Prove that multiplying the polynomial \((x + 1)^{n-1}\) by any polynomial different from zero, we obtain a polynomial having at least \(n\) nonzero coefficients.
In a group of friends, each two people have exactly five common acquaintances. Prove that the number of pairs of friends is divisible by 3.
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?
How many rational terms are contained in the expansion of
a) \((\sqrt 2 + \sqrt[4]{3})^{100}\);
b) \((\sqrt 2 + \sqrt[3]{3})^{300}\)?
Which term in the expansion \((1 + \sqrt 3)^{100}\) will be the largest by the Newton binomial formula?
Prove that in any infinite decimal fraction you can rearrange the numbers so that the resulting fraction becomes a rational number.