MYTS 2019 - Grade 8 - Round 2

Numbers $a,b$ have product 1. Prove that $(a+b)(a^5+b^5)\ge4$. When does equality hold?

Find all primes $p$ such that $p^2+26$ has at least two prime divisors greater than $p$.

Numbers $a_1,a_2,\ldots,a_{2019}$ have sum greater than 1. Prove that some positive integer $n\le2019$ has the property that $a_n$, $a_n+a_{n-1}$, ..., $a_n+a_{n-1}+\cdots+a_1$ are all positive.

Integers $a,b,c$ satisfy that $ab+bc+ca+1$ is divisible by 4. Prove that $a+b+c+abc$ is also divisible by 4.

Choose distinct points $C,D$ on segment $AB$ with $C$ between $A$ and $D$. Construct squares $ACMN$, $ADGH$, $BCPQ$ and $BDEF$ so that $M,P$ lie in one half-plane bounded by line $AB$, while $G,E$ lie in the other. Let $I$ be the intersection of $QH$ and $FN$. Prove that $IC=ID$.

Each of classes $8A,8B,8C$ has 36 students. Every student knows at least 42 students in the other two classes. Prove that there are two students in different classes who know each other and have at least six common acquaintances in the remaining class.

Note: Acquaintance is mutual.