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

Let $a,b,c,d$ satisfy $a+b+c+d=10$. Prove that $ab+bc+cd+da\ne50$.

Let $a,b,c,d$ be positive numbers with $abcd=1$. Prove:

a) $(ac+bd)(ad+bc)\ge(a+b)(c+d)$.

b) $\dfrac1{a(1+b)}+\dfrac1{b(1+c)}+\dfrac1{c(1+d)}+\dfrac1{d(1+a)}\ge2$.

Given a quadrilateral $ABCD$, let $M,N$ be the midpoints of $BC,AD$. Let $P=AM\cap BN$ and $Q=DM\cap CN$. Prove that $S_{PAB}+S_{QCD}=S_{MPNQ}$.

Find all real triples $(a,b,c)$ for which the following identity holds for every real $x,y,z$:
$$|ax+by+cz|+|ay+bz+cx|+|az+bx+cy|=|x|+|y|+|z|.$$

Solve $\sqrt{2x+1}+\sqrt{x}=\sqrt{6x+1}$.