Found problems: 966
Putnam 1939, A6
Do either $(1)$ or $(2)$:
$(1)$ A circle radius $r$ rolls around the inside of a circle radius $3r,$ so that a point on its circumference traces out a curvilinear triangle. Find the area inside this figure.
$(2)$ A frictionless shell is fired from the ground with speed $v$ at an unknown angle to the vertical. It hits a plane at a height $h.$ Show that the gun must be sited within a radius $\frac{v}{g} (v^2 - 2gh)^{\frac{1}{2}}$ of the point directly below the point of impact.
1993 India National Olympiad, 7
Let $A = \{ 1,2, 3 , \ldots, 100 \}$ and $B$ be a subset of $A$ having $53$ elements. Show that $B$ has 2 distinct elements $x$ and $y$ whose sum is divisible by $11$.
1995 Putnam, 6
For any $a>0$,set $\mathcal{S}(a)=\{\lfloor{na}\rfloor|n\in \mathbb{N}\}$. Show that there are no three positive reals $a,b,c$ such that
\[ \mathcal{S}(a)\cap \mathcal{S}(b)=\mathcal{S}(b)\cap \mathcal{S}(c)=\mathcal{S}(c)\cap \mathcal{S}(a)=\emptyset \]
\[ \mathcal{S}(a)\cup \mathcal{S}(b)\cup \mathcal{S}(c)=\mathbb{N} \]
1951 Putnam, B2
Two functions of $x$ are differentiable and not identically zero. Find an example of two such functions having the property that the derivative of their quotient is the quotient of their derivatives.
1973 Putnam, A5
A particle moves in $3$-space according to the equations:
$$ \frac{dx}{dt} =yz,\; \frac{dy}{dt} =xz,\; \frac{dz}{dt}= xy.$$
Show that:
(a) If two of $x(0), y(0), z(0)$ equal $0,$ then the particle never moves.
(b) If $x(0)=y(0)=1, z(0)=0,$ then the solution is
$$ x(t)= \sec t ,\; y(t) =\sec t ,\; z(t)= \tan t;$$
whereas if $x(0)=y(0)=1, z(0)=-1,$ then
$$ x(t) =\frac{1}{t+1} ,\; y(t)=\frac{1}{t+1}, z(t)=- \frac{1}{t+1}.$$
(c) If at least two of the values $x(0), y(0), z(0)$ are different from zero, then either the particle
moves to infinity at some finite time in the future, or it came from infinity at some finite
time in the past (a point $(x, y, z)$ in $3$-space "moves to infinity" if its distance from the
origin approaches infinity).
1941 Putnam, A7
Do either (1) or (2):
(1) Prove that the determinant of the matrix
$$\begin{pmatrix}
1+a^2 -b^2 -c^2 & 2(ab+c) & 2(ac-b)\\
2(ab-c) & 1-a^2 +b^2 -c^2 & 2(bc+a)\\
2(ac+b)& 2(bc-a) & 1-a^2 -b^2 +c^2
\end{pmatrix}$$
is given by $(1+a^2 +b^2 +c^2)^{3}$.
(2) A solid is formed by rotating the first quadrant of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ around the $x$-axis. Prove that this solid can rest in stable equilibrium on its vertex if and only if $\frac{a}{b}\leq \sqrt{\frac{8}{5}}$.
1964 Putnam, B5
Let $u_n$ denote the least common multiple of the first $n$ terms of a strictly increasing sequence of positive integers.
Prove that the series
$$\sum_{n=1}^{\infty} \frac{1}{ u_n }$$
is convergent
1948 Putnam, A1
What is the maximum of $|z^3 -z+2|$, where $z$ is a complex number with $|z|=1?$
1974 Putnam, A3
A well-known theorem asserts that a prime $p > 2$ can be written as the sum of two perfect squares ($p = m^2 +n^2$ , with $m$ and $n$ integers) if and only if $p \equiv 1$ (mod $4$). Assuming this result, find which
primes $p > 2$ can be written in each of the following forms, using integers $x$ and $y$:
a) $x^2 +16y^2, $
b) $4x^2 +4xy+ 5y^2.$
2017 Putnam, B2
Suppose that a positive integer $N$ can be expressed as the sum of $k$ consecutive positive integers
\[N=a+(a+1)+(a+2)+\cdots+(a+k-1)\]
for $k=2017$ but for no other values of $k>1.$ Considering all positive integers $N$ with this property, what is the smallest positive integer $a$ that occurs in any of these expressions?
