Found problems: 966
1988 Putnam, A6
If a linear transformation $A$ on an $n$-dimensional vector space has $n+1$ eigenvectors such that any $n$ of them are linearly independent, does it follow that $A$ is a scalar multiple of the identity? Prove your answer.
1993 Putnam, B2
A deck of $2n$ cards numbered from $1$ to $2n$ is shuffled and n cards are dealt to $A$ and $B$. $A$ and $B$ alternately discard a card face up, starting with $A$. The game when the sum of the discards is first divisible by $2n + 1$, and the last person to discard wins. What is the probability that $A$ wins if neither player makes a mistake?
1974 Putnam, B3
Prove that if $a$ is a real number such that
$$\cos \pi a= \frac{1}{3},$$
then $a$ is irrational.
1999 Putnam, 4
Let $f$ be a real function with a continuous third derivative such that $f(x)$, $f^\prime(x)$, $f^{\prime\prime}(x)$, $f^{\prime\prime\prime}(x)$ are positive for all $x$. Suppose that $f^{\prime\prime\prime}(x)\leq f(x)$ for all $x$. Show that $f^\prime(x)<2f(x)$ for all $x$.
2005 Putnam, B6
Let $S_n$ denote the set of all permutations of the numbers $1,2,\dots,n.$ For $\pi\in S_n,$ let $\sigma(\pi)=1$ if $\pi$ is an even permutation and $\sigma(\pi)=-1$ if $\pi$ is an odd permutation. Also, let $v(\pi)$ denote the number of fixed points of $\pi.$ Show that
\[ \sum_{\pi\in S_n}\frac{\sigma(\pi)}{v(\pi)+1}=(-1)^{n+1}\frac{n}{n+1}. \]
1962 Putnam, A2
Find every real-valued function $f$ whose domain is an interval $I$ (finite or infinite) having $0$ as a left-hand endpoint, such that for every positive $x\in I$ the average of $f$ over the closed interval $[0,x]$ is equal to $\sqrt{ f(0) f(x)}.$
1985 Putnam, B6
Let $G$ be a finite set of real $n \times n$ matrices $\left\{M_{i}\right\}, 1 \leq i \leq r,$ which form a group under matrix multiplication. Suppose that $\textstyle\sum_{i=1}^{r} \operatorname{tr}\left(M_{i}\right)=0,$ where $\operatorname{tr}(A)$ denotes the trace of the matrix $A .$ Prove that $\textstyle\sum_{i=1}^{r} M_{i}$ is the $n \times n$ zero matrix.
1956 Putnam, A2
Prove that every positive integer has a multiple whose decimal representation involves all ten digits.
1980 Putnam, A2
Let $r$ and $s$ be positive integers. Derive a formula for the number of ordered quadruples $(a,b,c,d)$ of positive integers such that
$$3^r \cdot 7^s = \text{lcm}(a,b,c)= \text{lcm}(a,b,d)=\text{lcm}(a,c,d)=\text{lcm}(b,c,d),$$
depending only on $r$ and $s.$
1941 Putnam, A2
Find the $n$-th derivative with respect to $x$ of
$$\int_{0}^{x} \left(1+\frac{x-t}{1!}+\frac{(x-t)^{2}}{2!}+\ldots+\frac{(x-t)^{n-1}}{(n-1)!}\right)e^{nt} dt.$$
1975 Putnam, A6
Given three points in space forming an acute-angled triangle, show that we can find two further points such that no three of the five points are collinear and the line through any two is normal to the plane through the other three.
1969 Putnam, A6
Let a sequence $(x_n)$ be given and let $y_n = x_{n-1} +2 x_n $ for $n>1.$ Suppose that the sequence $(y_n)$ converges. Prove that the sequence $(x_n)$ converges, too.
1962 Putnam, B3
Let $S$ be a convex region in the euclidean plane containing the origin. Assume that every ray from the origin has at least one point outside $S$. Prove that $S$ is bounded.
