$(a^{5}+a^{4}+a^{3}+a^{2}+a+1)\cdot$$(b^{5}+b^{4}+b^{3}+b^{2}+b+1)\cdot$$(c^{5}+c^{4}+c^{3}+c^{2}+c+1)\geq 8(a^{2}+a+1)(b^{2}+b+1)(c^{2}+c+1).$
Problem 2. Find all the prime numbers $p$ for which there exist positive integers $x$ and $y$ that satisfy the equation
$x(y^{2}-p)+ y(x^{2}-p)=5p.$
Problem 3. Let $n>3$ be a positive integer. An equilateral triangle $ABC$ is divided into $n^{2}$ identical equilateral "small" triangles using lines parallel to its sides. The figure below illustrates the case $n=4$. Let $m$ be the number of rhombi consisting of 2 "small" triangles. Let $d$ be the number of rhombi consisting of 8 "small" triangles. Find the difference $m-d$ in terms of $n$.
- Figure 1
- fig1.png (20.35 KiB) Vizualizat de 4320 ori
$S\leq\displaystyle\frac{AB\cdot CD+n(n-1)DA^{2}+n\cdot DA\cdot BC}{2n^{2}}.$
