VTMO 2026 - Grade 7 - Final - SET 4 - Self-study

Some cells of a $3\times4$ table are filled as shown. Which number should be removed from $0,2,3,4,5,7,8,10,11$ so that the remaining eight numbers can fill the empty cells with the same arithmetic mean in every row and every column? Explain why.

a) Prove that if $\frac ab=\frac cd$, then $\frac{a^2+b^2}{c^2+d^2}=\frac{ab}{cd}$.

b) Find the minimum value of $C=\frac6{|x|-3}$ for integer $x$.

Prove that if $\frac ab=\frac cd$, then $\frac{a^2+b^2}{c^2+d^2}=\frac{ab}{cd}$.

a) Let $A=0.1(2453)$ be the recurring decimal $0.1245324532453\ldots$. Find the smallest positive integer $k$ such that $kA$ is an integer.

b) An gives Binh four distinct positive integers and asks for the greatest common divisor of each of the six pairs. The results are $1,2,3,4,5,N$, with $N\gt 5$. Find the smallest possible $N$.

Find all pairs of positive integers $(a,b)$ such that $1\lt a\le b$ and $ab-1$ is divisible by $(a-1)(b-1)$.