Found problems: 233
2013 Harvard-MIT Mathematics Tournament, 4
Determine all real values of $A$ for which there exist distinct complex numbers $x_1$, $x_2$ such that the following three equations hold:
\begin{align*}x_1(x_1+1)&=A\\x_2(x_2+1)&=A\\x_1^4+3x_1^3+5x_1&=x_2^4+3x_2^3+5x_2.\end{align*}
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 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.
2011 HMNT, 7
Determine the number of angles $\theta$ between $0$ and $2 \pi$, other than integer multiples of $\pi /2$, such that the quantities $\sin \theta, \cos \theta, $ and $\tan \theta$ form a geometric sequence in some order.
2008 Harvard-MIT Mathematics Tournament, 7
Let $ C_1$ and $ C_2$ be externally tangent circles with radius 2 and 3, respectively. Let $ C_3$ be a circle internally tangent to both $ C_1$ and $ C_2$ at points $ A$ and $ B$, respectively. The tangents to $ C_3$ at $ A$ and $ B$ meet at $ T$, and $ TA \equal{} 4$. Determine the radius of $ C_3$.
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 $a_4 b_4$.
2015 HMIC, 2
Let $m,n$ be positive integers with $m \ge n$. Let $S$ be the set of pairs $(a,b)$ of relatively prime positive integers such that $a,b \le m$ and $a+b > m$.
For each pair $(a,b)\in S$, consider the nonnegative integer solution $(u,v)$ to the equation $au - bv = n$ chosen with $v \ge 0$ minimal, and let $I(a,b)$ denote the (open) interval $(v/a, u/b)$.
Prove that $I(a,b) \subseteq (0,1)$ for every $(a,b)\in S$, and that any fixed irrational number $\alpha\in(0,1)$ lies in $I(a,b)$ for exactly $n$ distinct pairs $(a,b)\in S$.
[i]Victor Wang, inspired by 2013 ISL N7[/i]
2019 HMNT, 3
The coefficients of the polynomial $P(x)$ are nonnegative integers, each less than 100. Given that $P(10) = 331633$ and $P(-10) = 273373$, compute $P(1)$.
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, 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 $$\prod_{i=0}^\infty a_i=a_0a_1a_2\ldots$$
2020 Harvest Math Invitational Team Round Problems, HMI Team #1
1. Let $f(n) = n^2+6n+11$ be a function defined on positive integers. Find the sum of the first three prime values $f(n)$ takes on.
[i]Proposed by winnertakeover[/i]
2014 HMNT, 5
Mark and William are playing a game with a stored value. On his turn, a player may either multiply the stored value by $2$ and add $1$ or he may multiply the stored value by $4$ and add $3$. The first player to make the stored value exceed $2^{100}$ wins. The stored value starts at $1$ and Mark goes first. Assuming both players play optimally, what is the maximum number of times that William can make a move?
(By optimal play, we mean that on any turn the player selects the move which leads to the best possible outcome given that the opponent is also playing optimally. If both moves lead to the same outcome, the player selects one of them arbitrarily.)
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?
2013 Harvard-MIT Mathematics Tournament, 21
Find the number of positive integers $j\leq 3^{2013}$ such that \[j=\sum_{k=0}^m\left((-1)^k\cdot 3^{a_k}\right)\] for some strictly increasing sequence of nonnegative integers $\{a_k\}$. For example, we may write $3=3^1$ and $55=3^0-3^3+3^4$, but $4$ cannot be written in this form.
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, $i=\sqrt{-1}$)
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 $(1, 2, \dots, n)$?
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, 9
Let $z$ be a non-real complex number with $z^{23}=1$. Compute \[\sum_{k=0}^{22}\dfrac{1}{1+z^k+z^{2k}}.\]
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.
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, 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.
2016 Harvard-MIT Mathematics Tournament, 4
A positive integer is written on each corner of a square such that numbers on opposite vertices are relatively prime while numbers on adjacent vertices are not relatively prime. What is the smallest possible value of the sum of these $4$ numbers?
2019 Harvard-MIT Mathematics Tournament, 7
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 HMNT, 4
A positive integer is written on each corner of a square such that numbers on opposite vertices are relatively prime while numbers on adjacent vertices are not relatively prime. What is the smallest possible value of the sum of these $4$ numbers?
2013 Harvard-MIT Mathematics Tournament, 10
Wesyu is a farmer, and she's building a cao (a relative of the cow) pasture. Shw starts with a triangle $A_0A_1A_2$ where angle $A_0$ is $90^\circ$, angle $A_1$ is $60^\circ$, and $A_0A_1$ is $1$. She then extends the pasture. FIrst, she extends $A_2A_0$ to $A_3$ such that $A_3A_0=\dfrac12A_2A_0$ and the new pasture is triangle $A_1A_2A_3$. Next, she extends $A_3A_1$ to $A_4$ such that $A_4A_1=\dfrac16A_3A_1$. She continues, each time extending $A_nA_{n-2}$ to $A_{n+1}$ such that $A_{n+1}A_{n-2}=\dfrac1{2^n-2}A_nA_{n-2}$. What is the smallest $K$ such that her pasture never exceeds an area of $K$?