Let $a$, $b$, and $c$ be positive integers for which the number \[\frac{a\sqrt2+b}{b\sqrt2+c}\]is rational. Show that the number $ab+bc+ca$ is divisible by $a+b+c$.

Point $D$ lies on the side $AB$ of triangle $ABC$, and point $E$ lies on the segment $CD$. Prove that if the sum of the areas of triangles $ACE$ and $BDE$ is equal to half the area of triangle $ABC$, then either point $D$ is the midpoint of $AB$ or point $E$ is the midpoint of $CD$.

Positive integers $a$ and $b$ are given such that each of the numbers $ab$ and $(a+1)(b+1)$ is a perfect square. Prove that there exists an integer $n>1$ such that the number $(a+n)(b+n)$ is a perfect square.

In each square of a $4\times 4$ board, we are to write an integer in such a way that the sums of the numbers in each column and in each row are nonnegative integral powers of $2$. Is it possible to do this in such a way that every two of these eight sums are different? Justify your answer.

Prove that if the diagonals of a certain trapezoid are perpendicular, then the sum of the lengths of the bases of this trapezoid is not greater than the sum of the lengths of the sides of this trapezoid.

Let $a$, $b$, and $d$ be positive integers. It is known that $a+b$ is divisible by $d$ and $a\cdot b$ is divisible by $d^2$. Prove that both $a$ and $b$ are divisible by $d$.

Rational numbers $a$, $b$, $c$ satisfy the equation \[(a+b+c)(a+b-c)=c^2\,.\]Show that $a+b=c=0$.

Consider an acute triangle $ABC$ with \[\angle ACB=45^\circ\,.\]Let $BCED$ and $ACFG$ be squares lying outside triangle $ABC$. Prove that the midpoint of segment $DG$ coincides with the circumcenter of triangle $ABC$.

In each square of an $11\times 11$ board, we are to write one of the numbers $-1$, $0$, or $1$ in such a way that the sum of the numbers in each column is nonnegative and the sum of the numbers in each row is nonpositive. What is the smallest number of zeros that can be written on the board? Justify your answer.