SMO 2019 Junior Round 2

In the triangle $ABC$, $AC=BC$, $\angle C=90^\circ$, $D$ is the midpoint of $BC$, $E$ is the point on $AB$ such that $AD$ is perpendicular to $CE$. Prove that $AE=2EB$.

There are 315 marbles divided into three piles of 81, 115 and 119. In each move Ah Meng can either merge several piles into a single pile or divide a pile with an even number of marbles into 2 equal piles. Can Ah Meng divide the marbles into 315 piles, each with a single marble?

Find all positive integers $m,n$ such that $\frac{2m-1}{n}$ and $\frac{2n-1}{m}$ are both integers.

Let $a>b>0$. Prove that $$\sqrt{2}a^3+\frac{3}{ab-b^2}\ge10.$$ When does equality hold?

Let $n$ be a positive integer and consider an arrangement of $2n$ blocks in a straight line, where $n$ of them are red and the rest blue. A swap refers to choosing two consecutive blocks and then swapping their positions. Let $A$ be the minimum number of swaps needed to make the first $n$ blocks all red and $B$ be the minimum number of swaps needed to make the first $n$ blocks all blue. Show that $A+B$ is independent of the starting arrangement and determine its value.