Found problems: 966
2013 Putnam, 2
Let $S$ be the set of all positive integers that are [i]not[/i] perfect squares. For $n$ in $S,$ consider choices of integers $a_1,a_2,\dots, a_r$ such that $n<a_1<a_2<\cdots<a_r$ and $n\cdot a_1\cdot a_2\cdots a_r$ is a perfect square, and let $f(n)$ be the minimum of $a_r$ over all such choices. For example, $2\cdot 3\cdot 6$ is a perfect square, while $2\cdot 3,2\cdot 4, 2\cdot 5, 2\cdot 3\cdot 4,$ $2\cdot 3\cdot 5, 2\cdot 4\cdot 5,$ and $2\cdot 3\cdot 4\cdot 5$ are not, and so $f(2)=6.$ Show that the function $f$ from $S$ to the integers is one-to-one.
1990 Putnam, A2
Is $ \sqrt{2} $ the limit of a sequence of numbers of the form $ \sqrt[3]{n} - \sqrt[3]{m} $, where $ n, m = 0, 1, 2, \cdots $.
2004 Putnam, A5
An $m\times n$ checkerboard is colored randomly: each square is independently assigned red or black with probability $\frac12.$ we say that two squares, $p$ and $q$, are in the same connected monochromatic region if there is a sequence of squares, all of the same color, starting at $p$ and ending at $q,$ in which successive squares in the sequence share a common side. Show that the expected number of connected monochromatic regions is greater than $\frac{mn}8.$
1978 Putnam, B4
Prove that for every real number $N$ the equation
$$ x_{1}^{2}+x_{2}^{2} +x_{3}^{2} +x_{4}^{2} = x_1 x_2 x_3 +x_1 x_2 x_4 + x_1 x_3 x_4 +x_2 x_3 x_4$$
has an integer solution $(x_1 , x_2 , x_3 , x_4)$ for which $x_1, x_2 , x_3 $ and $x_4$ are all larger than $N.$
1994 Putnam, 6
For $a\in \mathbb{Z}$ define \[ n_a=101a-100\cdot 2^a \]
Show that, for $0\le a,b,c,d\le 99$
\[ n_a+n_b\equiv n_c+n_d\pmod{10100}\implies \{a,b\}=\{c,d\} \]
2024 Putnam, B3
Let $r_n$ be the $n$th smallest positive solution to $\tan x=x$, where the argument of tangent is in radians. Prove that
\[
0<r_{n+1}-r_n-\pi<\frac{1}{(n^2+n)\pi}
\]
for $n\geq 1$.
Putnam 1938, A7
Do either $(1)$ or $(2)$
$(1)$ $S$ is a thin spherical shell of constant thickness and density with total mass $M$ and center $O.$ $P$ is a point outside $S.$ Prove that the gravitational attraction of $S$ at $P$ is the same as the gravitational attraction of a point mass $M$ at $O.$
$(2)$ $K$ is the surface $z = xy$ in Euclidean $3-$space. Find all straight lines lying in $S$. Draw a diagram to illustrate them.
1998 Putnam, 6
Let $A,B,C$ denote distinct points with integer coefficients in $\mathbb{R}^2$. Prove that if \[(|AB|+|BC|)^2<8\cdot[ABC]+1\] then $A,B,C$ are three vertices of a square. Here $|XY|$ is the length of segment $XY$ and $[ABC]$ is the area of triangle $ABC$.
1990 Putnam, A5
If $\mathbf{A}$ and $\mathbf{B}$ are square matrices of the same size such that $\mathbf{ABAB}=\mathbf{0}$, does it follow that $\mathbf{BABA}=\mathbf{0}$.
2017 Putnam, A5
Each of the integers from $1$ to $n$ is written on a separate card, and then the cards are combined into a deck and shuffled. Three players, $A,B,$ and $C,$ take turns in the order $A,B,C,A,\dots$ choosing one card at random from the deck. (Each card in the deck is equally likely to be chosen.) After a card is chosen, that card and all higher-numbered cards are removed from the deck, and the remaining cards are reshuffled before the next turn. Play continues until one of the three players wins the game by drawing the card numbered $1.$
Show that for each of the three players, there are arbitrarily large values of $n$ for which that player has the highest probability among the three players of winning the game.
2003 Putnam, 6
Let $f(x)$ be a continuous real-valued function defined on the interval $[0, 1]$. Show that \[\int_0^1\int_0^1|f(x)+f(y)|dx \; dy \ge \int_0^1 |f(x)|dx\]
2010 Contests, A3
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.
1968 Putnam, B6
Show that one cannot find compact sets $A_1, A_2, A_3, \ldots$ in $\mathbb{R}$ such that
(1) All elements of $A_n$ are rational.
(2) Any compact set $K\subset \mathbb{R}$ which only contains rational numbers is contained in some $A_{m}$.
