Found problems: 966
2018 Putnam, A2
Let $S_1, S_2, \dots, S_{2^n - 1}$ be the nonempty subsets of $\{1, 2, \dots, n\}$ in some order, and let $M$ be the $(2^n - 1) \times (2^n - 1)$ matrix whose $(i, j)$ entry is
\[m_{ij} = \left\{
\begin{array}{cl}
0 & \text{if $S_i \cap S_j = \emptyset$}, \\
1 & \text{otherwise}.
\end{array}
\right.\]
Calculate the determinant of $M$.
2001 Putnam, 3
For each integer $m$, consider the polynomial \[ P_m(x)=x^4-(2m+4)x^2+(m-2)^2. \] For what values of $m$ is $P_m(x)$ the product of two non-consant polynomials with integer coefficients?
2016 Putnam, B2
Define a positive integer $n$ to be [i]squarish[/i] if either $n$ is itself a perfect square or the distance from $n$ to the nearest perfect square is a perfect square. For example, $2016$ is squarish, because the nearest perfect square to $2016$ is $45^2=2025$ and $2025-2016=9$ is a perfect square. (Of the positive integers between $1$ and $10,$ only $6$ and $7$ are not squarish.)
For a positive integer $N,$ let $S(N)$ be the number of squarish integers between $1$ and $N,$ inclusive. Find positive constants $\alpha$ and $\beta$ such that
\[\lim_{N\to\infty}\frac{S(N)}{N^{\alpha}}=\beta,\]
or show that no such constants exist.
1963 Putnam, B3
Find every twice-differentiable function $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfies the functional equation
$$ f(x)^2 -f(y)^2 =f(x+y)f(x-y)$$
for all $x,y \in \mathbb{R}. $
1999 Putnam, 3
Let $A=\{(x,y): 0\le x,y < 1\}.$ For $(x,y)\in A,$ let
\[S(x,y)=\sum_{\frac12\le\frac mn\le2}x^my^n,\]
where the sum ranges over all pairs $(m,n)$ of positive integers satisfying the indicated inequalities. Evaluate
\[\lim_{(x,y)\to(1,1),(x,y)\in A}(1-xy^2)(1-x^2y)S(x,y).\]
1990 Putnam, A6
If $X$ is a finite set, let $X$ denote the number of elements in $X$. Call an ordered pair $(S,T)$ of subsets of $ \{ 1, 2, \cdots, n \} $ $ \emph {admissible} $ if $ s > |T| $ for each $ s \in S $, and $ t > |S| $ for each $ t \in T $. How many admissible ordered pairs of subsets $ \{ 1, 2, \cdots, 10 \} $ are there? Prove your answer.
1998 Putnam, 1
Find the minimum value of \[\dfrac{(x+1/x)^6-(x^6+1/x^6)-2}{(x+1/x)^3+(x^3+1/x^3)}\] for $x>0$.
1999 Putnam, 6
The sequence $(a_n)_{n\geq 1}$ is defined by $a_1=1,a_2=2,a_3=24,$ and, for $n\geq 4,$ \[a_n=\dfrac{6a_{n-1}^2a_{n-3}-8a_{n-1}a_{n-2}^2}{a_{n-2}a_{n-3}}.\] Show that, for all $n$, $a_n$ is an integer multiple of $n$.
2005 Putnam, B2
Find all positive integers $n,k_1,\dots,k_n$ such that $k_1+\cdots+k_n=5n-4$ and
\[ \frac1{k_1}+\cdots+\frac1{k_n}=1. \]
Putnam 1938, A5
$(1)$ Find $\lim_{x \to \infty} \frac{x^2}{e^x}$
$(2)$ Find $\lim_{k \to 0} \frac{1}{k} \int_{0}^{k} (1 + \sin 2x)^{\frac{1}{x}} dx$
2016 Putnam, B3
Suppose that $S$ is a finite set of points in the plane such that the area of triangle $\triangle ABC$ is at most $1$ whenever $A,B,$ and $C$ are in $S.$ Show that there exists a triangle of area $4$ that (together with its interior) covers the set $S.$
1949 Putnam, A6
Prove that for every real or complex $x$
$$\prod_{k=1}^{\infty} \frac{1+2\cos \frac{2x}{3^{k}}}{3} =\frac{\sin x}{x}.$$
2013 Putnam, 5
Let $X=\{1,2,\dots,n\},$ and let $k\in X.$ Show that there are exactly $k\cdot n^{n-1}$ functions $f:X\to X$ such that for every $x\in X$ there is a $j\ge 0$ such that $f^{(j)}(x)\le k.$
[Here $f^{(j)}$ denotes the $j$th iterate of $f,$ so that $f^{(0)}(x)=x$ and $f^{(j+1)}(x)=f\left(f^{(j)}(x)\right).$]
2014 Putnam, 6
Let $n$ be a positive integer. What is the largest $k$ for which there exist $n\times n$ matrices $M_1,\dots,M_k$ and $N_1,\dots,N_k$ with real entries such that for all $i$ and $j,$ the matrix product $M_iN_j$ has a zero entry somewhere on its diagonal if and only if $i\ne j?$
2007 Putnam, 2
Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola $ xy\equal{}1$ and both branches of the hyperbola $ xy\equal{}\minus{}1.$ (A set $ S$ in the plane is called [i]convex[/i] if for any two points in $ S$ the line segment connecting them is contained in $ S.$)
1948 Putnam, B4
For what $\lambda$ does the equation
$$ \int_{0}^{1} \min(x,y) f(y)\; dy =\lambda f(x)$$
have continuous solutions which do not vanish identically in $(0,1)?$ What are these solutions?
