Found problems: 966
2021 Putnam, A2
For every positive real number $x$, let
\[
g(x)=\lim_{r\to 0} ((x+1)^{r+1}-x^{r+1})^{\frac{1}{r}}.
\]
Find $\lim_{x\to \infty}\frac{g(x)}{x}$.
[hide=Solution]
By the Binomial Theorem one obtains\\
$\lim_{x \to \infty} \lim_{r \to 0} \left((1+r)+\frac{(1+r)r}{2}\cdot x^{-1}+\frac{(1+r)r(r-1)}{6} \cdot x^{-2}+\dots \right)^{\frac{1}{r}}$\\
$=\lim_{r \to 0}(1+r)^{\frac{1}{r}}=\boxed{e}$
[/hide]
1968 Putnam, B4
Suppose that $f:\mathbb{R} \rightarrow \mathbb{R}$ is continuous and $L=\int_{-\infty}^{\infty} f(x) dx$ exists. Show that $$\int_{-\infty}^{\infty}f\left(x-\frac{1}{x}\right)dx=L.$$
1951 Putnam, B5
A plane through the center of a torus is tangent to the torus. Prove that the intersection of the plane and the torus consists of two circles.
Putnam 1938, B4
The parabola $P$ has focus a distance $m$ from the directrix. The chord $AB$ is normal to $P$ at $A.$ What is the minimum length for $AB?$
1968 Putnam, B3
Given that a $60^{\circ}$ angle cannot be trisected with ruler and compass, prove that a $\frac{120^{\circ}}{n}$ angle cannot be trisected with ruler and compass for $n=1,2,\ldots$
2010 Putnam, A4
Prove that for each positive integer $n,$ the number $10^{10^{10^n}}+10^{10^n}+10^n-1$ is not prime.
1990 Putnam, A1
Let \[T_0=2, T_1=3, T_2=6,\] and for $n\ge 3$, \[T_n=(n+4)T_{n-1}-4nT_{n-2}+(4n-8)T_{n-3}.\] The first few terms are \[2, 3, 6, 14, 40, 152, 784, 5158, 40576, 363392.\] Find a formula for $T_n$ of the form \[T_n=A_n+B_n,\] where $\{A_n\}$ and $\{B_n\}$ are well known sequences.
1962 Putnam, A4
Assume that $|f(x)|\leq 1$ and $|f''(x)|\leq 1$ for all $x$ on an interval of length at least $2.$ Show that $|f'(x)|\leq 2 $ on the interval.
2010 Putnam, A5
Let $G$ be a group, with operation $*$. Suppose that
(i) $G$ is a subset of $\mathbb{R}^3$ (but $*$ need not be related to addition of vectors);
(ii) For each $\mathbf{a},\mathbf{b}\in G,$ either $\mathbf{a}\times\mathbf{b}=\mathbf{a}*\mathbf{b}$ or $\mathbf{a}\times\mathbf{b}=\mathbf{0}$ (or both), where $\times$ is the usual cross product in $\mathbb{R}^3.$
Prove that $\mathbf{a}\times\mathbf{b}=\mathbf{0}$ for all $\mathbf{a},\mathbf{b}\in G.$
2014 Putnam, 4
Show that for each positive integer $n,$ all the roots of the polynomial \[\sum_{k=0}^n 2^{k(n-k)}x^k\] are real numbers.
1995 Putnam, 5
Let $x_1,x_2,\cdots, x_n$ be real valued differentiable functions of a variable $t$ which satisfy
\begin{align*}
& \frac{\mathrm{d}x_1}{\mathrm{d}t}=a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n\\
& \frac{\mathrm{d}x_2}{\mathrm{d}t}=a_{21}x_1+a_{22}x_2+\cdots+a_{2n}x_n\\
& \;\qquad \vdots \\
& \frac{\mathrm{d}x_n}{\mathrm{d}t}=a_{n1}x_1+a_{n2}x_2+\cdots+a_{nn}x_n\\
\end{align*}
For some constants $a_{ij}>0$. Suppose that $\lim_{t \to \infty}x_i(t)=0$ for all $1\le i \le n$. Are the functions $x_i$ necessarily linearly dependent?
1969 Putnam, A5
Let $u(t)$ be a continuous function in the system of differential equations
$$\frac{dx}{dt} =-2y +u(t),\;\;\; \frac{dy}{dt}=-2x+ u(t).$$
Show that, regardless of the choice of $u(t)$, the solution of the system which satisfies $x=x_0 , y=y_0$
at $t=0$ will never pass through $(0, 0)$ unless $x_0 =y_0.$ When $x_0 =y_0 $, show that, for any positive value
$t_0$ of $t$, it is possible to choose $u(t)$ so the solution is equal to $(0,0)$ when $t=t_0 .$
1963 Putnam, A1
i) Show that a regular hexagon, six squares, and six equilateral triangles can be assembled without overlapping to form a regular dodecagon.
ii) Let $P_1 , P_2 ,\ldots, P_{12}$ be the vertices of a regular dodecagon. Prove that the three diagonals $P_{1}P_{9}, P_{2}P_{11}$ and $P_{4}P_{12}$ intersect.
