Problems

A rectangle is made up from six squares. Find side length of the largest square if side length of the smallest square is 1.

This shape below is made up from squares.

Find side length of the bottom square if side length of the smallest square is equal to 1.

You are given a convex quadrilateral. Is it always possible to cut out a parallelogram out of the quadrilateral such that three vertices of the new parallelogram are the vertices of the old quadrilateral?

The area of a rectangle is 1 cm${}^2$. Can its perimeter be greater than 1 km?

The number $x$ is such a number that exactly one of the four numbers $a=x-\sqrt{2}$, $b=x-1/x$, $c=x+1/x$, $d=x^2+2\sqrt{2}$ is not an integer. Find all such $x$.

A quadrilateral is given; $A$, $B$, $C$, $D$ are the successive midpoints of its sides, $P$ and $Q$ are the midpoints of its diagonals. Prove that the triangle $BCP$ is equal to the triangle $ADQ$.

In a regular 1981-gon 64 vertices were marked. Prove that there exists a trapezium with vertices at the marked points.

A square is cut by 18 straight lines, 9 of which are parallel to one side of the square and the other 9 parallel to the other – perpendicular to the first 9 – dividing the square into 100 rectangles. It turns out that exactly 9 of these rectangles are squares. Prove that among these 9 squares there will be two that are identical.

9 straight lines each divide a square into two quadrilaterals, with their areas having a ratio of $2:3$. Prove that at least three of the nine lines pass through the same point.

Prove that a convex quadrilateral $ICEF$ can contain a circle if and only if $IC+EH=CE+IF$.