1996 Putnam, 3
Suppose that each of $20$ students has made a choice of anywhere from $0$ to $6$ courses from a total of $6$ courses offered. Prove or disprove : there are $5$ students and $2$ courses such that all $5$ have chosen both courses or all $5$ have chosen neither course.
2011 Putnam, B3
Let $f$ and $g$ be (real-valued) functions defined on an open interval containing $0,$ with $g$ nonzero and continuous at $0.$ If $fg$ and $f/g$ are differentiable at $0,$ must $f$ be differentiable at $0?$
Putnam 1939, A2
Let $C$ be the curve $y = x^3$ (where $x$ takes all real values). The tangent at $A$ meets the curve again at $B.$ Prove that the gradient at $B$ is $4$ times the gradient at $A.$
1952 Putnam, B2
Find the surface generated by the solutions of \[ \frac {dx}{yz} = \frac {dy}{zx} = \frac{dz}{xy}, \] which intersects the circle $y^2+ z^2 = 1, x = 0.$
1959 Putnam, B7
For each positive integer $n$, let $f_n$ be a real-valued symmetric function of $n$ real variables. Suppose that for all $n$ and all real numbers $x_1,\ldots,x_n, x_{n+1},y$ it is true that
$\;(1)\; f_{n}(x_1 +y ,\ldots, x_n +y) = f_{n}(x_1 ,\ldots, x_n) +y,$
$\;(2)\;f_{n}(-x_1 ,\ldots, -x_n) =-f_{n}(x_1 ,\ldots, x_n),$
$\;(3)\; f_{n+1}(f_{n}(x_1,\ldots, x_n),\ldots, f_{n}(x_1,\ldots, x_n), x_{n+1}) =f_{n+1}(x_1 ,\ldots, x_{n}).$
Prove that $f_{n}(x_{1},\ldots, x_n) =\frac{x_{1}+\cdots +x_{n}}{n}.$
1949 Putnam, B4
Show that the coefficients $a_1 , a_2 , a_3 ,\ldots$ in the expansion
$$\frac{1}{4}\left(1+x-\frac{1}{\sqrt{1-6x+x^{2}}}\right) =a_{1} x+ a_2 x^2 + a_3 x^3 +\ldots$$
are positive integers.
1947 Putnam, B3
Let $x,y$ be cartesian coordinates in the plane. $I$ denotes the line segment $1\leq x\leq 3 , y=1.$ For every point $P$ on $I$, let $P'$ denote the point that lies on the segment joining the origin to $P$ and such that the distance $P P'$ is equal to $1 \slash 100.$ As $P$ describes $I$, the point $P'$ describes a curve $C$. Which of $I$ and $C$ has greater length?
1962 Putnam, B6
Let
$$f(x) =\sum_{k=0}^{n} a_{k} \sin kx +b_{k} \cos kx,$$
where $a_k$ and $b_k$ are constants. Show that if $|f(x)| \leq 1$ for $x \in [0, 2 \pi]$ and there exist $0\leq x_1 < x_2 <\ldots < x_{2n} < 2 \pi$ with $|f(x_i )|=1,$ then $f(x)= \cos(nx +a)$ for some constant $a.$
2007 Putnam, 3
Let $ x_0 \equal{} 1$ and for $ n\ge0,$ let $ x_{n \plus{} 1} \equal{} 3x_n \plus{} \left\lfloor x_n\sqrt {5}\right\rfloor.$ In particular, $ x_1 \equal{} 5,\ x_2 \equal{} 26,\ x_3 \equal{} 136,\ x_4 \equal{} 712.$ Find a closed-form expression for $ x_{2007}.$ ($ \lfloor a\rfloor$ means the largest integer $ \le a.$)
1954 Putnam, A7
Prove that there are no integers $x$ and $y$ for which
$$x^2 +3xy-2y^2 =122.$$
2004 Putnam, A3
Define a sequence $\{u_n\}_{n=0}^{\infty}$ by $u_0=u_1=u_2=1,$ and thereafter by the condition that
$\det\begin{vmatrix} u_n & u_{n+1} \\ u_{n+2} & u_{n+3} \end{vmatrix}=n!$
for all $n\ge 0.$ Show that $u_n$ is an integer for all $n.$ (By convention, $0!=1$.)
2004 Putnam, B4
Let $n$ be a positive integer, $n \ge 2$, and put $\theta=\frac{2\pi}{n}$. Define points $P_k=(k,0)$ in the [i]xy[/i]-plane, for $k=1,2,\dots,n$. Let $R_k$ be the map that rotates the plane counterclockwise by the angle $\theta$ about the point $P_k$. Let $R$ denote the map obtained by applying in order, $R_1$, then $R_2$, ..., then $R_n$. For an arbitrary point $(x,y)$, find and simplify the coordinates of $R(x,y)$.