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
$$x-y+z\equiv c\pmod p.$$