]
\[ b^2+bc+c^2=-2 \]
\[ c^2+ca+a^2=1, \]
compute $(ab+bc+ca)^2$. (Here, $i=\sqrt{-1}$)
2013 Harvard-MIT Mathematics Tournament, 17
The lines $y=x$, $y=2x$, and $y=3x$ are the three medians of a triangle with perimeter $1$. Find the length of the longest side of the triangle.
2019 Harvard-MIT Mathematics Tournament, 5
Find all positive integers $n$ such that the unit segments of an $n \times n$ grid of unit squares can be partitioned into groups of three such that the segments of each group share a common vertex.
2013 Harvard-MIT Mathematics Tournament, 19
An isosceles trapezoid $ABCD$ with bases $AB$ and $CD$ has $AB=13$, $CD=17$, and height $3$. Let $E$ be the intersection of $AC$ and $BD$. Circles $\Omega$ and $\omega$ are circumscribed about triangles $ABE$ and $CDE$. Compute the sum of the radii of $\Omega$ and $\omega$.
2011 Harvard-MIT Mathematics Tournament, 9
Let $\left\{ a_n \right\}$ and $\left\{ b_n \right\}$ be sequences defined recursively by $a_0 =2$; $b_0 = 2$, and $a_{n+1} = a_n \sqrt{1+a_n^2+b_n^2}-b_n$; $b_{n+1} = b_n\sqrt{1+a_n^2+b_n^2} + a_n$. Find the ternary (base 3) representation of $a_4$ and $b_4$.
2019 Harvard-MIT Mathematics Tournament, 5
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 $a_4 b_4$.
2013 Harvard-MIT Mathematics Tournament, 20
The polynomial $f(x)=x^3-3x^2-4x+4$ has three real roots $r_1$, $r_2$, and $r_3$. Let $g(x)=x^3+ax^2+bx+c$ be the polynomial which has roots $s_1$, $s_2$, and $s_3$, where $s_1=r_1+r_2z+r_3z^2$, $s_2=r_1z+r_2z^2+r_3$, $s_3=r_1z^2+r_2+r_3z$, and $z=\frac{-1+i\sqrt3}2$. Find the real part of the sum of the coefficients of $g(x)$.
2014 Harvard-MIT Mathematics Tournament, 6
Given $w$ and $z$ are complex numbers such that $|w+z|=1$ and $|w^2+z^2|=14$, find the smallest possible value of $|w^3+z^3|$. Here $| \cdot |$ denotes the absolute value of a complex number, given by $|a+bi|=\sqrt{a^2+b^2}$ whenever $a$ and $b$ are real numbers.
1999 Harvard-MIT Mathematics Tournament, 7
Carl and Bob can demolish a building in 6 days, Anne and Bob can do it in $3$, Anne and Carl in $5$. How many days does it take all of them working together if Carl gets injured at the end of the first day and can't come back?
2016 Harvard-MIT Mathematics Tournament, 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.
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 $\left\lfloor25\min\left(\left(\frac AC\right)^2,\left(\frac CA\right)^2\right)\right\rfloor$.
2016 HMNT, 4-6
4. A square can be divided into four congruent figures as shown: [asy]
size(2cm);
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
draw((1,0)--(1,2));
draw((0,1)--(2,1));
[/asy]
For how many $n$ with $1 \le n \le 100$ can a unit square be divided into $n$ congruent figures?
5. If $x + 2y - 3z = 7$ and $2x - y + 2z = 6$, determine $8x + y$.
6. Let $ABCD$ be a rectangle, and let $E$ and $F$ be points on segment $AB$ such that $AE = EF = FB$. If $CE$ intersects the line $AD$ at $P$, and $PF$ intersects $BC$ at $Q$, determine the ratio of $BQ$ to $CQ$.
2023 Harvard-MIT Mathematics Tournament, 8
Let $\triangle ABC$ be a triangle with $\angle BAC>90^{\circ}$, $AB=5$ and $AC=7$. Points $D$ and $E$ lie on segment $BC$ such that $BD=DE=EC$. If $\angle BAC+\angle DAE=180^{\circ}$, compute $BC$.
2020 Harvard-MIT Mathematics Tournament, 6
Alice writes $1001$ letters on a blackboard, each one chosen independently and uniformly at random from the set $S=\{a, b, c\}$. A move consists of erasing two distinct letters from the board and replacing them with the third letter in $S$. What is the probability that Alice can perform a sequence of moves which results in one letter remaining on the blackboard?
[i]Proposed by Daniel Zhu.[/i]
2016 Harvard-MIT Mathematics Tournament, 9
For any positive integer $n$, $S_{n}$ be the set of all permutations of $\{1,2,3,\dots,n\}$. For each permutation $\pi \in S_n$, let $f(\pi)$ be the number of ordered pairs $(j,k)$ for which $\pi(j)>\pi(k)$ and $1\leq j<k \leq n$. Further define $g(\pi)$ to be the number of positive integers $k \leq n$ such that $\pi(k)\equiv k \pm 1 \pmod{n}$. Compute \[ \sum_{\pi \in S_{999}} (-1)^{f(\pi)+g(\pi)}. \]
2019 Harvard-MIT Mathematics Tournament, 7
Find the value of
\[\sum_{a = 1}^{\infty} \sum_{b = 1}^{\infty} \sum_{c = 1}^{\infty} \frac{ab(3a + c)}{4^{a+b+c} (a+b)(b+c)(c+a)}.\]