Found problems: 966
1992 Putnam, B6
Let $M$ be a set of real $n \times n$ matrices such that
i) $I_{n} \in M$, where $I_n$ is the identity matrix.
ii) If $A\in M$ and $B\in M$, then either $AB\in M$ or $-AB\in M$, but not both
iii) If $A\in M$ and $B \in M$, then either $AB=BA$ or $AB=-BA$.
iv) If $A\in M$ and $A \ne I_n$, there is at least one $B\in M$ such that $AB=-BA$.
Prove that $M$ contains at most $n^2 $ matrices.
2016 Putnam, A2
Given a positive integer $n,$ let $M(n)$ be the largest integer $m$ such that
\[\binom{m}{n-1}>\binom{m-1}{n}.\]
Evaluate
\[\lim_{n\to\infty}\frac{M(n)}{n}.\]
1947 Putnam, B2
Let $f(x)$ be a differentiable function defined on the interval $(0,1)$ such that $|f'(x)| \leq M$ for $0<x<1$ and a positive real number $M.$ Prove that
$$\left| \int_{0}^{1} f(x)\; dx - \frac{1}{n} \sum_{k=1}^{n} f\left(\frac{k}{n} \right) \right | \leq \frac{M}{n}.$$
1953 Putnam, A7
Assuming that the roots of $x^3 +px^2 +qx +r=0$ are all real and positive, find the relation between $p,q,r$ which is a necessary and sufficient condition that the roots are the cosines of the angles of a triangle.
1965 Putnam, B5
Consider collections of unordered pairs of $V$ different objects $a$, $b$, $c$, $\ldots$, $k$. Three pairs such as $ab$, $bc$, $ab$ are said to form a triangle. Prove that, if $4E\leq V^2$, it is possible to choose $E$ pairs so that no triangle is formed.
1968 Putnam, B2
Let $G$ be a finite group with $n$ elements and $K$ a subset of $G$ with more than $\frac{n}{2}$ elements. Show that for any $g\in G$ one can find $h,k\in K$ such that $g=h\cdot k$.
1973 Putnam, A4
How many zeroes does the function $f(x)=2^x -1 -x^2 $ have on the real line?
1999 Putnam, 6
Let $S$ be a finite set of integers, each greater than $1$. Suppose that for each integer $n$ there is some $s\in S$ such that $\gcd(s,n)=1$ or $\gcd(s,n)=s$. Show that there exist $s,t\in S$ such that $\gcd(s,t)$ is prime.
1957 Putnam, B7
Let $C$ consist of a regular polygon and its interior. Show that for each positive integer $n$, there exists a set of points $S(n)$ in the plane such that every $n$ points can be covered by $C$, but $S(n)$ cannot be covered by $C.$
1942 Putnam, A6
Any circle in the $xy$-plane is "represented" by a point on the vertical line through the center of the circle and at a distance "above" the plane of the circle equal to the radius of the circle.
Show that the locus of the representations of all the circles which cut a fixed circle at a constant angle is a portion of a one-sheeted hyperboloid.
By consideration of a suitable family of circles in the plane, demonstrate the existence of two families of rulings on the hyperboloid.
1984 Putnam, A5
Putnam 1984/A5) Let $R$ be the region consisting of all triples $(x,y,z)$ of nonnegative real numbers satisfying $x+y+z\leq 1$. Let $w=1-x-y-z$. Express the value of the triple integral
\[\iiint_{R}xy^{9}z^{8}w^{4}\ dx\ dy\ dz\]
in the form $a!b!c!d!/n!$ where $a,b,c,d$ and $n$ are positive integers.
[hide="A solution"]\[\iiint_{R}xy^{9}z^{8}w^{4}\ dx dy dz = 4\iiint_{R}\int_{0}^{1-x-y-z}xy^{9}z^{8}w^{3}\ dw dx dy dz = 4\iiiint_{Q}xy^{9}z^{8}w^{3}\ dw dx dy dz\]
where $Q=\left\{ (x,y,z,w)\in\mathbb{R}^{4}|\ x,y,z,w\geq 0, x+y+z+w\leq 1\right\}$, which is a Dirichlet integral giving
\[4\iiiint_{Q}x^{1}y^{9}z^{8}w^{3}\ dw dx dy dz = 4\cdot\frac{1!9!8!3!}{(2+10+9+4)!}= \frac{1!9!8!4!}{25!}\][/hide]
1958 February Putnam, A7
Show that ten equal-sized squares cannot be placed on a plane in such a way that no two have an interior point in common and the first touches each of the others.
1972 Putnam, B6
Let $ n_1<n_2<n_3<\cdots <n_k$ be a set of positive integers. Prove that the polynomial $ 1\plus{}z^{n_1}\plus{}z^{n_2}\plus{}\cdots \plus{}z^{n_k}$ has no roots inside the circle $ |z|<\frac{\sqrt{5}\minus{}1}{2}$.
