VN-TST 2021

Define the sequence $(a_n)$ as $a_1 = 1$ , $a_{2n} = a_n$ and $a_{2n+1} = a_n + 1$ for all $n\geq 1$ .

a) Find all positive integers $n$ such that $a_{kn} = a_n$ for all integers $1 \leq k \leq n$ . b) Prove that there exist infinitely many positive integers $m$ such that $a_{km} \geq a_m$ for all positive integers $k$ .

In a board of $2021 \times 2021$ grids, we pick $k$ unit squares such that every picked square shares vertice(s) with at most $1$ other picked square. Determine the maximum of $k$ .

Let $ABC$ be a triangle and $N$ be a point that differs from $A,B,C$ . Let $A_b$ be the reflection of $A$ through $NB$ , and $B_a$ be the reflection of $B$ through $NA$ . Similarly, we define $B_c, C_b, A_c, C_a$ . Let $m_a$ be the line through $N$ and perpendicular to $B_cC_b$ . Define similarly $m_b, m_c$ .

a) Assume that $N$ is the orthocenter of $\triangle ABC$ , show that the respective reflection of $m_a, m_b, m_c$ through the bisector of angles $\angle BNC, \angle CNA, \angle ANB$ are the same line. b) Assume that $N$ is the nine-point center of $\triangle ABC$ , show that the respective reflection of $m_a, m_b, m_c$ through $BC, CA, AB$ concur.

Let $a,b,c$ are non-negative numbers such that $$2(a^2+b^2+c^2)+3(ab+bc+ca)=5(a+b+c)$$ then prove that $4(a^2+b^2+c^2)+2(ab+bc+ca)+7abc\le 25$

Given a fixed circle $(O)$ and two fixed points $B, C$ on that circle, let $A$ be a moving point on $(O)$ such that $\triangle ABC$ is acute and scalene. Let $I$ be the midpoint of $BC$ and let $AD, BE, CF$ be the three heights of $\triangle ABC$ . In two rays $\overrightarrow{FA}, \overrightarrow{EA}$ , we pick respectively $M,N$ such that $FM = CE, EN = BF$ . Let $L$ be the intersection of $MN$ and $EF$ , and let $G \neq L$ be the second intersection of $(LEN)$ and $(LFM)$ .

a) Show that the circle $(MNG)$ always goes through a fixed point.

b) Let $AD$ intersects $(O)$ at $K \neq A$ . In the tangent line through $D$ of $(DKI)$ , we pick $P,Q$ such that $GP \parallel AB, GQ \parallel AC$ . Let $T$ be the center of $(GPQ)$ . Show that $GT$ always goes through a fixed point.

Let $n \geq 3$ be a positive integers and $p$ be a prime number such that $p > 6^{n-1} - 2^n + 1$ . Let $S$ be the set of $n$ positive integers with different residues modulo $p$ . Show that there exists a positive integer $c$ such that there are exactly two ordered triples $(x,y,z) \in S^3$ with distinct elements, such that $x-y+z-c$ is divisible by $p$ .