Let $(x_n)$ be a real sequence with $x_1\in(0,1/2)$ and
$$x_{n+1}=3x_n^2-2n x_n^3\qquad(n\ge1).$$
a) Prove that $\lim_{n\to\infty}x_n=0$.
b) For $n\ge1$, put $y_n=x_1+2x_2+\cdots+nx_n$. Prove that $(y_n)$ has a finite limit.
Let $ABC$ be an acute, non-isosceles triangle with orthocenter $H$, and let $D,E,F$ be the feet of the altitudes from $A,B,C$, respectively. Let $(I)$ be the circumcircle of $HEF$, with center $I$, and let $K,J$ be the midpoints of $BC,EF$. The line $HJ$ meets $(I)$ again at $G$, and $GK$ meets $(I)$ again at $L$.
a) Prove that $AL\perp EF$.
b) Let $M=AL\cap EF$. The line $IM$ meets the circumcircle of $IEF$ again at $N$, and $DN$ meets $AB,AC$ at $P,Q$, respectively. Prove that $PE,QF,AK$ are concurrent.
For an integer $n\ge2$, let $s(n)$ be the sum of the positive integers at most $n$ that are not relatively prime to $n$.
a) Prove that $s(n)=\frac n2\bigl(n+1-\varphi(n)\bigr)$, where $\varphi(n)$ counts the positive integers at most $n$ relatively prime to $n$.
b) Prove that no integer $n\ge2$ satisfies $s(n)=s(n+2021)$.
Let $P(x)=a_{21}x^{21}+a_{20}x^{20}+\cdots+a_1x+a_0$, where all coefficients belong to $[1011,2021]$. Assume that $P$ has an integer root and that $c\gt 0$ satisfies $|a_{k+2}-a_k|\le c$ for every $k\in\{0,1,\ldots,19\}$.
a) Prove that $P$ has exactly one integer root.
b) Prove that $\sum_{k=0}^{10}(a_{2k+1}-a_{2k})^2\le440c^2$.
A student divides all $30$ marbles among five boxes labeled $1,2,3,4,5$; empty boxes are allowed.
a) How many divisions are there, where two divisions differ if some box has a different number of marbles?
b) The student then paints every marble with one color. A color may be used on several marbles, but no two marbles in one box have the same color, and from any two boxes one cannot select eight marbles painted in four colors. Prove that at least ten colors are required, regardless of the division.
c) Exhibit a division and coloring satisfying b) with exactly ten colors.
Let $ABC$ be an acute, non-isosceles triangle inscribed in $(O)$. The tangents to $(O)$ at $B,C$ meet at $D$. The circle through $A$ tangent to $BC$ at $B$ meets the median from $A$ again at $G$. The lines $BG,CG$ meet $CD,BD$ at $E,F$, respectively.
a) The line through the midpoints of $BE,CF$ meets $BF,CE$ at $N,M$, respectively. Prove that $A,D,M,N$ are concyclic.
b) The lines $AD,AG$ meet the circumcircles of $DBC,GBC$ again at $H,K$, respectively. The perpendicular bisectors of $HK,HE,HF$ meet $BC,CA,AB$ at $R,P,Q$, respectively. Prove that $R,P,Q$ are collinear.