1997 Putnam, 5
Let $N_k$ denote number of ordered $n$-tuples of positive integers $(a_1,a_2, \cdots ,a_k)$ such that
\[ \frac{1}{a_1}+\frac{1}{a_2}+\ldots +\frac{1}{a_k}=1 \]
Determine $N_{10}$ is odd or even.
2015 Putnam, B5
Let $P_n$ be the number of permutations $\pi$ of $\{1,2,\dots,n\}$ such that \[|i-j|=1\text{ implies }|\pi(i)-\pi(j)|\le 2\] for all $i,j$ in $\{1,2,\dots,n\}.$ Show that for $n\ge 2,$ the quantity \[P_{n+5}-P_{n+4}-P_{n+3}+P_n\] does not depend on $n,$ and find its value.
2010 Putnam, A6
Let $f:[0,\infty)\to\mathbb{R}$ be a strictly decreasing continuous function such that $\lim_{x\to\infty}f(x)=0.$ Prove that $\displaystyle\int_0^{\infty}\frac{f(x)-f(x+1)}{f(x)}\,dx$ diverges.
1947 Putnam, B6
Let $OX, OY, OZ$ be mutually orthogonal lines in space. Let $C$ be a fixed point on $OZ$ and $U,V$ variable points on $OX, OY,$ respectively. Find the locus of a point $P$ such that $PU, PV, PC$ are mutually orthogonal.
1992 Putnam, B1
Let $S$ be a set of $n$ distinct real numbers. Let $A_{S}$ be the set of numbers that occur as averages of two distinct
elements of $S$. For a given $n \geq 2$, what is the smallest possible number of elements in $A_{S}$?
1961 Putnam, A6
Prove that $p(x)=1+x+x^2 +\ldots+x^n$ is reducible over $\mathbb{F}_{2}$ in case $n+1$ is composite. If $n+1$ is prime, is $p(x)$ irreducible over $\mathbb{F}_{2}$ ?
1990 Putnam, A3
Prove that any convex pentagon whose vertices (no three of which are collinear) have integer coordinates must have area greater than or equal to $ \dfrac {5}{2} $.
2011 Putnam, A1
Define a [i]growing spiral[/i] in the plane to be a sequence of points with integer coordinates $P_0=(0,0),P_1,\dots,P_n$ such that $n\ge 2$ and:
• The directed line segments $P_0P_1,P_1P_2,\dots,P_{n-1}P_n$ are in successive coordinate directions east (for $P_0P_1$), north, west, south, east, etc.
• The lengths of these line segments are positive and strictly increasing.
\[\begin{picture}(200,180)
\put(20,100){\line(1,0){160}}
\put(100,10){\line(0,1){170}}
\put(0,97){West}
\put(180,97){East}
\put(90,0){South}
\put(90,180){North}
\put(100,100){\circle{1}}\put(100,100){\circle{2}}\put(100,100){\circle{3}}
\put(115,100){\circle{1}}\put(115,100){\circle{2}}\put(115,100){\circle{3}}
\put(115,130){\circle{1}}\put(115,130){\circle{2}}\put(115,130){\circle{3}}
\put(40,130){\circle{1}}\put(40,130){\circle{2}}\put(40,130){\circle{3}}
\put(40,20){\circle{1}}\put(40,20){\circle{2}}\put(40,20){\circle{3}}
\put(170,20){\circle{1}}\put(170,20){\circle{2}}\put(170,20){\circle{3}}
\multiput(100,99.5)(0,.5){3}{\line(1,0){15}}
\multiput(114.5,100)(.5,0){3}{\line(0,1){30}}
\multiput(40,129.5)(0,.5){3}{\line(1,0){75}}
\multiput(39.5,20)(.5,0){3}{\line(0,1){110}}
\multiput(40,19.5)(0,.5){3}{\line(1,0){130}}
\put(102,90){P0}
\put(117,90){P1}
\put(117,132){P2}
\put(28,132){P3}
\put(30,10){P4}
\put(172,10){P5}
\end{picture}\]
How many of the points $(x,y)$ with integer coordinates $0\le x\le 2011,0\le y\le 2011$ [i]cannot[/i] be the last point, $P_n,$ of any growing spiral?
2010 Putnam, B4
Find all pairs of polynomials $p(x)$ and $q(x)$ with real coefficients for which
\[p(x)q(x+1)-p(x+1)q(x)=1.\]
1974 Putnam, A5
Consider the two mutually tangent parabolas $y=x^2$ and $y=-x^2$. The upper parabola rolls without slipping around the fixed lower parabola. Find the locus of the focus of the moving parabola.
2014 Putnam, 2
Suppose that $f$ is a function on the interval $[1,3]$ such that $-1\le f(x)\le 1$ for all $x$ and $\displaystyle \int_1^3f(x)\,dx=0.$ How large can $\displaystyle\int_1^3\frac{f(x)}x\,dx$ be?
1956 Putnam, A4
Suppose that the $n$ times differentiable real function $f(x)$ has at least $n+1$ distinct zeros in the closed interval $[a,b]$ and that the polynomial $P(z)=z^n +c_{n-1}z^{n-1}+\ldots+c_1 x +c_0$ has only real zeroes. Show that
$f^{(n)}(x)+ c_{n-1} f^{(n-1)}(x) +\ldots +c_1 f'(x)+ c_0 f(x)$ has at least one zero in $[a,b]$, where $f^{(n)}$ denotes the $n$-th derivative of $f.$