AZ-MO 2020

$13$ fractions are corrected by using each of the numbers $1,2,...,26$ once.Example: $\frac{12}{5},\frac{18}{26}....$ What is the maximum number of fractions which are integers?

$a,b,c$ are positive integer. Solve the equation: $2^{a!}+2^{b!}=c^3$

$a,b,c$ are positive numbers. $a+b+c=3$ Prove that: $\sum \frac{a^2+6}{2a^2+2b^2+2c^2+2a-1}\leq 3$

There is a non-equilateral triangle $ABC$ .Let $ABC$ 's Incentri $I$ .Point $D$ is on the $BC$ side.The circle drawn outside the triangle $IBD$ and $ICD$ intersects the sides $AB$ and $AC$ at points $E$ and $F.$ The circle drawn outside the triangle $DEF$ intersects the sides $AB$ and $AC$ at $N$ and $M$ .Prove that $EM\parallel FN$ .