Found problems: 966
1993 Putnam, B3
$x$ and $y$ are chosen at random (with uniform density) from the interval $(0, 1)$. What is the probability that the closest integer to $x/y$ is even?
1960 Putnam, B3
The motion of the particles of a fluid in the plane is specified by the following components of velocity
$$\frac{dx}{dt}=y+2x(1-x^2 -y^2),\;\; \frac{dy}{dt}=-x.$$
Sketch the shape of the trajectories near the origin. Discuss what happens to an individual particle as $t\to \infty$, and justify your conclusion.
2024 Putnam, A3
Let $S$ be the set of bijections
\[
T\colon\{1,\,2,\,3\}\times\{1,\,2,\,\ldots,\,2024\}\to\{1,\,2,\,\ldots,\,6072\}
\]
such that $T(1,\,j)<T(2,\,j)<T(3,\,j)$ for all $j\in\{1,\,2,\,\ldots,\,2024\}$ and $T(i,\,j)<T(i,\,j+1)$ for all $i\in\{1,\,2,\,3\}$ and $j\in\{1,\,2,\,\ldots,\,2023\}$. Do there exist $a$ and $c$ in $\{1,\,2,\,3\}$ and $b$ and $d$ in $\{1,\,2,\,\ldots,\,2024\}$ such that the fraction of elements $T$ in $S$ for which $T(a,\,b)<T(c,\,d)$ is at least $1/3$ and at most $2/3$.
1960 Putnam, B1
Find all integer solutions $(m,n)$ to $m^{n}=n^{m}.$
2017 Putnam, A6
The $30$ edges of a regular icosahedron are distinguished by labeling them $1,2,\dots,30.$ How many different ways are there to paint each edge red, white, or blue such that each of the 20 triangular faces of the icosahedron has two edges of the same color and a third edge of a different color?
2015 Putnam, A5
Let $q$ be an odd positive integer, and let $N_q$ denote the number of integers $a$ such that $0<a<q/4$ and $\gcd(a,q)=1.$ Show that $N_q$ is odd if and only if $q$ is of the form $p^k$ with $k$ a positive integer and $p$ a prime congruent to $5$ or $7$ modulo $8.$
2007 Putnam, 4
Let $ n$ be a positive integer. Find the number of pairs $ P,Q$ of polynomials with real coefficients such that
\[ (P(X))^2\plus{}(Q(X))^2\equal{}X^{2n}\plus{}1\]
and $ \text{deg}P<\text{deg}{Q}.$
Putnam 1938, A6
A swimmer is standing at a corner of a square swimming pool. She swims at a fixed speed and runs at a fixed speed (possibly different). No time is taken entering or leaving the pool. What path should she follow to reach the opposite corner of the pool in the shortest possible time?
1973 Putnam, A6
Prove that it is impossible for seven distinct straight lines to be situated in the euclidean plane so as to have at least six points where exactly three of these lines intersect and at least four points where exactly two of these lines intersect.
2018 Putnam, B3
Find all positive integers $n < 10^{100}$ for which simultaneously $n$ divides $2^n$, $n-1$ divides $2^n - 1$, and $n-2$ divides $2^n - 2$.
2002 Putnam, 5
Define a sequence by $a_0=1$, together with the rules $a_{2n+1}=a_n$ and $a_{2n+2}=a_n+a_{n+1}$ for each integer $n\ge0$. Prove that every positive rational number appears in the set $ \left\{ \tfrac {a_{n-1}}{a_n}: n \ge 1 \right\} = \left\{ \tfrac {1}{1}, \tfrac {1}{2}, \tfrac {2}{1}, \tfrac {1}{3}, \tfrac {3}{2}, \cdots \right\} $.
1965 Putnam, A5
In how many ways can the integers from $1$ to $n$ be ordered subject to the condition that, except for the first integer on the left, every integer differs by $1$ from some integer to the left of it?
1986 Putnam, B5
Let $f(x,y,z) = x^2+y^2+z^2+xyz$. Let $p(x,y,z), q(x,y,z)$, $r(x,y,z)$ be polynomials with real coefficients satisfying
\[
f(p(x,y,z), q(x,y,z), r(x,y,z)) = f(x,y,z).
