Found problems: 966
1980 Putnam, B5
For each $t \geq 0$ let $S_t$ be the set of all nonnegative, increasing, convex, continuous, real-valued functions $f(x)$ defined on the closed interval $[0,1]$ for which
$$f(1) -2 f(2 \slash 3) +f (1 \slash 3) \geq t( f( 2 \slash 3) -2 f(1 \slash 3) +f(0)).$$
Define necessary and sufficient conditions on $ t$ for $S_t $ to be closed under multiplication.
2008 Putnam, B1
What is the maximum number of rational points that can lie on a circle in $ \mathbb{R}^2$ whose center is not a rational point? (A [i]rational point[/i] is a point both of whose coordinates are rational numbers.)
1956 Putnam, A3
A particle falls in a vertical plane from rest under the influence of gravity and a force perpendicular to and proportional to its velocity. Obtain the equations of the trajectory and identify the curve.
2022 Putnam, B5
For $0 \leq p \leq 1/2,$ let $X_1, X_2, \ldots$ be independent random variables such that
$$X_i=\begin{cases}
1 & \text{with probability } p, \\
-1 & \text{with probability } p, \\
0 & \text{with probability } 1-2p,
\end{cases}
$$
for all $i \geq 1.$ Given a positive integer $n$ and integers $b,a_1, \ldots, a_n,$ let $P(b, a_1, \ldots, a_n)$ denote the probability that $a_1X_1+ \ldots + a_nX_n=b.$ For which values of $p$ is it the case that $$P(0, a_1, \ldots, a_n) \geq P(b, a_1, \ldots, a_n)$$ for all positive integers $n$ and all integers $b, a_1,\ldots, a_n?$
1963 Putnam, B5
Let $(a_n )$ be a sequence of real numbers satisfying the inequalities
$$ 0 \leq a_k \leq 100a_n \;\; \text{for} \;\, n \leq k \leq 2n \;\; \text{and} \;\; n=1,2,\ldots,$$
and such that the series
$$\sum_{n=0}^{\infty} a_n $$
converges. Prove that
$$\lim_{n\to \infty} n a_n = 0.$$
2024 Putnam, B6
For a real number $a$, let $F_a(x)=\sum_{n\geq 1}n^ae^{2n}x^{n^2}$ for $0\leq x<1$. Find a real number $c$ such that
\begin{align*}
\lim_{x\to 1^-}F_a(x)e^{-1/(1-x)}&=0 \ \ \ \text{for all $a<c$, and}\\
\lim_{x\to 1^-}F_a(x)e^{-1/(1-x)}&=\infty \ \ \ \text{for all $a>c$.}
\end{align*}
1949 Putnam, A3
Assume that the complex numbers $a_1 , a_2, \ldots$ are all different from $0$, and that $|a_r - a_s| >1$ for $r\ne s.$
Show that the series
$$\sum_{n=1}^{\infty} \frac{1}{a_{n}^{3}}$$
converges.
1995 Putnam, 5
A game starts with four heaps of beans, containing 3, 4, 5 and 6 beans. The two players move alternately. A move consists of taking [list]
(a) $\text{either}$ one bean from a heap, provided at least two beans are left behind in that heap,
(b) $\text{or}$ a complete heap of two or three beans.[/list]
The player who takes the last heap wins. To win the game, do you want to move first or second? Give a winning strategy.
1940 Putnam, A1
Prove that if $f(x)$ is a polynomial with integer coefficients and there exists an integer $k$ such that none of $f(1),\ldots,f(k)$ is divisible by $k$, then $f(x)$ has no integral root.
1970 Putnam, A1
Show that the power series for the function
$$e^{ax} \cos bx,$$
where $a,b >0$, has either no zero coefficients or infinitely many zero coefficients.
1963 Putnam, A4
Let $(a_n)$ be a sequence of positive real numbers. Show that
$$ \limsup_{n \to \infty} n \left(\frac{1 +a_{n+1}}{a_n } -1 \right) \geq 1$$
and prove that $1$ cannot be replaced by any larger number.
1948 Putnam, A2
Two spheres in contact have a common tangent cone. These three surfaces divide the space into various parts, only one of which is bounded by all three surfaces, it is "ring-shaped." Being given the radii of the spheres, $r$ and $R$, find the volume of the "ring-shaped" part. (The desired expression is a rational function of $r$ and $R.$)
1960 Putnam, A2
Show that if three points are inside are closed square of unit side, then some pair of them are within $\sqrt{6}-\sqrt{2}$ units apart.
