The sequence $(a_n)$ is defined by $a_1=1$ and, for every integer $n\ge1$,
$$a_{2n}=a_n,\qquad a_{2n+1}=a_n+1.$$
(a) Find all positive integers $n$ such that $a_{kn}=a_n$ for every positive integer $k\le n$.
(b) Prove that there are infinitely many positive integers $m$ such that $a_{km}\ge a_m$ for every positive integer $k$.
Annual practice duration is 540 minutes, combining the two source-stated 270-minute exam days. Question point values reproduce the supplied source labels; no automatic proof grading is implied. Importer scoring multipliers: correct 1, incorrect 0, blank 0.
A square board has $2021\times2021$ cells. Find the largest possible $k$ such that $k$ cells can be marked and every marked cell shares a vertex with at most one other marked cell.
Let $ABC$ be a triangle and let $N$ be a point different from $A,B,C$. Let $A_b$ be the reflection of $A$ across $NB$, and let $B_a$ be the reflection of $B$ across $NA$. Define the other two pairs $B_c,C_b$ and $C_a,A_c$ similarly. Let $m_a$ pass through $N$ and be perpendicular to $B_cC_b$, and define $m_b,m_c$ cyclically.
(a) Suppose $N$ is the orthocenter of $ABC$. Prove that the reflections of $m_a,m_b,m_c$ across the bisectors of $\angle BNC,\angle CNA,\angle ANB$, respectively, coincide.
(b) Suppose $N$ is the center of the nine-point (Euler) circle of $ABC$. Prove that the reflections of $m_a,m_b,m_c$ across $BC,CA,AB$, respectively, are concurrent at one point.
A circle $(O)$ and a chord $BC$ are fixed. A point $A$ moves on $(O)$ so that $ABC$ is acute and scalene. Let $I$ be the midpoint of $BC$, and let $AD,BE,CF$ be the altitudes. On rays $FA,EA$, respectively, choose $M,N$ such that $FM=CE$ and $FN = BF$. Suppose $MN$ meets $EF$ at $L$, and circles $(LEN)$ and $(LFM)$ meet again at $G$.
(a) Prove that circle $(MNG)$ always passes through a fixed point.
(b) Suppose $AD$ meets $(O)$ again at $K$. On the tangent at $D$ to circle $(DKI)$, take $P,Q$ such that $GP\parallel AB$ and $GQ\parallel AC$. Let $T$ be the center of circle $(GPQ)$. Prove that line $GT$ always passes through a fixed point.
The supplied scan states FN = BF. The external booklet states EN = BF. The following reference solution applies only to the corrected EN = BF version; it is not a proof of the literal FN = BF statement.
Corrected reference statement (booklet):
A circle $(O)$ and a chord $BC$ are fixed. A point $A$ moves on $(O)$ so that $ABC$ is acute and scalene. Let $I$ be the midpoint of $BC$, and let $AD,BE,CF$ be the altitudes. On rays $FA,EA$, respectively, choose $M,N$ such that $FM=CE$ and $EN = BF$. Suppose $MN$ meets $EF$ at $L$, and circles $(LEN)$ and $(LFM)$ meet again at $G$.
(a) Prove that circle $(MNG)$ always passes through a fixed point.
(b) Suppose $AD$ meets $(O)$ again at $K$. On the tangent at $D$ to circle $(DKI)$, take $P,Q$ such that $GP\parallel AB$ and $GQ\parallel AC$. Let $T$ be the center of circle $(GPQ)$. Prove that line $GT$ always passes through a fixed point.
Let $n\ge3$ be an integer and let $p$ be a prime such that $p>6^{n-1}-2^n+1$. Let $S$ consist of $n$ positive integers with pairwise distinct residues modulo $p$. Prove that there is a positive integer $c$ such that exactly two ordered triples $(x,y,z)\in S^3$, with pairwise distinct components, satisfy
$$x-y+z\equiv c\pmod p.$$
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.