2002 Putnam, 2
Consider a polyhedron with at least five faces such that exactly three edges emerge from each of its vertices. Two players play the following game: Each, in turn, signs his or her name on a previously unsigned face. The winner is the player who first succeeds in signing three faces that share a common vertex. Show that the player who signs first will always win by playing as well as possible.
1960 Putnam, A6
A player repeatedly throwing a die is to play until their score reaches or passes a total $n$. Denote by $p(n)$ the probability of making exactly the total $n,$ and find the value of $\lim_{n \to \infty} p(n).$
1968 Putnam, A3
Let $S$ be a finite set and $P$ the set of all subsets of $S$. Show that one can label the elements of $P$ as $A_i$ such that
(1) $A_1 =\emptyset$.
(2) For each $n\geq1 $ we either have $A_{n-1}\subset A_{n}$ and $|A_{n} \setminus A_{n-1}|=1$ or $A_{n}\subset A_{n-1}$ and $|A_{n-1} \setminus A_{n}|=1.$
2005 Putnam, B5
Let $P(x_1,\dots,x_n)$ denote a polynomial with real coefficients in the variables $x_1,\dots,x_n,$ and suppose that
(a) $\left(\frac{\partial^2}{\partial x_1^2}+\cdots+\frac{\partial^2}{\partial x_n^2} \right)P(x_1,\dots,x_n)=0$ (identically)
and that
(b) $x_1^2+\cdots+x_n^2$ divides $P(x_1,\dots,x_n).$
Show that $P=0$ identically.
1954 Putnam, B3
Let $[a_1 , b_1 ] , \ldots, [a_n ,b_n ]$ be a collection of closed intervals such that any of these closed intervals have a point in common. Prove that there exists a point contained in every one of these intervals.
1986 Putnam, B6
Suppose $A,B,C,D$ are $n \times n$ matrices with entries in a field $F$, satisfying the conditions that $AB^T$ and $CD^T$ are symmetric and $AD^T - BC^T = I$. Here $I$ is the $n \times n$ identity matrix, and if $M$ is an $n \times n$ matrix, $M^T$ is its transpose. Prove that $A^T D - C^T B = I$.
2011 Mongolia Team Selection Test, 2
Mongolia TST 2011 Test 1 #2
Let $p$ be a prime number. Prove that:
$\sum_{k=0}^p (-1)^k \dbinom{p}{k} \dbinom{p+k}{k} \equiv -1 (\mod p^3)$
(proposed by B. Batbayasgalan, inspired by Putnam olympiad problem)
Note: I believe they meant to say $p>2$ as well.
1947 Putnam, B1
Let $f(x)$ be a function such that $f(1)=1$ and for $x \geq 1$
$$f'(x)= \frac{1}{x^2 +f(x)^{2}}.$$
Prove that
$$\lim_{x\to \infty} f(x)$$
exists and is less than $1+ \frac{\pi}{4}.$
2018 Putnam, B5
Let $f = (f_1, f_2)$ be a function from $\mathbb{R}^2$ to $\mathbb{R}^2$ with continuous partial derivatives $\tfrac{\partial f_i}{\partial x_j}$ that are positive everywhere. Suppose that
\[\frac{\partial f_1}{\partial x_1} \frac{\partial f_2}{\partial x_2} - \frac{1}{4} \left(\frac{\partial f_1}{\partial x_2} + \frac{\partial f_2}{\partial x_1} \right)^2 > 0\]
everywhere. Prove that $f$ is one-to-one.
2013 Putnam, 1
Recall that a regular icosahedron is a convex polyhedron having 12 vertices and 20 faces; the faces are congruent equilateral triangles. On each face of a regular icosahedron is written a nonnegative integer such that the sum of all $20$ integers is $39.$ Show that there are two faces that share a vertex and have the same integer written on them.
1951 Putnam, B1
Find the conditions that the functions $M(x, y)$ and $N (x, y)$ must satisfy in order that the differential equation $Mdx + Ndy =0$ shall have an integrating factor of the form $f(xy).$ You may assume that $M$ and $N$ have continuous partial derivatives of all orders.
2014 Putnam, 6
Let $f:[0,1]\to\mathbb{R}$ be a function for which there exists a constant $K>0$ such that $|f(x)-f(y)|\le K|x-y|$ for all $x,y\in [0,1].$ Suppose also that for each rational number $r\in [0,1],$ there exist integers $a$ and $b$ such that $f(r)=a+br.$ Prove that there exist finitely many intervals $I_1,\dots,I_n$ such that $f$ is a linear function on each $I_i$ and $[0,1]=\bigcup_{i=1}^nI_i.$