Olympiad problems

2017 Putnam, B4

Evaluate the sum \[\sum_{k=0}^{\infty}\left(3\cdot\frac{\ln(4k+2)}{4k+2}-\frac{\ln(4k+3)}{4k+3}-\frac{\ln(4k+4)}{4k+4}-\frac{\ln(4k+5)}{4k+5}\right)\] \[=3\cdot\frac{\ln 2}2-\frac{\ln 3}3-\frac{\ln 4}4-\frac{\ln 5}5+3\cdot\frac{\ln 6}6-\frac{\ln 7}7-\frac{\ln 8}8-\frac{\ln 9}9+3\cdot\frac{\ln 10}{10}-\cdots.\] (As usual, $\ln x$ denotes the natural logarithm of $x.$)

2002 Putnam, 1

Tags: Putnam , probability , induction , conditional probability , college contests

Shanille O'Keal shoots free throws on a basketball court. She hits the first and misses the second, and thereafter the probability that she hits the next shot is equal to the proportion of shots she has hit so far. What is the probability she hits exactly $50$ of her first $100$ shots?

1955 Putnam, A7

Tags: Putnam

Consider the function $f$ defined by the differential equation \[ f'' (x) = (x^3 + ax) f(x) \] and the initial conditions $f(0) = 1, f'(0) = 0.$ Prove that the roots of $f$ are bounded above but unbounded below.

1953 Putnam, B3

Tags: Putnam , differential equation

Solve the equations $$ \frac{dy}{dx}=z(y+z)^n, \;\; \; \frac{dz}{dx} = y(y+z)^n,$$ given the initial conditions $y=1$ and $z=0$ when $x=0.$

1962 Putnam, B4

Tags: Putnam , Coloring , circles

The euclidean plane is divided into regions by drawing a finite number of circles. Show that it is possible to color each of these regions either red or blue in such a way that no two adjacent regions have the same color.

1990 Putnam, B3

Tags: linear algebra , matrix , pigeonhole principle , Putnam , college contests

Let $S$ be a set of $ 2 \times 2 $ integer matrices whose entries $a_{ij}(1)$ are all squares of integers and, $(2)$ satisfy $a_{ij} \le 200$. Show that $S$ has more than $ 50387 (=15^4-15^2-15+2) $ elements, then it has two elements that commute.

1995 Putnam, 4

Tags: Putnam , induction , continued fraction , college contests

Evaluate : \[ \sqrt[8]{2207-\frac{1}{2207-\frac{1}{2207-\cdots}}} \] Express your expression in the form $\frac{a+b\sqrt{c}}{d}$ where $a,b,c,d\in \Bbb{Z}$.

1957 Putnam, A1

Tags: Putnam , surfaces , Normal vector

The normals to a surface all intersect a fixed straight line. Show that the surface is a portion of a surface of revolution.

1967 Putnam, B1

Tags: Putnam , geometry , euclidean geometry , hexagon

Let $ABCDEF$ be a hexagon inscribed in a circle of radius $r.$ Show that if $AB=CD=EF=r,$ then the midpoints of $BC, DE$ and $FA$ are the vertices of an equilateral triangle.

1982 Putnam, A5

Tags: Putnam , number theory unsolved , number theory

$a, b, c, d$ are positive integers, and $r=1-\frac{a}{b}-\frac{c}{d}$. And, $a+c \le 1982, r \ge 0$. Prove that $r>\frac{1}{1983^3}$.

2010 Putnam, B3

Tags: Putnam , number theory , greatest common divisor , pigeonhole principle , relatively prime , college contests

There are 2010 boxes labeled $B_1,B_2,\dots,B_{2010},$ and $2010n$ balls have been distributed among them, for some positive integer $n.$ You may redistribute the balls by a sequence of moves, each of which consists of choosing an $i$ and moving [i]exactly[/i] $i$ balls from box $B_i$ into any one other box. For which values of $n$ is it possible to reach the distribution with exactly $n$ balls in each box, regardless of the initial distribution of balls?

1994 Putnam, 4

Tags: Putnam , linear algebra , matrix , limit , induction , college contests

For $n\ge 1$ let $d_n$ be the $\gcd$ of the entries of $A^n-\mathcal{I}_2$ where \[ A=\begin{pmatrix} 3&2\\ 4&3\end{pmatrix}\quad \text{ and }\quad \mathcal{I}_2=\begin{pmatrix}1&0\\ 0&1\\\end{pmatrix}\] Show that $\lim_{n\to \infty}d_n=\infty$.

2021 Putnam, B5

Tags: Putnam , Putnam 2021

Say that an $n$-by-$n$ matrix $A=(a_{ij})_{1\le i,j \le n}$ with integer entries is very odd if, for every nonempty subset $S$ of $\{1,2,\dots,n \}$, the $|S|$-by-$|S|$ submatrix $(a_{ij})_{i,j \in S}$ has odd determinant. Prove that if $A$ is very odd, then $A^k$ is very odd for every $k \ge 1$.

