VTMO 2026 - Grade 8 - Final - SET 9 - Self-study

The numbers $2$ and $4$ are on a board. Hung repeatedly chooses numbers $a,b$ already on the board and writes a new number of one of the forms $3a+1$, $5b+2$, or $2ab+a+b$. Can $2024$ ever be among the numbers he writes? Explain.

Let $a$ be any natural number. Prove that $A=(a+2013)(a+2011)(a+2009)(a+2007)+16$ is a perfect square.

(a) Let $ABCD$ be a convex quadrilateral, $I$ the midpoint of $AC$, and $M$ a point on $AC$ such that $AM=\sqrt2\,AI$. The line through $M$ parallel to $BC$ meets $AB$ at $E$, and the line through $M$ parallel to $DC$ meets $AD$ at $F$. Prove $S_{IEF}=\tfrac12S_{MDB}$. (b) In acute triangle $ABC$, $AM$ is a median ($M\in BC$), $BE$ is an altitude ($E\in AC$), and $K=AM\cap BE$. Given $AM=BE$, prove $AC=\sqrt3\,BK$.

Consider the sequence of strings $0;\ 01;\ 0110;\ 01101001;\ 0110100110010110;\ldots$. Each new string is formed by appending the complement of the previous string, obtained by swapping $0$ and $1$. For example, the complement of $0110$ is $1001$, giving the next string $01101001$. The twelfth string has $2048$ digits. What is its $2016$th digit, counted from the left?

Let $f(x)=x^{2016}+2x^{2015}-3x+4$ and $g(x)=x^2+3x+2$. Find the remainder on dividing $f(x)$ by $g(x)$.