Found problems: 966
1958 February Putnam, A2
Two uniform solid spheres of equal radii are so placed that one is directly above the other. The bottom sphere is fixed, and the top sphere, initially at rest, rolls off. At what point will contact between the two spheres be "lost"? Assume the coefficient of friction is such that no slipping occurs.
1957 Putnam, A7
Each member of a set of circles in the $xy$-plane is tangent to the $x$-axis and no two of the circles intersect. Show that
(a) the points of tangency can include all rational points on the axis.
(b) the points of tangency cannot include all the irrational points.
1940 Putnam, B3
Let $p>0$ be a real constant. From any point $(a,b)$ in the cartesian plane, show that
i) Three normals, real or imaginary, can be drawn to the parabola $y^2=4px$.
ii) These are real and distinct if $4(2-p)^3 +27pb^2<0$.
iii) Two of them coincide if $(a,b)$ lies on the curve $27py^2=4(x-2p)^3$.
iv) All three coincide only if $a=2p$ and $b=0$.
1981 Putnam, B5
Let $B(n)$ be the number of ones in the base two expression for the positive integer $n.$ Determine whether
$$\exp \left( \sum_{n=1}^{\infty} \frac{ B(n)}{n(n+1)} \right)$$
is a rational number.
1957 Putnam, A3
Let $a,b$ be real numbers and $k$ a positive integer. Show that
$$ \left| \frac{ \cos kb \cos a - \cos ka \cos b}{\cos b -\cos a} \right|<k^2 -1$$
whenever the left side is defined.
1958 February Putnam, B2
Prove that the product of four consecutive positive integers cannot be a perfect square or cube.
1980 Putnam, B1
For which real numbers $c$ is
$$\frac{e^x +e^{-x} }{2} \leq e^{c x^2 }$$
for all real $x?$
2011 Putnam, B4
In a tournament, 2011 players meet 2011 times to play a multiplayer game. Every game is played by all 2011 players together and ends with each of the players either winning or losing. The standings are kept in two $2011\times 2011$ matrices, $T=(T_{hk})$ and $W=(W_{hk}).$ Initially, $T=W=0.$ After every game, for every $(h,k)$ (including for $h=k),$ if players $h$ and $k$ tied (that is, both won or both lost), the entry $T_{hk}$ is increased by $1,$ while if player $h$ won and player $k$ lost, the entry $W_{hk}$ is increased by $1$ and $W_{kh}$ is decreased by $1.$
Prove that at the end of the tournament, $\det(T+iW)$ is a non-negative integer divisible by $2^{2010}.$
1963 Putnam, A3
Find an integral formula for the solution of the differential equation
$$\delta (\delta-1)(\delta-2) \cdots(\delta -n +1) y= f(x), \;\;\, x\geq 1,$$
for $y$ as a function of $f$ satisfying the initial conditions $y(1)=y'(1)=\ldots= y^{(n-1)}(1)=0$, where $f$ is continuous and $\delta$ is the differential operator $ x \frac{d}{dx}.$
1995 Putnam, 6
Suppose that each of $n$ people writes down the numbers $1, 2, 3$ in random order in one column of a $3\times n$ matrix, with all orders equally likely and with the orders for different columns independent of each other. Let the row sums $a, b, c$ of the resulting matrix be rearranged (if necessary) so that $a \le b \le c$. Show that for some $n \ge 1995$ ,it is at least four times as likely that both $b = a+1$ and $c = a+2$ as that $a = b = c$.
2022 Putnam, B1
Suppose that $P(x)=a_1x+a_2x^2+\ldots+a_nx^n$ is a polynomial with integer coefficients, with $a_1$ odd. Suppose that $e^{P(x)}=b_0+b_1x+b_2x^2+\ldots$ for all $x.$ Prove that $b_k$ is nonzero for all $k \geq 0.$
2020 Putnam, B1
For a positive integer $n$, define $d(n)$ to be the sum of the digits of $n$ when written in binary (for example, $d(13)=1+1+0+1=3$). Let
\[
S=\sum_{k=1}^{2020}(-1)^{d(k)}k^3.
\]
Determine $S$ modulo $2020$.
