Olympiad problems

2006 Harvard-MIT Mathematics Tournament, 1

Larry can swim from Harvard to MIT (with the current of the Charles River) in $40$ minutes, or back (against the current) in $45$ minutes. How long does it take him to row from Harvard to MIT, if he rows the return trip in $15$ minutes? (Assume that the speed of the current and Larry’s swimming and rowing speeds relative to the current are all constant.) Express your answer in the format mm:ss.

2008 Harvard-MIT Mathematics Tournament, 7

Tags: trigonometry , HMMT , Harvard , college , MIT , trig identities , geometry

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$.

2018 HMIC, 2

Tags: Harvard , college , MIT , combinatorics , HMIC

Consider a finite set of points $T\in\mathbb{R}^n$ contained in the $n$-dimensional unit ball centered at the origin, and let $X$ be the convex hull of $T$. Prove that for all positive integers $k$ and all points $x\in X$, there exist points $t_1, t_2, \dots, t_k\in T$, not necessarily distinct, such that their centroid \[\frac{t_1+t_2+\dots+t_k}{k}\]has Euclidean distance at most $\frac{1}{\sqrt{k}}$ from $x$. (The $n$-dimensional unit ball centered at the origin is the set of points in $\mathbb{R}^n$ with Euclidean distance at most $1$ from the origin. The convex hull of a set of points $T\in\mathbb{R}^n$ is the smallest set of points $X$ containing $T$ such that each line segment between two points in $X$ lies completely inside $X$.)

2007 Harvard-MIT Mathematics Tournament, 7

Tags: Harvard , college , modular arithmetic

A student at Harvard named Kevin Was counting his stones by $11$ He messed up $n$ times And instead counted $9$s And wound up at $2007$. How many values of $n$ could make this limerick true?

2018 HMIC, 1

Tags: Harvard , college , MIT , combinatorics , HMIC

Let $m>1$ be a fixed positive integer. For a nonempty string of base-ten digits $S$, let $c(S)$ be the number of ways to split $S$ into contiguous nonempty strings of digits such that the base-ten number represented by each string is divisible by $m$. These strings are allowed to have leading zeroes. In terms of $m$, what are the possible values that $c(S)$ can take? For example, if $m=2$, then $c(1234)=2$ as the splits $1234$ and $12|34$ are valid, while the other six splits are invalid.

2008 Harvard-MIT Mathematics Tournament, 1

Tags: Harvard , college , MIT , counting , distinguishability

Four students from Harvard, one of them named Jack, and five students from MIT, one of them named Jill, are going to see a Boston Celtics game. However, they found out that only $ 5$ tickets remain, so $ 4$ of them must go back. Suppose that at least one student from each school must go see the game, and at least one of Jack and Jill must go see the game, how many ways are there of choosing which $ 5$ people can see the game?

2006 Putnam, B6

Tags: Putnam , limit , factorial , Harvard , college , inequalities , integration

Let $k$ be an integer greater than $1.$ Suppose $a_{0}>0$ and define \[a_{n+1}=a_{n}+\frac1{\sqrt[k]{a_{n}}}\] for $n\ge 0.$ Evaluate \[\lim_{n\to\infty}\frac{a_{n}^{k+1}}{n^{k}}.\]

2018 HMIC, 5

Tags: Harvard , college , MIT , combinatorics , HMIC

Let $G$ be an undirected simple graph. Let $f(G)$ be the number of ways to orient all of the edges of $G$ in one of the two possible directions so that the resulting directed graph has no directed cycles. Show that $f(G)$ is a multiple of $3$ if and only if $G$ has a cycle of odd length.

2018 HMIC, 3

Tags: Harvard , college , MIT , combinatorics , HMIC

A polygon in the plane (with no self-intersections) is called $\emph{equitable}$ if every line passing through the origin divides the polygon into two (possibly disconnected) regions of equal area. Does there exist an equitable polygon which is not centrally symmetric about the origin? (A polygon is centrally symmetric about the origin if a $180$-degree rotation about the origin sends the polygon to itself.)

2018 HMIC, 4

Tags: Harvard , college , MIT , combinatorics , HMIC

Find all functions $f: \mathbb{R}^+\to\mathbb{R}^+$ such that \[f(x+f(y+xy))=(y+1)f(x+1)-1\]for all $x,y\in\mathbb{R}^+$. ($\mathbb{R}^+$ denotes the set of positive real numbers.)

2007 ITest, 26

Tags: Harvard , college , Stanford

Julie runs a website where she sells university themed clothing. On Monday, she sells thirteen Stanford sweatshirts and nine Harvard sweatshirts for a total of $\$370$. On Tuesday, she sells nine Stanford sweatshirts and two Harvard sweatshirts for a total of $\$180$. On Wednesday, she sells twelve Stanford sweatshirts and six Harvard sweatshirts. If Julie didn't change the prices of any items all week, how much money did she take in (total number of dollars) from the sale of Stanford and Harvard sweatshirts on Wednesday?