Found problems: 12
2021 Putnam, B4
Let $F_0,F_1,\dots$ be the sequence of Fibonacci numbers, with $F_0=0,F_1=1$, and $F_n=F_{n-1}+F_{n-2}$ for $n \ge 2$. For $m>2$, let $R_m$ be the remainder when the product $\prod_{k=1}^{F_m-1} k^k$ is divided by $F_m$. Prove that $R_m$ is also a Fibonacci number.
2021 Putnam, B2
Determine the maximum value of the sum
\[
S=\sum_{n=1}^{\infty}\frac{n}{2^n}(a_1 a_2 \dots a_n)^{\frac{1}{n}}
\]
over all sequences $a_1,a_2,a_3,\dots$ of nonnegative real numbers satisfying
\[
\sum_{k=1}^{\infty}a_k=1.
\]
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]
2021 Putnam, B3
Let $h(x,y)$ be a real-valued function that is twice continuously differentiable throughout $\mathbb{R}^2$, and define
\[
\rho (x,y)=yh_x -xh_y .
\]
Prove or disprove: For any positive constants $d$ and $r$ with $d>r$, there is a circle $S$ of radius $r$ whose center is a distance $d$ away from the origin such that the integral of $\rho$ over the interior of $S$ is zero.
2021 Putnam, A4
Let
\[
I(R)=\iint\limits_{x^2+y^2 \le R^2}\left(\frac{1+2x^2}{1+x^4+6x^2y^2+y^4}-\frac{1+y^2}{2+x^4+y^4}\right) dx dy.
\]
Find
\[
\lim_{R \to \infty}I(R),
\]
or show that this limit does not exist.
2021 Putnam, B6
Given an ordered list of $3N$ real numbers, we can trim it to form a list of $N$ numbers as follows: We divide the list into $N$ groups of $3$ consecutive numbers, and within each group, discard the highest and lowest numbers, keeping only the median. \\
Consider generating a random number $X$ by the following procedure: Start with a list of $3^{2021}$ numbers, drawn independently and unfiformly at random between $0$ and $1$. Then trim this list as defined above, leaving a list of $3^{2020}$ numbers. Then trim again repeatedly until just one number remains; let $X$ be this number. Let $\mu$ be the expected value of $\left|X-\frac{1}{2} \right|$. Show that
\[
\mu \ge \frac{1}{4}\left(\frac{2}{3} \right)^{2021}.
\]
2021 Putnam, A5
Let $A$ be the set of all integers $n$ such that $1 \le n \le 2021$ and $\text{gcd}(n,2021)=1$. For every nonnegative integer $j$, let
\[
S(j)=\sum_{n \in A}n^j.
\]
Determine all values of $j$ such that $S(j)$ is a multiple of $2021$.
2021 Putnam, B5
Say that an $n$-by-$n$ matrix $A=(a_{ij})_{1\le i,j \le n}$ with integer entries is very odd if, for every nonempty subset $S$ of $\{1,2,\dots,n \}$, the $|S|$-by-$|S|$ submatrix $(a_{ij})_{i,j \in S}$ has odd determinant. Prove that if $A$ is very odd, then $A^k$ is very odd for every $k \ge 1$.
2021 Putnam, A1
A grasshopper starts at the origin in the coordinate plane and makes a sequence of hops. Each hop has length $5$, and after each hop the grasshopper is at a point whose coordinates are both integers; thus, there are $12$ possible locations for the grasshopper after the first hop. What is the smallest number of hops needed for the grasshopper to reach the point $(2021,2021)$?
2021 Putnam, A6
Let $P(x)$ be a polynomial whose coefficients are all either $0$ or $1$. Suppose that $P(x)$ can be written as the product of two nonconstant polynomials with integer coefficients. Does it follow that $P(2)$ is a composite integer?
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]
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.