2008 Putnam, B2
Let $ F_0\equal{}\ln x.$ For $ n\ge 0$ and $ x>0,$ let $ \displaystyle F_{n\plus{}1}(x)\equal{}\int_0^xF_n(t)\,dt.$ Evaluate $ \displaystyle\lim_{n\to\infty}\frac{n!F_n(1)}{\ln n}.$
1987 Putnam, B3
Let $F$ be a field in which $1+1 \neq 0$. Show that the set of solutions to the equation $x^2+y^2=1$ with $x$ and $y$ in $F$ is given by $(x,y)=(1,0)$ and
\[
(x,y) = \left( \frac{r^2-1}{r^2+1}, \frac{2r}{r^2+1} \right)
\]
where $r$ runs through the elements of $F$ such that $r^2\neq -1$.
1995 Putnam, 3
To each number with $n^2$ digits, we associate the $n\times n$ determinant of the matrix obtained by writing the digits of the number in order along the rows. For example : $8617\mapsto \det \left(\begin{matrix}{\;8}& 6\;\\ \;1 &{ 7\;}\end{matrix}\right)=50$.
Find, as a function of $n$, the sum of all the determinants associated with $n^2$-digit integers. (Leading digits are assumed to be nonzero; for example, for $n = 2$, there are $9000$ determinants.)
2021 Putnam, B1
Suppose that the plane is tiled with an infinite checkerboard of unit squares. If another unit square is dropped on the plane at random with position and orientation independent of the checkerboard tiling, what is the probability that it does not cover any of the corners of the squares of the checkerboard?
[hide=Solution]
With probability $1$ the number of corners covered is $0$, $1$, or $2$ for example by the diameter of a square being $\sqrt{2}$ so it suffices to compute the probability that the square covers $2$ corners. This is due to the fact that density implies the mean number of captured corners is $1$. For the lattice with offset angle $\theta \in \left[0,\frac{\pi}{2}\right]$ consider placing a lattice uniformly randomly on to it and in particular say without loss of generality consider the square which covers the horizontal lattice midpoint $\left(\frac{1}{2},0 \right)$. The locus of such midpoint locations so that the square captures the $2$ points $(0,0),(1,0)$, is a rectangle. As capturing horizontally adjacent points does not occur when capturing vertically adjacent points one computes twice that probability as $\frac{4}{\pi} \int_0^{\frac{\pi}{2}} (1-\sin(\theta))(1-\cos(\theta)) d\theta=\boxed{\frac{2(\pi-3)}{\pi}}$ \\
[asy]
draw((0,0)--(80,40));
draw((0,0)--(-40,80));
draw((80,40)--(40,120));
draw((-40,80)--(40,120));
draw((80,40)--(-20,40));
draw((-40,80)--(60,80));
draw((32*sqrt(5),16*sqrt(5))--(-8*sqrt(5),16*sqrt(5)));
draw((40+8*sqrt(5),120-16*sqrt(5))--(40-32*sqrt(5),120-16*sqrt(5)));
draw((12*sqrt(5),16*sqrt(5))--(12*sqrt(5)+2*(40-16*sqrt(5)),16*sqrt(5)+(40-16*sqrt(5))));
draw((12*sqrt(5),16*sqrt(5))--(12*sqrt(5)-(80-16*sqrt(5))/2,16*sqrt(5)+(80-16*sqrt(5))));
draw((40-12*sqrt(5),120-16*sqrt(5))--(40-12*sqrt(5)+(120-16*sqrt(5)-40)/2,120-16*sqrt(5)-(120-16*sqrt(5)-40)));
draw((40-12*sqrt(5),120-16*sqrt(5))--(40-12*sqrt(5)-2*(120-16*sqrt(5)-80),120-16*sqrt(5)-(120-16*sqrt(5)-80)));
[/asy]
[/hide]
1957 Putnam, B3
For $f(x)$ a positive , monotone decreasing function defined in $[0,1],$ prove that
$$ \int_{0}^{1} f(x) dx \cdot \int_{0}^{1} xf(x)^{2} dx \leq \int_{0}^{1} f(x)^{2} dx \cdot \int_{0}^{1} xf(x) dx.$$
1968 Putnam, B1
The random variables $X, Y$ can each take a finite number of integer values. They are not necessarily independent. Express $P(\min(X,Y)=k)$ in terms of $p_1=P(X=k)$, $p_2=P(Y=k)$ and $p_3=P(\max(X,Y)=k)$.
1965 Putnam, B2
In a round-robin tournament with $n$ players $P_1$, $P_2$, $\ldots$, $P_n$ (where $n > 1$), each player plays one game with each of the other players and the rules are such that no ties can occur. Let $w_r$ and $l_r$ be the number of games won and lost, respectively, by $P_r$. Show that
\[
\sum_{r=1}^nw_r^2 = \sum_{r=1}^nl_r^2.
\]
1954 Putnam, B4
Given the focus $F$ and the directrix $D$ of a parabola $P$ and a line $L$, describe a euclidean construction for the point or points of intersection of $P$ and $L.$ Be sure to identify the case for which there are no points of intersection.
1963 Putnam, B1
For what integers $a$ does $x^2 -x+a$ divide $x^{13}+ x +90$ ?