Find all non-negative integers $x, y$ and primes $p$ such that $$3^x+p^2=7 \cdot 2^y.$$

Let $a, b, c$ be positive reals such that $ab+bc+ca=\frac{3}{4}$. Show that $$(a+b+c)^6 \geq (\frac{9} {8})^3(1+(a+b)^2)(1+(b+c)^2)(1+(c+a)^2).$$When does equality hold?

a) Is it possible to place $2024$ checkers on a board $70 \times 70$ so that any square $2 \times 2$ contains even number of checkers? b) Is it possible to place $2023$ checkers on a board $70 \times 70$ so that any square $2 \times 2$ contains odd number of checkers?

Let $I$ be the incenter of a triangle $ABC$ with $AB \neq AC$. Let $M$ be the midpoint of $BC$, $M' \in BC$ be such that $IM'=IM$ and $K$ be the midpoint of the arc $BAC$. If $AK \cap BC=L$, show that $KLIM'$ is cyclic.