Let $f:\mathbb R\to(0,+\infty)$ be continuous and satisfy $\lim_{x\to-\infty}f(x)=\lim_{x\to+\infty}f(x)=0$.
(a) Prove that $f$ attains a greatest value on $\mathbb R$.
(b) Prove that there exist sequences $(x_n),(y_n)$ with $x_n\lt y_n$ for every $n\ge1$, converging to the same limit, such that $f(x_n)=f(y_n)$ for every $n$.
An integer sequence $(x_n)$ satisfies $0\le x_0\lt x_1\le100$ and
$$x_{n+2}=7x_{n+1}-x_n+280\qquad(n\ge0).$$
(a) If $x_0=2,x_1=3$, prove that for every natural number $n$, the sum of the positive divisors of $x_nx_{n+1}+x_{n+1}x_{n+2}+x_{n+2}x_{n+3}+2018$ is divisible by $24$.
(b) Find all pairs $(x_0,x_1)$ for which $x_nx_{n+1}+2019$ is a perfect square for infinitely many natural numbers $n$.
For every polynomial $f(x)=a_0+a_1x+\cdots+a_mx^m$, define $\Gamma(f)=a_0^2+a_1^2+\cdots+a_m^2$. Let $P(x)=(x+1)(x+2)\cdots(x+2020)$. Prove that there exist at least $2^{2019}$ pairwise distinct polynomials $Q_k(x)$, all with positive coefficients, satisfying $\deg Q_k=2020$ and $\Gamma(Q_k(x)^n)=\Gamma(P(x)^n)$ for every positive integer $n$.
Triangle $ABC$ has orthocenter $H$ and incenter $I$. On rays $AB,AC,BC,BA,CA,CB$, respectively, choose $A_1,A_2,B_1,B_2,C_1,C_2$ so that
$$AA_1=AA_2=BC,\quad BB_1=BB_2=CA,\quad CC_1=CC_2=AB.$$
The pairs of lines $(B_1B_2,C_1C_2)$, $(C_1C_2,A_1A_2)$, $(A_1A_2,B_1B_2)$ meet at $A',B',C'$, respectively.
(a) Prove that the area of $A'B'C'$ is at most the area of $ABC$.
(b) Let $J$ be the circumcenter of $A'B'C'$. Lines $AJ,BJ,CJ$ meet $BC,CA,AB$ at $R,S,T$, respectively. The circumcircles of $AST,BTR,CRS$ have a common point $K$. Prove that if $ABC$ is scalene, then $IHJK$ is a parallelogram.
Let $f(x)=x^2-\alpha x+1$, where $\alpha\in\mathbb R$.
(a) For $\alpha=\sqrt{15}/2$, express $f$ as the quotient of two polynomials with nonnegative coefficients.
(b) Find all real $\alpha$ for which such a representation exists.
Let $ABC$ be an acute scalene triangle inscribed in $(O)$, with orthocenter $H$. Let $M,N,P$ be the midpoints of $BC,CA,AB$, and let $D,E,F$ be the feet of the altitudes from $A,B,C$, respectively. Let $K$ be the reflection of $H$ in $BC$. Lines $DE,MP$ meet at $X$, and lines $DF,MN$ meet at $Y$.
(a) Line $XY$ meets the minor arc $BC$ of $(O)$ at $Z$. Prove that $K,Z,E,F$ are concyclic.
(b) Lines $KE,KF$ meet $(O)$ again at $S,T$, respectively, with $S,T\ne K$. Prove that $BS,CT,XY$ are concurrent.
Identical square sheets are divided into a $5\times5$ grid on both faces. Color the sheets using $n$ colors so that each cell has the same color on its two faces. Two colored sheets are considered identical if they can be placed one on top of the other so that all corresponding cells have the same colors. Prove that there are at most
$$\frac18\left(n^{25}+4n^{15}+n^{13}+2n^7\right)$$
pairwise distinct colored sheets.