Putnam 1999

Find polynomials $f(x)$,$g(x)$, and $h(x)$, if they exist, such that for all $x$, $$|f(x)|-|g(x)|+h(x) = \begin{cases} -1 & \text{if $x<-1$} \\ 3x+2 & \text{if $-1 \leq x \leq 0$} \\ -2x+2 & \text{if $x>0$.} \end{cases}$$

Let $p(x)$ be a polynomial that is nonnegative for all real $x$. Prove that for some $k$, there are polynomials $f_1(x),\dots,f_k(x$) such that $$p(x) = \sum_{j=1}^k (f_j(x))^2.$$

Consider the power series expansion $$\frac{1}{1-2x-x^2} = \sum_{n=0}^\infty a_n x^n.$$ Prove that, for each integer $n\geq 0$, there is an integer $m$ such that $$a_n^2 + a_{n+1}^2 = a_m .$$

Sum the series $$\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{m^2 n}{3^m(n3^m+m3^n)}.$$

Prove that there is a constant $C$ such that, if $p(x)$ is a polynomial of degree 1999, then $$|p(0)|\leq C \int_{-1}^1 |p(x)| dx.$$

The sequence $(a_n)_{n\geq 1}$ is defined by $a_1=1, a_2=2, a_3=24,$ and, for $n\geq 4$, $$a_n = \frac{6a_{n-1}^2a_{n-3} - 8a_{n-1}a_{n-2}^2}{a_{n-2}a_{n-3}}.$$ Show that, for all n, $a_n$ is an integer multiple of $n$.

Right triangle $ABC$ has right angle at $C$ and $\angle BAC =\theta$; the point $D$ is chosen on $AB$ so that $|AC|=|AD|=1$; the point $E$ is chosen on $BC$ so that $\angle CDE = \theta$. The perpendicular to $BC$ at $E$ meets $AB$ at $F$. Evaluate $\lim_{\theta\rightarrow 0} |EF|$.

Let $P(x)$ be a polynomial of degree $n$ such that $P(x)=Q(x)P''(x)$, where $Q(x)$ is a quadratic polynomial and $P''(x)$ is the second derivative of $P(x)$. Show that if $P(x)$ has at least two distinct roots then it must have $n$ distinct roots.

Let $A=\{(x,y):0\leq x,y<1\}$. For $(x,y)\in A$, let $$S(x,y) = \sum_{\frac{1}{2}\leq \frac{m}{n}\leq 2} x^m y^n,$$ where the sum ranges over all pairs $(m,n)$ of positive integers satisfying the indicated inequalities. Evaluate $$\lim_{(x,y)\rightarrow (1,1), (x,y)\in A} (1-xy^2)(1-x^2y)S(x,y).$$

Let $f$ be a real function with a continuous third derivative such that $f(x), f'(x), f''(x), f'''(x)$ are positive for all $x$. Suppose that $f'''(x)\leq f(x)$ for all $x$. Show that $f'(x)<2f(x)$ for all $x$.

For an integer $n\geq 3$, let $\theta=2\pi/n$. Evaluate the determinant of the $n\times n$ matrix $I+A$, where $I$ is the $n\times n$ identity matrix and $A=(a_{jk})$ has entries $a_{jk}=\cos(j\theta+k\theta)$ for all $j,k$.

Let $S$ be a finite set of integers, each greater than 1. Suppose that for each integer $n$ there is some $s\in S$ such that $\gcd(s,n)=1$ or $\gcd(s,n)=s$. Show that there exist $s,t\in S$ such that $\gcd(s,t)$ is prime.