Let $a,b$ be real numbers. Prove the inequality \[ 2(a^4+a^2b^2+b^4)\ge 3(a^3b+ab^3).\]

In trapezoid $ABCD$, $AD$ is parallel to $BC$. Knowing that $AB=AD+BC$, prove that the bisector of $\angle A$ also bisects $CD$.

In an $m\times n$ rectangular chessboard,there is a stone in the lower leftmost square. Two persons A,B move the stone alternately. In each step one can move the stone upward or rightward any number of squares. The one who moves it into the upper rightmost square wins. Find all $(m,n)$ such that the first person has a winning strategy.

Decimal digits $a,b,c$ satisfy \[ 37\mid (a0a0\ldots a0b0c0c\ldots 0c)_{10} \] where there are $1001$ a's and $1001$ c's. Prove that $b=a+c$.

$a,b$ are real numbers such that: \[ a^3+b^3=8-6ab. \] Find the maximal and minimal value of $a+b$.

Let $I$ be the incenter of a triangle $ABC$. $D,E,F$ are the symmetric points of $I$ with respect to $BC,AC,AB$ respectively. Knowing that $D,E,F,B$ are concyclic,find all possible values of $\angle B$.

In an $m\times n$ rectangular chessboard,there is a stone in the lower leftmost square. Two persons A,B move the stone alternately. In each step one can move the stone upward or rightward any number of squares. The one who moves it into the upper rightmost square wins. Find all $(m,n)$ such that the first person has a winning strategy.

Arrange arbitrarily $1,2,\ldots ,25$ on a circumference. We consider all $25$ sums obtained by adding $5$ consecutive numbers. If the number of distinct residues of those sums modulo $5$ is $d$ $(0\le d\le 5)$,find all possible values of $d$.