Problems

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Found: 13

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?

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

A square of side 15 contains 20 non-overlapping unit squares. Prove that it is possible to place a circle of radius 1 inside the large square, so that it does not overlap with any of the unit squares.

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 \(x\).

Ben is going to bend a square sheet of paper \(ABCD\). Ben calls the fold beautiful, if the side \(AB\) crosses the side \(CD\) and the four resulting rectangular triangles are equal. Before that, Jack selects a random point on the sheet \(F\). Find the probability that Ben will be able to make a beautiful fold through the point \(F\).

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 \(ADQ\).