Found problems: 966
2010 Contests, A1
Given a positive integer $n,$ what is the largest $k$ such that the numbers $1,2,\dots,n$ can be put into $k$ boxes so that the sum of the numbers in each box is the same?
[When $n=8,$ the example $\{1,2,3,6\},\{4,8\},\{5,7\}$ shows that the largest $k$ is [i]at least[/i] 3.]
1972 Putnam, B2
A particle moves in a straight line with monotonically decreasing acceleration. It starts from rest and has velocity $v$ a distance $d$ from the start. What is the maximum time it could have taken to travel the distance $d$?
1960 Putnam, A7
Let $N(n)$ denote the smallest positive integer $N$ such that $x^N =e$ for every element $x$ of the symmetric group $S_n$, where $e$ denotes the identity permutation. Prove that if $n>1,$
$$\frac{N(n)}{N(n-1)} =\begin{cases} p \;\text{if}\; n\; \text{is a power of a prime } p\\
1\; \text{otherwise}.
\end{cases}$$
1974 Putnam, B5
Show that
$$1+\frac{n}{1!} + \frac{n^{2}}{2!} +\ldots+ \frac{n^{n}}{n!} > \frac{e^{n}}{2}$$
for every integer $n\geq 0.$
2006 Putnam, B1
Show that the curve $x^{3}+3xy+y^{3}=1$ contains only one set of three distinct points, $A,B,$ and $C,$ which are the vertices of an equilateral triangle.
2006 Putnam, B4
Let $Z$ denote the set of points in $\mathbb{R}^{n}$ whose coordinates are $0$ or $1.$ (Thus $Z$ has $2^{n}$ elements, which are the vertices of a unit hypercube in $\mathbb{R}^{n}$.) Given a vector subspace $V$ of $\mathbb{R}^{n},$ let $Z(V)$ denote the number of members of $Z$ that lie in $V.$ Let $k$ be given, $0\le k\le n.$ Find the maximum, over all vector subspaces $V\subseteq\mathbb{R}^{n}$ of dimension $k,$ of the number of points in $V\cap Z.$
2004 Putnam, B2
Let $m$ and $n$ be positive integers. Show that
$\frac{(m+n)!}{(m+n)^{m+n}} < \frac{m!}{m^m}\cdot\frac{n!}{n^n}$
1949 Putnam, A2
We consider three vectors drawn from the same initial point $O,$ of lengths $a,b$ and $c$, respectively. Let $E$ be the parallelepiped with vertex $O$ of which the given vectors are the edges and $H$ the parallelepiped with vertex $O$ of which the given vectors are the altitudes. Show that the product of the volumes of $E$ and $H$ equals $(abc)^{2}$ and generalize this result to $n$ dimensions.
1950 Putnam, B3
In the Gregorian calendar:
(i) years not divisible by $4$ are common years;
(ii) years divisible by $4$ but not by $100$ are leap years;
(iii) years divisible by $100$ but not by $400$ are common years;
(iv) years divisible by $400$ are leap years;
(v) a leap year contains $366$ days; a common year $365$ days.
Prove that the probability that Christmas falls on a Wednesday is not $1/7.$
1958 November Putnam, A3
Under the assumption that the following set of relations has a unique solution for $u(t),$ determine it.
$$ \frac{d u(t) }{dt} = u(t) + \int_{0}^{t} u(s)\, ds, \;\;\; u(0)=1.$$
1969 Putnam, B5
Let $a_1 <a_2 < \ldots$ be an increasing sequence of positive integers. Let the series
$$\sum_{i=1}^{\infty} \frac{1}{a_i }$$
be convergent. For any real number $x$, let $k(x)$ be the number of the $a_i$ which do not exceed $x$. Show
that $\lim_{x\to \infty} \frac{k(x)}{x}=0.$
1967 Putnam, B6
Let $f$ be a real-valued function having partial derivatives and which is defined for $x^2 +y^2 \leq1$ and is such that $|f(x,y)|\leq 1.$ Show that there exists a point $(x_0, y_0 )$ in the interior of the unit circle such that
$$\left( \frac{ \partial f}{\partial x}(x_0 ,y_0 ) \right)^{2}+ \left( \frac{ \partial f}{\partial y}(x_0 ,y_0 ) \right)^{2} \leq 16.$$
1958 November Putnam, B4
Let $C$ be a real number, and let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a three times differentiable function such that
$$ \lim_{x \to \infty} f(x)=C, \;\; \; \lim_{x \to \infty} f'''(x)=0.$$
Prove that
$$ \lim_{x \to \infty} f'(x) =0 \;\; \text{and} \;\; \lim_{x \to \infty} f''(x)=0.$$
1969 Putnam, A3
Let $P$ be a non-selfintersecting closed polygon with $n$ sides. Let its vertices be $P_1 , P_2 ,\ldots, P_n .$
Let $m$ other points,$Q_1 , Q_2 ,\ldots, Q_m $ , interior to $P$, be given. Let the figure be triangulated.
