Found problems: 966
1951 Putnam, A6
Determine the position of a normal chord of a parabola such that it cuts off of the parabola a segment of minimum area.
2004 Putnam, A6
Suppose that $f(x,y)$ is a continuous real-valued function on the unit square $0\le x\le1,0\le y\le1.$ Show that
$\int_0^1\left(\int_0^1f(x,y)dx\right)^2dy + \int_0^1\left(\int_0^1f(x,y)dy\right)^2dx$
$\le\left(\int_0^1\int_0^1f(x,y)dxdy\right)^2 + \int_0^1\int_0^1\left[f(x,y)\right]^2dxdy.$
1959 Putnam, B5
Find the equation of the smallest sphere which is tangent to both of the lines
$$\begin{pmatrix}
x\\y\\z \end{pmatrix} =\begin{pmatrix}
t+1\\
2t+4\\
-3t +5
\end{pmatrix},\;\;\;\begin{pmatrix}
x\\y\\z \end{pmatrix} =\begin{pmatrix}
4t-12\\
-t+8\\
t+17
\end{pmatrix}.$$
1958 February Putnam, A6
What is the smallest amount that may be invested at interest rate $i$, compounded annually, in order that one may withdraw $1$ dollar at the end of the first year, $4$ dollars at the end of the second year, $\ldots$ , $n^2$ dollars at the end of the $n$-th year, in perpetuity?
1986 Putnam, A2
What is the units (i.e., rightmost) digit of
\[
\left\lfloor \frac{10^{20000}}{10^{100}+3}\right\rfloor ?
\]
2020 Putnam, B4
Let $n$ be a positive integer, and let $V_n$ be the set of integer $(2n+1)$-tuples $\mathbf{v}=(s_0,s_1,\cdots,s_{2n-1},s_{2n})$ for which $s_0=s_{2n}=0$ and $|s_j-s_{j-1}|=1$ for $j=1,2,\cdots,2n$. Define
\[
q(\mathbf{v})=1+\sum_{j=1}^{2n-1}3^{s_j},
\]
and let $M(n)$ be the average of $\frac{1}{q(\mathbf{v})}$ over all $\mathbf{v}\in V_n$. Evaluate $M(2020)$.
1942 Putnam, B5
Sketch the curve
$$y= \frac{x}{1+x^6 (\sin x)^{2}},$$
and show that
$$ \int_{0}^{\infty} \frac{x}{1+x^6 (\sin x)^{2}}\; dx$$
exists.
1941 Putnam, A4
Let the roots $a,b,c$ of
$$f(x)=x^3 +p x^2 + qx+r$$
be real, and let $a\leq b\leq c$. Prove that $f'(x)$ has a root in the interval $\left[\frac{b+c}{2}, \frac{b+2c}{3}\right]$. What will be the form of $f(x)$ if the root in question falls at either end of the interval?
2017 Putnam, A4
A class with $2N$ students took a quiz, on which the possible scores were $0,1,\dots,10.$ Each of these scores occurred at least once, and the average score was exactly $7.4.$ Show that the class can be divided into two groups of $N$ students in such a way that the average score for each group was exactly $7.4.$
1996 Putnam, 4
$S$ be a set of ordered triples $(a,b,c)$ of distinct elements of a finite set $A$. Suppose that
[list=1]
[*] $(a,b,c)\in S\iff (b,c,a)\in S$
[*] $(a,b,c)\in S\iff (c,b,a)\not\in S$
[*] $(a,b,c),(c,d,a)\text{ both }\in S\iff (b,c,d),(d,a,b)\text{ both }\in S$[/list]
Prove there exists $g: A\to \mathbb{R}$, such that $g$ is one-one and $g(a)<g(b)<g(c)\implies (a,b,c)\in S$
1946 Putnam, B6
A particle moves on a circle with center $O$, starting from rest at a point $P$ and coming to rest again at a point $Q$, without coming to rest at any intermediate point. Prove that the acceleration vector of the particle does not vanish at any point between $P$ and $ Q$ and that, at some point $R$ between $P$ and $Q$, the acceleration vector points in along the radius $RO.$
1942 Putnam, A3
Is the series
$$\sum_{n=0}^{\infty} \frac{n!}{(n+1)^{n}}\cdot \left(\frac{19}{7}\right)^{n}$$
convergent or divergent?