1942 Putnam, B4

Tags: Putnam , physics

A particle moves under a central force inversely proportional to the $k$-th power of the distance. If the particle describes a circle ( the central force proceeding from a point on the circumference of the circle ), find $k$.

2003 Putnam, 3

Tags: Putnam , number theory , least common multiple , floor function , prime factorization , college contests

Show that for each positive integer n, \[n!=\prod_{i=1}^n \; \text{lcm} \; \{1, 2, \ldots, \left\lfloor\frac{n}{i} \right\rfloor\}\] (Here lcm denotes the least common multiple, and $\lfloor x\rfloor$ denotes the greatest integer $\le x$.)

2002 Putnam, 1

Tags: Putnam , calculus , derivative , function , limit , college contests

Let $k$ be a fixed positive integer. The $n$th derivative of $\tfrac{1}{x^k-1}$ has the form $\tfrac{P_n(x)}{(x^k-1)^{n+1}}$, where $P_n(x)$ is a polynomial. Find $P_n(1)$.

1980 Putnam, A3

Tags: Putnam , trigonometry , integration

Evaluate $$\int_{0}^{ \pi \slash 2} \frac{ dx}{1+( \tan x)^{\sqrt{2}} }\;.$$

1985 Putnam, A3

Tags: Putnam

Let $d$ be a real number. For each integer $m \geq 0,$ define a sequence $\left\{a_{m}(j)\right\}, j=0,1,2, \ldots$ by the condition \begin{align*} a_{m}(0)&=d / 2^{m},\\ a_{m}(j+1)&=\left(a_{m}(j)\right)^{2}+2 a_{m}(j), \quad j \geq 0. \end{align*} Evaluate $\lim _{n \rightarrow \infty} a_{n}(n).$

1952 Putnam, A6

Tags: Putnam

A man has a rectangular block of wood $m$ by $n$ by $r$ inches ($m, n,$ and $r$ are integers). He paints the entire surface of the block, cuts the block into inch cubes, and notices that exactly half the cubes are completely unpainted. Prove that the number of essentially different blocks with this property is finite. (Do [i] not [/i] attempt to enumerate them.)

2010 Putnam, A3

Tags: Putnam , function , calculus , derivative , complex analysis , probability , vector

Suppose that the function $h:\mathbb{R}^2\to\mathbb{R}$ has continuous partial derivatives and satisfies the equation \[h(x,y)=a\frac{\partial h}{\partial x}(x,y)+b\frac{\partial h}{\partial y}(x,y)\] for some constants $a,b.$ Prove that if there is a constant $M$ such that $|h(x,y)|\le M$ for all $(x,y)$ in $\mathbb{R}^2,$ then $h$ is identically zero.

1994 Putnam, 5

Tags: Putnam , floor function , college contests

For each $\alpha\in \mathbb{R}$ define $f_{\alpha}(x)=\lfloor{\alpha x}\rfloor$. Let $n\in \mathbb{N}$. Show there exists a real $\alpha$ such that for $1\le \ell \le n$ : \[ f_{\alpha}^{\ell}(n^2)=n^2-\ell=f_{\alpha^{\ell}}(n^2).\] Here $f^{\ell}_{\alpha}(x)=(f_{\alpha}\circ f_{\alpha}\circ \cdots \circ f_{\alpha})(x)$ where the composition is carried out $\ell$ times.

1967 Putnam, A1

Tags: Putnam , trigonometry , inequalities

Let $f(x)= a_1 \sin x + a_2 \sin 2x+\cdots +a_{n} \sin nx $, where $a_1 ,a_2 ,\ldots,a_n $ are real numbers and where $n$ is a positive integer. Given that $|f(x)| \leq | \sin x |$ for all real $x,$ prove that $$|a_1 +2a_2 +\cdots +na_{n}|\leq 1.$$

1978 Putnam, B2

Tags: Putnam , Summation , Double Sum

Express $$\sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{m^2 n +m n^2 +2mn }$$ as a rational number.

1955 Putnam, B5

Tags: Putnam

Given an infinite sequence of $0$'s and $1$'s and a fixed integer $k,$ suppose that there are no more than $k$ distinct blocks of $k$ consecutive terms. Show that the sequence is eventually periodic. (For example, the sequence $11011010101$ followed by alternating $0$'s and $1$'s indefinitely, which is periodic beginning with the fifth term.)

1963 Putnam, A6

Tags: Putnam , conics , ellipse , geometry

Let $U$ and $V$ be any two distinct points on an ellipse, let $M$ be the midpoint of the chord $UV$, and let $AB$ and $CD$ be any two other chords through $M$. If the line $UV$ meets the line $AC$ in the point $P$ and the line $BD$ in the point $Q$, prove that $M$ is the midpoint of the segment $PQ.$