2024 Putnam, B5
Let $k$ and $m$ be positive integers. For a positive integer $n$, let $f(n)$ be the number of integer sequences $x_1,\,\ldots,\,x_k,\,y_1,\,\ldots,\,y_m,\,z$ satisfying $1\leq x_1\leq\cdots\leq x_k\leq z\leq n$ and $1\leq y_1\leq\cdots\leq y_m\leq z\leq n$. Show that $f(n)$ can be expressed as a polynomial in $n$ with nonnegative coefficients.
1965 Putnam, A6
In the plane with orthogonal Cartesian coordinates $x$ and $y$, prove that the line whose equation is $ux+vy = 1$ will be tangent to the cirve $x^m+y^m=1$ (where $m>1$) if and only if $u^n + v^n = 1$ and $m^{-1} + n^{-1} = 1$.
2015 Putnam, B4
Let $T$ be the set of all triples $(a,b,c)$ of positive integers for which there exist triangles with side lengths $a,b,c.$ Express \[\sum_{(a,b,c)\in T}\frac{2^a}{3^b5^c}\] as a rational number in lowest terms.
1995 Putnam, 2
An ellipse, whose semi-axes have length $a$ and $b$, rolls without slipping on the curve $y=c\sin{\left(\frac{x}{a}\right)}$. How are $a,b,c$ related, given that the ellipse completes one revolution when it traverses one period of the curve?
1959 Putnam, A5
A sparrow, flying horizontally in a straight line, is $50$ feet directly below an eagle and $100$ feet directly above a hawk. Both hawk and eagle fly directly toward the sparrow, reaching it simultaneously. The hawk flies twice as fast as the sparrow. How far does each bird fly? At what rate does the eagle fly?
1978 Putnam, A1
Let $A$ be any set of $20$ distinct integers chosen from the arithmetic progression $1, 4, 7,\ldots,100$. Prove that there must be two distinct integers in $A$ whose sum is $104$.
2000 Putnam, 1
Let $a_j$, $b_j$, $c_j$ be integers for $1 \le j \le N$. Assume for each $j$, at least one of $a_j$, $b_j$, $c_j$ is odd. Show that there exists integers $r, s, t$ such that $ra_j+sb_j+tc_j$ is odd for at least $\tfrac{4N}{7}$ values of $j$, $1 \le j \le N$.
2005 Putnam, A4
Let $H$ be an $n\times n$ matrix all of whose entries are $\pm1$ and whose rows are mutually orthogonal. Suppose $H$ has an $a\times b$ submatrix whose entries are all $1.$ Show that $ab\le n.$
1992 Putnam, A2
Define $C(\alpha)$ to be the coefficient of $x^{1992}$ in the power series about $x = 0$ of $(1 + x)^{\alpha}$ . Evaluate
$$\int_{0}^{1} \left( C(-y-1) \sum_{k=1}^{1992} \frac{1}{y+k} \right)\, dy.$$
1988 Putnam, B5
For positive integers $n$, let $M_n$ be the $2n+1$ by $2n+1$ skew-symmetric matrix for which each entry in the first $n$ subdiagonals below the main diagonal is 1 and each of the remaining entries below the main diagonal is -1. Find, with proof, the rank of $M_n$. (According to one definition, the rank of a matrix is the largest $k$ such that there is a $k \times k$ submatrix with nonzero determinant.)
One may note that
\begin{align*}
M_1 &= \left( \begin{array}{ccc} 0 & -1 & 1 \\ 1 & 0 & -1 \\ -1 & 1 & 0
\end{array}\right) \\
M_2 &= \left( \begin{array}{ccccc} 0 & -1 & -1 & 1
& 1 \\ 1 & 0 & -1 & -1 & 1 \\ 1 & 1 & 0 & -1 & -1 \\ -1 & 1 & 1 & 0 &
-1 \\ -1 & -1 & 1 & 1 & 0 \end{array} \right).
\end{align*}
2005 Germany Team Selection Test, 1
Find the smallest positive integer $n$ with the following property:
For any integer $m$ with $0 < m < 2004$, there exists an integer $k$ such that
\[\frac{m}{2004}<\frac{k}{n}<\frac{m+1}{2005}.\]
2007 Putnam, 5
Suppose that a finite group has exactly $ n$ elements of order $ p,$ where $ p$ is a prime. Prove that either $ n\equal{}0$ or $ p$ divides $ n\plus{}1.$