1997 Putnam, 3
Evaluate the following :
\[ \int_{0}^{\infty}\left(x-\frac{x^3}{2}+\frac{x^5}{2\cdot 4}-\frac{x^7}{2\cdot 4\cdot 6}+\cdots \right)\;\left(1+\frac{x^2}{2^2}+\frac{x^4}{2^2\cdot 4^2}+\frac{x^6}{2^2\cdot 4^2\cdot 6^2}+\cdots \right)\,\mathrm{d}x \]
1999 Putnam, 1
Right triangle $ABC$ has right angle at $C$ and $\angle BAC=\theta$; the point $D$ is chosen on $AB$ so that $|AC|=|AD|=1$; the point $E$ is chosen on $BC$ so that $\angle CDE=\theta$. The perpendicular to $BC$ at $E$ meets $AB$ at $F$. Evaluate $\lim_{\theta\to 0}|EF|$.
1975 Putnam, B2
A [i]slab[/i] is the set of points strictly between two parallel planes. Prove that a countable sequence of slabs, the sum of whose thicknesses converges, cannot fill space.
Putnam 1938, B5
Find the locus of the foot of the perpendicular from the center of a rectangular hyperbola to a tangent. Obtain its equation in polar coordinates and sketch it.
1993 Putnam, A2
The sequence an of non-zero reals satisfies $a_n^2 - a_{n-1}a_{n+1} = 1$ for $n \geq 1$. Prove that there exists a real number $\alpha$ such that $a_{n+1} = \alpha a_n - a_{n-1}$ for $n \geq 1$.
2020 Putnam, A6
For a positive integer $N$, let $f_N$ be the function defined by
\[ f_N (x)=\sum_{n=0}^N \frac{N+1/2-n}{(N+1)(2n+1)} \sin\left((2n+1)x \right). \]
Determine the smallest constant $M$ such that $f_N (x)\le M$ for all $N$ and all real $x$.
1974 Putnam, B4
A function $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is said to be [i]continuous in each variable separately [/i] if, for each fixed value $y_0$ of $y$, the function $f(x, y_0)$ is contnuous in the usual sense as a function in $x,$ and similarly $f(x_0 , y)$ is continuous as a function of $y$ for each fixed $x_0$.
Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be continuous in each variable separately. Show that there exists a sequence of continuous functions $g_n: \mathbb{R}^{2} \rightarrow \mathbb{R}$ such that
$$f(x,y) =\lim_{n\to \infty}g_{n}(x,y)$$
for all $(x,y)\in \mathbb{R}^{2}.$
1952 Putnam, B4
A homogeneous solid body is made by joining a base of a circular cylinder of height $h$ and radius $r,$ and the base of a hemisphere of radius $r.$ This body is placed with the hemispherical end on a horizontal table, with the axis of the cylinder in a vertical position, and then slightly oscillated. It is intuitively evident that if $r$ is large as compared to $h$, the equilibrium will be stable; but if $r$ is small compared to $h$, the equilibrium will be unstable. What is the critical value of the ratio $r\slash h$ which enables the body to rest in neutral equilibrium in any position?
1949 Putnam, A1
Answer either (i) or (ii):
(i) Let $a>0.$ Three straight lines pass through the three points $(0,-a,a), (a,0,-a)$ and $(-a,a,0),$ parallel to the $x-,y-$ and $z-$axis, respectively. A variable straight line moves so that it has one point in common with each of the three given lines. Find the equation of the surface described by the variable line.
(II) Which planes cut the surface $xy+yz+xz=0$ in (1) circles, (2) parabolas?
1952 Putnam, B6
Prove the necessary and sufficient condition that a triangle inscribed in an ellipse shall have maximum area is that its centroid coincides with the center of the ellipse.
2009 Putnam, A3
Let $ d_n$ be the determinant of the $ n\times n$ matrix whose entries, from left to right and then from top to bottom, are $ \cos 1,\cos 2,\dots,\cos n^2.$ (For example, $ d_3 \equal{} \begin{vmatrix}\cos 1 & \cos2 & \cos3 \\
\cos4 & \cos5 & \cos 6 \\
\cos7 & \cos8 & \cos 9\end{vmatrix}.$ The argument of $ \cos$ is always in radians, not degrees.)
Evaluate $ \lim_{n\to\infty}d_n.$
1958 February Putnam, B7
Prove that if $f(x)$ is continuous for $a\leq x \leq b$ and
$$\int_{a}^{b} x^n f(x) \, dx =0$$
for $n=0,1,2, \ldots,$ then $f(x)$ is identically zero on $a \leq x \leq b.$