\]
Prove or disprove the assertion that the sequence $p,q,r$ consists of some permutation of $\pm x, \pm y, \pm z$, where the number of minus signs is $0$ or $2.$
1955 Putnam, A1
Prove that there is no set of integers $m, n, p$ except $0, 0, 0$ for which \[m + n \sqrt2 + p \sqrt3 = 0.\]
1953 Putnam, B5
Show that the roots of $x^4 +ax^3 +bx^2 +cx +d$, if suitably numbered, satisfy the relation $\frac{r_1 }{r_2 } = \frac{ r_3 }{r _4},$ provided $a^2 d=c^2 \ne 0.$
2019 Putnam, A5
Let $p$ be an odd prime number, and let $\mathbb{F}_p$ denote the field of integers modulo $p$. Let $\mathbb{F}_p[x]$ be the ring of polynomials over $\mathbb{F}_p$, and let $q(x) \in \mathbb{F}_p[x]$ be given by $q(x) = \sum_{k=1}^{p-1} a_k x^k$ where $a_k = k^{(p-1)/2}$ mod $p$. Find the greatest nonnegative integer $n$ such that $(x-1)^n$ divides $q(x)$ in $\mathbb{F}_p[x]$.
2001 Putnam, 6
Assume that $(a_n)_{n \ge 1}$ is an increasing sequence of positive real numbers such that $\lim \tfrac{a_n}{n}=0$. Must there exist infinitely many positive integers $n$ such that $a_{n-i}+a_{n+i}<2a_n$ for $i=1,2,\cdots,n-1$?
PEN G Problems, 4
Let $a, b, c$ be integers, not all zero and each of absolute value less than one million. Prove that \[\left\vert a+b\sqrt{2}+c\sqrt{3}\right\vert > \frac{1}{10^{21}}.\]
1994 Putnam, 1
Find all positive integers that are within $250$ of exactly $15$ perfect squares.
2013 Putnam, 4
A finite collection of digits $0$ and $1$ is written around a circle. An [i]arc[/i] of length $L\ge 0$ consists of $L$ consecutive digits around the circle. For each arc $w,$ let $Z(w)$ and $N(w)$ denote the number of $0$'s in $w$ and the number of $1$'s in $w,$ respectively. Assume that $|Z(w)-Z(w')|\le 1$ for any two arcs $w,w'$ of the same length. Suppose that some arcs $w_1,\dots,w_k$ have the property that \[Z=\frac1k\sum_{j=1}^kZ(w_j)\text{ and }N=\frac1k\sum_{j=1}^k N(w_j)\] are both integers. Prove that there exists an arc $w$ with $Z(w)=Z$ and $N(w)=N.$
1973 Putnam, B3
Consider an integer $p>1$ with the property that the polynomial $x^2 - x + p$ takes prime values for all integers $x$ such that $0\leq x <p$. Show that there is exactly one triple of integers $a, b, c$ satisfying the conditions:
$$b^2 -4ac = 1-4p,\;\; 0<a \leq c,\;\; -a\leq b<a.$$
Putnam 1939, A5
Do either $(1)$ or $(2)$
$(1)$ $x$ and $y$ are functions of $t.$ Solve $x' = x + y - 3, y' = -2x + 3y + 1,$ given that $x(0) = y(0) = 0.$
$(2)$ A weightless rod is hinged at $O$ so that it can rotate without friction in a vertical plane. A mass $m$ is attached to the end of the rod $A,$ which is balanced vertically above $O.$ At time $t = 0,$ the rod moves away from the vertical with negligible initial angular velocity. Prove that the mass first reaches the position under $O$ at $t = \sqrt{(\frac{OA}{g})} \ln{(1 + sqrt(2))}.$
2009 Putnam, A5
Is there a finite abelian group $ G$ such that the product of the orders of all its elements is $ 2^{2009}?$
2016 Putnam, B6
Evaluate
\[\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k}\sum_{n=0}^{\infty}\frac{1}{k2^n+1}.\]
1996 Putnam, 5
Let $p$ be a prime greater than $3$. Prove that
\[ p^2\Big| \sum_{i=1}^{\left\lfloor\frac{2p}{3}\right\rfloor}\dbinom{p}{i}. \]