1962 Putnam, B2
Let $S$ be the set of all subsets of the positive integers. Construct a function $f \colon \mathbb{R} \rightarrow S$ such that $f(a)$ is a proper subset of $f(b)$ whenever $a <b.$
1988 Putnam, B6
Prove that there exist an infinite number of ordered pairs $(a,b)$ of integers such that for every positive integer $t$, the number $at+b$ is a triangular number if and only if $t$ is a triangular number. (The triangular numbers are the $t_n = n(n+1)/2$ with $n$ in $\{0,1,2,\dots\}$.)
2002 Putnam, 3
Show that for all integers $n>1$, \[ \dfrac {1}{2ne} < \dfrac {1}{e} - \left( 1 - \dfrac {1}{n} \right)^n < \dfrac {1}{ne}. \]
1996 Putnam, 6
Let $(a_1,b_1),(a_2,b_2),\ldots ,(a_n,b_n)$ be the vertices of a convex polygon containing the origin in its interior. Prove that there are positive real numbers $x,y$ such that :
\[ (a_1,b_1)x^{a_1}y^{b_1}+(a_2,b_2)x^{a_2}y^{b_2}+\ldots +(a_n,b_n)x^{a_n}y^{b_n}=(0,0) \]
2003 Putnam, 6
For a set $S$ of nonnegative integers, let $r_S(n)$ denote the number of ordered pairs $(s_1, s_2)$ such that $s_1 \in S$, $s_2 \in S$, $s_1 \neq s_2$, and $s_1 + s_2 = n$. Is it possible to partition the nonnegative integers into two sets $A$ and $B$ in such a way that $r_A(n) = r_B(n)$ for all $n$?
1975 Putnam, A5
Let $I\subset \mathbb{R}$ be an interval and $f(x)$ a continuous real-valued function on $I$. Let $y_1$ and $y_2$ be linearly independent solutions of $y''=f(x)y$ taking positive values on $I$. Show that for some positive number $k$ the function $k\cdot\sqrt{y_1 y_2}$ is a solution of $y''+\frac{1}{y^{3}}=f(x)y$.
1955 Putnam, A4
On a circle, $n$ points are selected and the chords joining them in pairs are drawn. Assuming that no three of these chords are concurrent (except at the endpoints), how many points of intersection are there?
1982 Putnam, A6
Let $\sigma$ be a bijection on the positive integers. Let $x_1,x_2,x_3,\ldots$ be a sequence of real numbers with the following three properties:
$(\text i)$ $|x_n|$ is a strictly decreasing function of $n$;
$(\text{ii})$ $|\sigma(n)-n|\cdot|x_n|\to0$ as $n\to\infty$;
$(\text{iii})$ $\lim_{n\to\infty}\sum_{k=1}^nx_k=1$.
Prove or disprove that these conditions imply that
$$\lim_{n\to\infty}\sum_{k=1}^nx_{\sigma(k)}=1.$$
2012 Putnam, 3
A round-robin tournament among $2n$ teams lasted for $2n-1$ days, as follows. On each day, every team played one game against another team, with one team winning and one team losing in each of the $n$ games. Over the course of the tournament, each team played every other team exactly once. Can one necessarily choose one winning team from each day without choosing any team more than once?
Putnam 1939, A3
The roots of $x^3 + a x^2 + b x + c = 0$ are $\alpha, \beta$ and $\gamma.$ Find the cubic whose roots are $\alpha^3, \beta^3, \gamma^3.$
1960 Putnam, B6
Any positive integer $n$ can be written in the form $n=2^{k}(2l+1)$ with $k,l$ positive integers. Let $a_n =e^{-k}$ and $b_n = a_1 a_2 a_3 \cdots a_n.$ Prove that
$$\sum_{n=1}^{\infty} b_n$$
converges.
2018 Putnam, B2
Let $n$ be a positive integer, and let $f_n(z) = n + (n-1)z + (n-2)z^2 + \dots + z^{n-1}$. Prove that $f_n$ has no roots in the closed unit disk $\{z \in \mathbb{C}: |z| \le 1\}$.
1973 Putnam, A1
(a) Let $ABC$ be any triangle. Let $X, Y, Z$ be points on the sides $BC, CA, AB$ respectively.
Suppose that $BX \leq XC, CY \leq YA, AZ \leq ZB$. Show that the area of the triangle $XYZ$ $\geq 1\slash 4$ times the area of $ABC.$
(b) Let $ABC$ be any triangle, and let $X, Y, Z$ be points on the sides $BC, CA, AB$ respectively. Using (a) or by any other method, show: One of the three corner triangles $AZY, BXZ, CYX$ has an area $\leq$ area of the triangle $XYZ.$