This means that certain pairs of the $(n+m)$ points $P_1 ,\ldots , Q_m$ are connected by line
segments such that (i) the resulting figure consists exclusively of a set $T$ of triangles, (ii) if two
different triangles in $T$ have more than a vertex in common then they have exactly a side in
common, and (iii) the set of vertices of the triangles in $T$ is precisely the set of the $(n+m)$ points
$P_1 ,\ldots , Q_m.$ How many triangles are in $T$?
2015 Putnam, B1
Let $f$ be a three times differentiable function (defined on $\mathbb{R}$ and real-valued) such that $f$ has at least five distinct real zeros. Prove that $f+6f'+12f''+8f'''$ has at least two distinct real zeros.
2020 Putnam, A3
Let $a_0=\pi /2$, and let $a_n=\sin (a_{n-1})$ for $n\ge 1$. Determine whether
\[ \sum_{n=1}^{\infty}a_n^2 \]
converges.
1958 February Putnam, B3
In a round-robin tournament with $n$ players in which there are no draws, the numbers of wins scored by the players are $s_1 , s_2 , \ldots, s_n$. Prove that a necessary and sufficient condition for the existence of three players $A,B,C$ such that $A$ beats $B$, $B$ beats $C$, and $C$ beats $A$ is
$$s_{1}^{2} +s_{2}^{2} + \ldots +s_{n}^{2} < \frac{(2n-1)(n-1)n}{6}.$$
1953 Putnam, B4
Determine the equations of a surface in three-dimensional cartesian space which has the following properties: (a) it passes through the point $(1,1,1)$ and (b) if the tangent plane is drawn at any point $P$ and $X,Y, Z$ are the intersections of this plane with the $x, y$ and $z-$axis respectively, then $P$ is the orthocenter of the triangle $XYZ.$
2001 Putnam, 1
Consider a set $S$ and a binary operation $*$, i.e. for each $a,b\in S$, $a*b\in S$. Assume $(a*b)*a=b$ for all $a,b\in S$. Prove that $a*(b*a)=b$ for all $a,b \in S$.
1964 Putnam, B1
Let $u_k$ be a sequence of integers, and let $V_n$ be the number of those which are less than or equal to $n$. Show that if
$$\sum_{k=1}^{\infty} \frac{1}{u_k } < \infty,$$
then
$$\lim_{n \to \infty} \frac{ V_{n}}{n}=0.$$
1946 Putnam, A4
Let $g(x)$ be a function that has a continuous first derivative $g'(x)$. Suppose that $g(0)=0$ and $|g'(x)| \leq |g(x)|$ for all values of $x.$ Prove that $g(x)$ vanishes identically.
1970 Putnam, B1
Evaluate
$$\lim_{n\to \infty} \frac{1}{n^4 } \prod_{i=1}^{2n} (n^2 +i^2 )^{\frac{1}{n}}.$$
2001 Putnam, 6
Can an arc of a parabola inside a circle of radius $1$ have a length greater than $4$?
1958 February Putnam, A3
Real numbers are chosen at random from the interval $[0,1].$ If after choosing the $n$-th number the sum of the numbers so chosen first exceeds $1$, show that the expected value for $n$ is $e$.
1967 Putnam, A3
Consider polynomial functions $ax^2 -bx +c$ with integer coefficients which have two distinct zeros in the open interval $(0,1).$ Exhibit with proof the least positive integer value of $a$ for which such a polynomial exists.