Let \(f (x)\) be a polynomial about which it is known that the equation \(f (x) = x\) has no roots. Prove that then the equation \(f (f (x)) = x\) does not have any roots.
Prove that there exist numbers, that can be presented in no fewer than 100 ways in the form of a summation of 20001 terms, each of which is the 2000th power of a whole number.
Let \(f\) be a continuous function defined on the interval \([0; 1]\) such that \(f (0) = f (1) = 0\). Prove that on the segment \([0; 1]\) there are 2 points at a distance of 0.1 at which the function \(f 4(x)\) takes equal values.
A convex figure and point \(A\) inside it are given. Prove that there is a chord (that is, a segment joining two boundary points of a convex figure) passing through point \(A\) and dividing it in half at point \(A\).
Calculate \(\int_0^{\pi/2} (\sin^2 (\sin x) + \cos^2 (\cos x))\,dx\).
Prove that rational numbers from \([0; 1]\) can be covered by a system of intervals of total length no greater than \(1/1000\).
Does a continuous function that takes every real value exactly 3 times exist?
Prove that a convex quadrilateral \(ICEF\) can contain a circle if and only if \(IC+EH = CE+IF\).
Two perpendicular straight lines are drawn through the centre of the square. Prove that their intersection points with the sides of a square form a square.
Two circles \(c\) and \(d\) are tangent at point \(B\). Two straight lines intersecting the first circle at points \(D\) and \(E\) and the second circle at points \(G\) and \(F\) are drawn through the point \(B\). Prove that \(ED\) is parallel to \(FG\).