Found problems: 11
2019 Putnam, B3
Let $Q$ be an $n$-by-$n$ real orthogonal matrix, and let $u\in \mathbb{R}^n$ be a unit column vector (that is, $u^Tu=1$). Let $P=I-2uu^T$, where $I$ is the $n$-by-$n$ identity matrix. Show that if $1$ is not an eigenvalue of $Q$, then $1$ is an eigenvalue of $PQ$.
2019 Putnam, A3
Given real numbers $b_0,b_1,\ldots, b_{2019}$ with $b_{2019}\neq 0$, let $z_1,z_2,\ldots, z_{2019}$ be the roots in the complex plane of the polynomial
\[
P(z) = \sum_{k=0}^{2019}b_kz^k.
\]
Let $\mu = (|z_1|+ \cdots + |z_{2019}|)/2019$ be the average of the distances from $z_1,z_2,\ldots, z_{2019}$ to the origin. Determine the largest constant $M$ such that $\mu\geq M$ for all choices of $b_0,b_1,\ldots, b_{2019}$ that satisfy
\[
1\leq b_0 < b_1 < b_2 < \cdots < b_{2019} \leq 2019.
\]
2019 Putnam, A2
In the triangle $\triangle ABC$, let $G$ be the centroid, and let $I$ be the center of the inscribed circle. Let $\alpha$ and $\beta$ be the angles at the vertices $A$ and $B$, respectively. Suppose that the segment $IG$ is parallel to $AB$ and that $\beta = 2\tan^{-1}(1/3)$. Find $\alpha$.
2019 Putnam, A4
Let $f$ be a continuous real-valued function on $\mathbb R^3$. Suppose that for every sphere $S$ of radius $1$, the integral of $f(x,y,z)$ over the surface of $S$ equals zero. Must $f(x,y,z)$ be identically zero?
2019 Putnam, B2
For all $n\ge 1$, let $a_n=\sum_{k=1}^{n-1}\frac{\sin(\frac{(2k-1)\pi}{2n})}{\cos^2(\frac{(k-1)\pi}{2n})\cos^2(\frac{k\pi}{2n})}$. Determine $\lim_{n\rightarrow \infty}\frac{a_n}{n^3}$.
2019 Putnam, A5
Let $p$ be an odd prime number, and let $\mathbb{F}_p$ denote the field of integers modulo $p$. Let $\mathbb{F}_p[x]$ be the ring of polynomials over $\mathbb{F}_p$, and let $q(x) \in \mathbb{F}_p[x]$ be given by $q(x) = \sum_{k=1}^{p-1} a_k x^k$ where $a_k = k^{(p-1)/2}$ mod $p$. Find the greatest nonnegative integer $n$ such that $(x-1)^n$ divides $q(x)$ in $\mathbb{F}_p[x]$.
2019 Putnam, A1
Determine all possible values of $A^3+B^3+C^3-3ABC$ where $A$, $B$, and $C$ are nonnegative integers.
2019 Putnam, B4
Let $\mathcal F$ be the set of functions $f(x,y)$ that are twice continuously differentiable for $x\geq 1$, $y\geq 1$ and that satisfy the following two equations (where subscripts denote partial derivatives):
\[xf_x + yf_y = xy\ln(xy),\] \[x^2f_{xx} + y^2f_{yy} = xy.\]
For each $f\in\mathcal F$, let
\[
m(f) = \min_{s\geq 1}\left(f(s+1,s+1) - f(s+1,s)-f(s,s+1) + f(s,s)\right).
\]
Determine $m(f)$, and show that it is independent of the choice of $f$.
2019 Putnam, A6
Let $g$ be a real-valued function that is continuous on the closed interval $[0,1]$ and twice differentiable on the open interval $(0,1)$. Suppose that for some real number $r>1$,
\[
\lim_{x\to 0^+}\frac{g(x)}{x^r} = 0.
\]
Prove that either
\[
\lim_{x\to 0^+}g'(x) = 0\qquad\text{or}\qquad \limsup_{x\to 0^+}x^r|g''(x)|= \infty.
\]
2019 Putnam, B1
Denote by $\mathbb Z^2$ the set of all points $(x,y)$ in the plane with integer coordinates. For each integer $n\geq 0$, let $P_n$ be the subset of $\mathbb Z^2$ consisting of the point $(0,0)$ together with all points $(x,y)$ such that $x^2+y^2=2^k$ for some integer $k\leq n$. Determine, as a function of $n$, the number of four-point subsets of $P_n$ whose elements are the vertices of a square.
2019 Putnam, B6
Let $\mathbb{Z}^n$ be the integer lattice in $\mathbb{R}^n$. Two points in $\mathbb{Z}^n$ are called {\em neighbors} if they differ by exactly 1 in one coordinate and are equal in all other coordinates. For which integers $n \geq 1$ does there exist a set of points $S \subset \mathbb{Z}^n$ satisfying the following two conditions? \\
(1) If $p$ is in $S$, then none of the neighbors of $p$ is in $S$. \\
(2) If $p \in \mathbb{Z}^n$ is not in $S$, then exactly one of the neighbors of $p$ is in $S$.