2012 Putnam, 1
Let $S$ be a class of functions from $[0,\infty)$ to $[0,\infty)$ that satisfies:
(i) The functions $f_1(x)=e^x-1$ and $f_2(x)=\ln(x+1)$ are in $S;$
(ii) If $f(x)$ and $g(x)$ are in $S,$ the functions $f(x)+g(x)$ and $f(g(x))$ are in $S;$
(iii) If $f(x)$ and $g(x)$ are in $S$ and $f(x)\ge g(x)$ for all $x\ge 0,$ then the function $f(x)-g(x)$ is in $S.$
Prove that if $f(x)$ and $g(x)$ are in $S,$ then the function $f(x)g(x)$ is also in $S.$
1965 Putnam, B4
Consider the function
\[
f(x,n) = \frac{\binom n0 + \binom n2 x + \binom n4x^2 + \cdots}{\binom n1 + \binom n3 x + \binom n5 x^2 + \cdots},
\]
where $n$ is a positive integer. Express $f(x,n+1)$ rationally in terms of $f(x,n)$ and $x$. Hence, or otherwise, evaluate $\textstyle\lim_{n\to\infty}f(x,n)$ for suitable fixed values of $x$. (The symbols $\textstyle\binom nr$ represent the binomial coefficients.)
2011 Putnam, B1
Let $h$ and $k$ be positive integers. Prove that for every $\varepsilon >0,$ there are positive integers $m$ and $n$ such that \[\varepsilon < \left|h\sqrt{m}-k\sqrt{n}\right|<2\varepsilon.\]
1941 Putnam, A1
Prove that the polynomial
$$(a-x)^6 -3a(a-x)^5 +\frac{5}{2} a^2 (a-x)^4 -\frac{1}{2} a^4 (a-x)^2 $$
takes only negative values for $0<x<a$.
1951 Putnam, A3
Find the sum to infinity of the series: \[ 1 - \frac 14 + \frac 17 - \frac 1{10} + \cdots + \frac{(-1)^{n + 1}}{3n - 2} + \cdots.\]
1981 Putnam, A3
Find
$$ \lim_{t\to \infty} e^{-t} \int_{0}^{t} \int_{0}^{t} \frac{e^x -e^y }{x-y} \,dx\,dy,$$
or show that the limit does not exist.
1951 Putnam, B3
Show that if $x$ is positive, then \[ \log_e (1 + 1/x) > 1 / (1 + x).\]
1958 November Putnam, A7
Let $a$ and $b$ be relatively prime positive integers, $b$ even. For each positive integer $q$, let $p=p(q)$ be chosen so that
$$ \left| \frac{p}{q} - \frac{a}{b} \right|$$
is a minimum. Prove that
$$ \lim_{n \to \infty} \sum_{q=1 }^{n} \frac{ q\left| \frac{p}{q} - \frac{a}{b} \right|}{n} = \frac{1}{4}.$$
1951 Putnam, B4
Investigate, in any way which yields significant results, the existence, in the plane, of the configuration consisting of an ellipse simultaneously tangent to four distinct concentric circles.
1995 Putnam, 3
The number $d_1d_2\cdots d_9$ has nine (not necessarily distinct) decimal digits. The number $e_1e_2\cdots e_9$ is such that each of the nine $9$-digit numbers formed by replacing just one of the digits $d_i$ in $d_1d_2\cdots d_9$ by the corresponding digit $e_i \;\;(1 \le i \le 9)$ is divisible by $7$. The number $f_1f_2\cdots f_9$ is related to $e_1e_2\cdots e_9$ is the same way: that is, each of the nine numbers formed by replacing one of the $e_i$ by the corresponding $f_i$ is divisible by $7$. Show that, for each $i$, $d_i-f_i$ is divisible by $7$. [For example, if $d_1d_2\cdots d_9 = 199501996$, then $e_6$ may be $2$ or $9$, since $199502996$ and $199509996$ are multiples of $7$.]
2011 Putnam, B2
Let $S$ be the set of all ordered triples $(p,q,r)$ of prime numbers for which at least one rational number $x$ satisfies $px^2+qx+r=0.$ Which primes appear in seven or more elements of $S?$
1953 Putnam, B1
Is the infinite series
$$\sum_{n=1}^{\infty} \frac{1}{n^{1+\frac{1}{n}}}$$
convergent?
1952 Putnam, A1
Let \[ f(x) = \sum_{i=0}^{i=n} a_i x^{n - i}\] be a polynomial of degree $n$ with integral coefficients. If $a_0, a_n,$ and $f(1)$ are odd, prove that $f(x) = 0$ has no rational roots.
1970 Putnam, B5
Let $u_n$ denote the ramp function
$$ u_n (x) =\begin{cases}
-n \;\; \text{for} \;\; x \leq -n, \\
\; x \;\;\; \text{for} \;\; -n \leq x \leq n,\\
\;n \;\; \; \text{for} \;\; n \leq x,
\end{cases}$$
and let $f$ be a real function of a real variable. Show that $f$ is continuous if and only if $u_n \circ f$ is continuous for all $n.$
1955 Putnam, A3
Suppose that $\sum^\infty_{i=1} x_i$ is a convergent series of positive terms which monotonically decrease (that is, $x_1 \ge x_2 \ge x_3 \ge \cdots$). Let $P$ denote the set of all numbers which are sums of some (finite or infinite) subseries of $\sum^\infty_{i= 1} x_i.$ Show that $P$ is an interval if and only if \[ x_n \le \sum^\infty_{i = n + 1} x_i\] for every integer $n.$