Given a,b c are lenth of a triangle (If ABC is a triangle then AC=b, BC=a, AC=b) and $a+b+c=2$. Prove that $1+abc<ab+bc+ca\leq \frac{28}{27}+abc$

Solve in integer the equation $\frac{1}{2}(x+y)(y+z)(x+z)+(x+y+z)^{3}=1-xyz$

Find the last five digits of $1^{100}+2^{100}+3^{100}+...+999999^{100}$

Let $ABCD$ is a cyclic. $K,L,M,N$ are midpoints of segments $AB$, $BC$ $CD$ and $DA$. $H_{1},H_{2},H_{3},H_{4}$ are orthocenters of $AKN$ $KBL$ $LCM$ and $MND$. Prove that $H_{1}H_{2}H_{3}H_{4}$ is a paralelogram.