Found problems: 966
1986 Putnam, A5
Suppose $f_1(x), f_2(x), \dots, f_n(x)$ are functions of $n$ real variables $x = (x_1, \dots, x_n)$ with continuous second-order partial derivatives everywhere on $\mathbb{R}^n$. Suppose further that there are constants $c_{ij}$ such that
\[
\frac{\partial f_i}{\partial x_j} - \frac{\partial f_j}{\partial x_i}
= c_{ij}
\]
for all $i$ and $j$, $1\leq i \leq n$, $1 \leq j \leq n$. Prove that there is a function $g(x)$ on $\mathbb{R}^n$ such that $f_i + \partial g/\partial x_i$ is linear for all $i$, $1 \leq i \leq n$. (A linear function is one of the form \(
a_0 + a_1 x_1 + a_2 x_2 + \cdots + a_n x_n.)
\)
2008 Putnam, A2
Alan and Barbara play a game in which they take turns filling entries of an initially empty $ 2008\times 2008$ array. Alan plays first. At each turn, a player chooses a real number and places it in a vacant entry. The game ends when all entries are filled. Alan wins if the determinant of the resulting matrix is nonzero; Barbara wins if it is zero. Which player has a winning strategy?
2005 Putnam, A6
Let $n$ be given, $n\ge 4,$ and suppose that $P_1,P_2,\dots,P_n$ are $n$ randomly, independently and uniformly, chosen points on a circle. Consider the convex $n$-gon whose vertices are the $P_i.$ What is the probability that at least one of the vertex angles of this polygon is acute.?
1996 Putnam, 3
Let $S_n$ be the set of all permutations of $(1,2,\ldots,n)$. Then find :
\[ \max_{\sigma \in S_n} \left(\sum_{i=1}^{n} \sigma(i)\sigma(i+1)\right) \]
where $\sigma(n+1)=\sigma(1)$.
2013 Putnam, 2
Let $C=\bigcup_{N=1}^{\infty}C_N,$ where $C_N$ denotes the set of 'cosine polynomials' of the form \[f(x)=1+\sum_{n=1}^Na_n\cos(2\pi nx)\] for which:
(i) $f(x)\ge 0$ for all real $x,$ and
(ii) $a_n=0$ whenever $n$ is a multiple of $3.$
Determine the maximum value of $f(0)$ as $f$ ranges through $C,$ and prove that this maximum is attained.
1997 Putnam, 5
Let us define a sequence $\{a_n\}_{n\ge 1}$. Define as follows:
\[ a_1=2\text{ and }a_{n+1}=2^{a_n}\text{ for }n\ge 1 \]
Show this :
\[ a_{n}\equiv a_{n-1}\pmod n \]
2011 Putnam, A5
Let $F:\mathbb{R}^2\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be twice continuously differentiable functions with the following properties:
• $F(u,u)=0$ for every $u\in\mathbb{R};$
• for every $x\in\mathbb{R},g(x)>0$ and $x^2g(x)\le 1;$
• for every $(u,v)\in\mathbb{R}^2,$ the vector $\nabla F(u,v)$ is either $\mathbf{0}$ or parallel to the vector $\langle g(u),-g(v)\rangle.$
Prove that there exists a constant $C$ such that for every $n\ge 2$ and any $x_1,\dots,x_{n+1}\in\mathbb{R},$ we have
\[\min_{i\ne j}|F(x_i,x_j)|\le\frac{C}{n}.\]
Putnam 1938, B3
A horizontal disk diameter $3$ inches rotates once every $15$ seconds. An insect starts at the southernmost point of the disk facing due north. Always facing due north, it crawls over the disk at $1$ inch per second. Where does it again reach the edge of the disk?
1997 Putnam, 2
$f$ be a twice differentiable real valued function satisfying
\[ f(x)+f^{\prime\prime}(x)=-xg(x)f^{\prime}(x) \]
where $g(x)\ge 0$ for all real $x$. Show that $|f(x)|$ is bounded.
1984 Putnam, A3
Let $n$ be a positive integer. Let $a,b,x$ be real numbers, with $a \neq b$ and let $M_n$ denote the $2n x 2n $ matrix whose $(i,j)$ entry $m_{ij}$ is given by
$m_{ij}=x$ if $i=j$,
$m_{ij}=a$ if $i \not= j$ and $i+j$ is even,
$m_{ij}=b$ if $i \not= j$ and $i+j$ is odd.
For example
$ M_2=\begin{vmatrix}x& b& a & b\\ b& x & b &a\\ a
& b& x & b\\ b & a & b & x \end{vmatrix}$.
Express $\lim_{x\to\ 0} \frac{ det M_n}{ (x-a)^{(2n-2)} }$ as a polynomial in $a,b $ and $n$ .
P.S. How write in latex $m_{ij}=...$ with symbol for the system (because is multiform function?)
2000 Putnam, 5
Let $S_0$ be a finite set of positive integers. We define finite sets $S_1, S_2, \cdots$ of positive integers as follows: the integer $a$ in $S_{n+1}$ if and only if exactly one of $a-1$ or $a$ is in $S_n$. Show that there exist infinitely many integers $N$ for which $S_N = S_0 \cup \{ N + a: a \in S_0 \}$.
1990 Putnam, A4
Consider a paper punch that can be centered at any point
of the plane and that, when operated, removes from the
plane precisely those points whose distance from the
center is irrational. How many punches are needed to
remove every point?
