VN-TST 2022

Given a real number $\alpha$ and consider function $\varphi(x)=x^2e^{\alpha x}$ for $x\in\mathbb R$ . Find all function $f:\mathbb R\to\mathbb R$ that satisfy: $$f(\varphi(x)+f(y))=y+\varphi(f(x))$$ forall $x,y\in\mathbb R$

Given a convex polyhedron with 2022 faces. In 3 arbitary faces, there are already number $26; 4$ and $2022$ (each face contains 1 number). They want to fill in each other face a real number that is an arithmetic mean of every numbers in faces that have a common edge with that face. Prove that there is only one way to fill all the numbers in that polyhedron.

Let $ABCD$ be a parallelogram, $AC$ intersects $BD$ at $I$ . Consider point $G$ inside $\triangle ABC$ that satisfy $\angle IAG=\angle IBG\neq 45^{\circ}-\frac{\angle AIB}{4}$ . Let $E,G$ be projections of $C$ on $AG$ and $D$ on $BG$ . The $E-$ median line of $\triangle BEF$ and $F-$ median line of $\triangle AEF$ intersects at $H$ . $a)$ Prove that $AF,BE$ and $IH$ concurrent. Call the concurrent point $L$ . $b)$ Let $K$ be the intersection of $CE$ and $DF$ . Let $J$ circumcenter of $(LAB)$ and $M,N$ are respectively be circumcenters of $(EIJ)$ and $(FIJ)$ . Prove that $EM,FN$ and the line go through circumcenters of $(GAB),(KCD)$ are concurrent.

An acute, non-isosceles triangle $ABC$ is inscribed in a circle with centre $O$ . A line go through $O$ and midpoint $I$ of $BC$ intersects $AB, AC$ at $E, F$ respectively. Let $D, G$ be reflections to $A$ over $O$ and circumcentre of $(AEF)$ , respectively. Let $K$ be the reflection of $O$ over circumcentre of $(OBC)$ . $a)$ Prove that $D, G, K$ are collinear. $b)$ Let $M, N$ are points on $KB, KC$ that $IM\perp AC$ , $IN\perp AB$ . The midperpendiculars of $IK$ intersects $MN$ at $H$ . Assume that $IH$ intersects $AB, AC$ at $P, Q$ respectively. Prove that the circumcircle of $\triangle APQ$ intersects $(O)$ the second time at a point on $AI$ .

A fractional number $x$ is called pretty if it has finite expression in base $-b$ numeral system, $b$ is a positive integer in $[2;2022]$ . Prove that there exists finite positive integers $n\geq 4$ that with every $m$ in $(\frac{2n}{3}; n)$ then there is at least one pretty number between $\frac{m}{n-m}$ and $\frac{n-m}{m}$

Given a set $A=\{1;2;...;4044\}$ . They color $2022$ numbers of them by white and the rest of them by black. With each $i\in A$ , called the important number of $i$ be the number of all white numbers smaller than $i$ and black numbers larger than $i$ . With every natural number $m$ , find all positive integers $k$ that exist a way to color the numbers that can get $k$ important numbers equal to $m$ .