2018 Putnam, B6
Let $S$ be the set of sequences of length 2018 whose terms are in the set $\{1, 2, 3, 4, 5, 6, 10\}$ and sum to 3860. Prove that the cardinality of $S$ is at most
\[2^{3860} \cdot \left(\frac{2018}{2048}\right)^{2018}.\]
2001 Putnam, 3
For any positive integer $n$, let $ \left< n \right> $ denote the closest integer to $ \sqrt {n} $. Evaluate: \[ \displaystyle\sum_{n=1}^{\infty} \dfrac {2^{\left< n \right>} + 2^{- \left< n \right>}}{2^n} \]
1993 Putnam, B5
Show that given any $4$ points in the plane we can find two whose distance apart is not an odd integer.
1993 Putnam, A4
Given a sequence of $19$ positive (not necessarily distinct) integers not greater than $93$, and a set of $93$ positive (not necessarily distinct) integers not greater than $19$. Show that we can find non-empty subsequences of the two sequences with equal sum.
1946 Putnam, A2
If $a(x), b(x), c(x)$ and $d(x)$ are polynomials in $ x$, show that
$$ \int_{1}^{x} a(x) c(x)\; dx\; \cdot \int_{1}^{x} b(x) d(x) \; dx - \int_{1}^{x} a(x) d(x)\; dx\; \cdot \int_{1}^{x} b(x) c(x)\; dx$$
is divisible by $(x-1)^4.$
2000 Putnam, 5
Three distinct points with integer coordinates lie in the plane on a circle of radius $r>0$. Show that two of these points are separated by a distance of at least $r^{1/3}$.
1950 Putnam, B4
The cross-section of a right cylinder is an ellipse, with semi-axes $a$ and $b,$ where $a > b.$ The cylinder is very long, made of very light homogeneous material. The cylinder rests on the horizontal ground which it touches along the straight line joining the lower endpoints of the minor axes of its several cross-sections. Along the upper endpoints of these minor axes lies a very heavy homogeneous wire, straight and just as long as the cylinder. The wire and the cylinder are rigidly connected. We neglect the weight of the cylinder, the breadth of the wire, and the friction of the ground.
The system described is in equilibrium, because of its symmetry. This equilibrium seems to be stable when the ratio $b/a$ is very small, but unstable when this ratio comes close to $1.$ Examine this assertion and find the value of the ratio $b/a$ which separates the cases of stable and unstable equilibrium.
2004 Putnam, B6
Let $A$ be a nonempty set of positive integers, and let $N(x)$ denote the number of elements of $A$ not exceeding $x$. Let $B$ denote the set of positive integers $b$ that can be written in the form $b=a-a^{\prime}$ with $a\in A$ and $a^{\prime}\in A$. Let $b_1<b_2<\cdots$ be the members of $B$, listed in increasing order. Show that if the sequence $b_{i+1}-b_i$ is unbounded, then $\lim_{x\to \infty}\frac{N(x)}{x}=0$.
1987 Putnam, A5
Let
\[
\vec{G}(x,y) = \left( \frac{-y}{x^2+4y^2}, \frac{x}{x^2+4y^2},0
\right).
\]
Prove or disprove that there is a vector-valued function
\[
\vec{F}(x,y,z) = (M(x,y,z), N(x,y,z), P(x,y,z))
\]
with the following properties:
(i) $M,N,P$ have continuous partial derivatives for all $(x,y,z) \neq (0,0,0)$;
(ii) $\mathrm{Curl}\,\vec{F} = \vec{0}$ for all $(x,y,z) \neq (0,0,0)$;
(iii) $\vec{F}(x,y,0) = \vec{G}(x,y)$.
1964 Putnam, A5
Prove that there exists a constant $K$ such that the following inequality holds for any sequence of positive numbers $a_1 , a_2 , a_3 , \ldots:$
$$\sum_{n=1}^{\infty} \frac{n}{a_1 + a_2 +\ldots + a_n } \leq K \sum_{n=1}^{\infty} \frac{1}{a_{n}}.$$
2021 Putnam, A3
Determine all positive integers $N$ for which the sphere
\[
x^2+y^2+z^2=N
\]
has an inscribed regular tetrahedron whose vertices have integer coordinates.
2013 IMC, 2
Let $\displaystyle{p,q}$ be relatively prime positive integers. Prove that
\[\displaystyle{ \sum_{k=0}^{pq-1} (-1)^{\left\lfloor \frac{k}{p}\right\rfloor + \left\lfloor \frac{k}{q}\right\rfloor} = \begin{cases} 0 & \textnormal{ if } pq \textnormal{ is even}\\ 1 & \textnormal{if } pq \textnormal{ odd}\end{cases}}\]
[i]Proposed by Alexander Bolbot, State University, Novosibirsk.[/i]
1960 Putnam, A5
Find all polynomials $f(x)$ with real coefficients having the property $f(g(x))=g(f(x))$ for every polynomial $g(x)$ with real coefficients.