2020 Putnam, A5
Let $a_n$ be the number of sets $S$ of positive integers for which
\[ \sum_{k\in S}F_k=n,\]
where the Fibonacci sequence $(F_k)_{k\ge 1}$ satisfies $F_{k+2}=F_{k+1}+F_k$ and begins $F_1=1$, $F_2=1$, $F_3=2$, $F_4=3$. Find the largest number $n$ such that $a_n=2020$.
2012 Putnam, 6
Let $f(x,y)$ be a continuous, real-valued function on $\mathbb{R}^2.$ Suppose that, for every rectangular region $R$ of area $1,$ the double integral of $f(x,y)$ over $R$ equals $0.$ Must $f(x,y)$ be identically $0?$
2024 Putnam, A2
For which real polynomials $p$ is there a real polynomial $q$ such that
\[
p(p(x))-x=(p(x)-x)^2q(x)
\]
for all real $x$?
1964 Putnam, A2
Find all continuous positive functions $f(x)$, for $0\leq x \leq 1$, such that
$$\int_{0}^{1} f(x)\; dx =1, $$
$$\int_{0}^{1} xf(x)\; dx =\alpha,$$
$$\int_{0}^{1} x^2 f(x)\; dx =\alpha^2, $$
where $\alpha$ is a given real number.
2013 Putnam, 6
Let $n\ge 1$ be an odd integer. Alice and Bob play the following game, taking alternating turns, with Alice playing first. The playing area consists of $n$ spaces, arranged in a line. Initially all spaces are empty. At each turn, a player either
• places a stone in an empty space, or
• removes a stone from a nonempty space $s,$ places a stone in the nearest empty space to the left of $s$ (if such a space exists), and places a stone in the nearest empty space to the right of $s$ (if such a space exists).
Furthermore, a move is permitted only if the resulting position has not occurred previously in the game. A player loses if he or she is unable to move. Assuming that both players play optimally throughout the game, what moves may Alice make on her first turn?
Putnam 1939, B7
Do either $(1)$ or $(2)$:
$(1)$ Let $ai = \sum_{n=0}^{\infty} \dfrac{x^{3n+i}}{(3n+i)!}$ Prove that $a_0^3 + a_1^3 + a_2^3 - 3 a_0a_1a_2 = 1.$
$(2)$ Let $O$ be the origin, $\lambda$ a positive real number, $C$ be the conic $ax^2 + by^2 + cx + dy + e = 0,$ and $C\lambda$ the conic $ax^2 + by^2 + \lambda cx + \lambda dy + \lambda 2e = 0.$ Given a point $P$ and a non-zero real number $k,$ define the transformation $D(P,k)$ as follows. Take coordinates $(x',y')$ with $P$ as the origin. Then $D(P,k)$ takes $(x',y')$ to $(kx',ky').$ Show that $D(O,\lambda)$ and $D(A,-\lambda)$ both take $C$ into $C\lambda,$ where $A$ is the point $(\dfrac{-c \lambda} {(a(1 + \lambda))}, \dfrac{-d \lambda} {(b(1 + \lambda))}) $. Comment on the case $\lambda = 1.$
2021 Putnam, A4
Let
\[
I(R)=\iint\limits_{x^2+y^2 \le R^2}\left(\frac{1+2x^2}{1+x^4+6x^2y^2+y^4}-\frac{1+y^2}{2+x^4+y^4}\right) dx dy.
\]
Find
\[
\lim_{R \to \infty}I(R),
\]
or show that this limit does not exist.
1940 Putnam, A6
Let $f(x)$ be a polynomial of degree $n$ such that $f(x)^{p}$ is divisible by $f'(x)^{q}$ for some positive integers $p,q$.
Prove that $f(x)$ is divisible by $f'(x)$ and that $f(x)$ has a single root of multiplicity $n$.
2003 Putnam, 4
Suppose that $a, b, c, A, B, C$ are real numbers, $a \not= 0$ and $A \not= 0$, such that \[|ax^2+ bx + c| \le |Ax^2+ Bx + C|\] for all real numbers $x$. Show that \[|b^2- 4ac| \le |B^2- 4AC|\]
2012 Putnam, 2
Let $P$ be a given (non-degenerate) polyhedron. Prove that there is a constant $c(P)>0$ with the following property: If a collection of $n$ balls whose volumes sum to $V$ contains the entire surface of $P,$ then $n>c(P)/V^2.$
1998 Putnam, 4
Let $A_1=0$ and $A_2=1$. For $n>2$, the number $A_n$ is defined by concatenating the decimal expansions of $A_{n-1}$ and $A_{n-2}$ from left to right. For example $A_3=A_2A_1=10$, $A_4=A_3A_2=101$, $A_5=A_4A_3=10110$, and so forth. Determine all $n$ such that $11$ divides $A_n$.
2023 Putnam, A4
Let $v_1, \ldots, v_{12}$ be unit vectors in $\mathbb{R}^3$ from the origin to the vertices of a regular icosahedron. Show that for every vector $v \in \mathbb{R}^3$ and every $\varepsilon>0$, there exist integers $a_1, \ldots, a_{12}$ such that $\left\|a_1 v_1+\cdots+a_{12} v_{12}-v\right\|<\varepsilon$.
1966 Putnam, B5
Given $n(\geq 3)$ distinct points in the plane, no three of which are on the same straight line, prove that there exists a simple closed polygon with these points as vertices.