The function \(f (x)\) is defined on the positive real \(x\) and takes only positive values. It is known that \(f (1) + f (2) = 10\) and \(f(a+b) = f(a) + f(b) + 2\sqrt{f(a)f(b)}\) for any \(a\) and \(b\). Find \(f (2^{2011})\).
On a chessboard, \(n\) white and \(n\) black rooks are arranged so that the rooks of different colours cannot capture one another. Find the greatest possible value of \(n\).
Does there exist a real number \({\alpha}\) such that the number \(\cos {\alpha}\) is irrational, and all the numbers \(\cos 2{\alpha}\), \(\cos 3{\alpha}\), \(\cos 4{\alpha}\), \(\cos 5{\alpha}\) are rational?
We create some segments in a regular \(n\)-gon by joining endpoints of the \(n\)-gon. What’s the maximum number of such segments while ensuring that no two segments are parallel? The segments are allowed to be sides of the \(n\)-gon - that is, joining adjacent vertices of the polygon.
Solve the inequality: \(\lfloor x\rfloor \times \{x\} < x - 1\).
We are given a polynomial \(P(x)\) and numbers \(a_1\), \(a_2\), \(a_3\), \(b_1\), \(b_2\), \(b_3\) such that \(a_1a_2a_3 \ne 0\). It turned out that \(P (a_1x + b_1) + P (a_2x + b_2) = P (a_3x + b_3)\) for any real \(x\). Prove that \(P (x)\) has at least one real root.
Let \(x_1, x_2, \dots , x_n\) be some numbers belonging to the interval \([0, 1]\). Prove that on this segment there is a number \(x\) such that \[\frac{1}{n} (|x - x_1| + |x - x_2| + \dots + |x - x_n|) = 1/2.\]
It is known that a certain polynomial at rational points takes rational values. Prove that all its coefficients are rational.
On the selection to the government of the planet of liars and truth tellers \(12\) candidates gave a speech about themselves. After a while, one said: “before me only once did someone lie” Another said: “And now-twice.” “And now – thrice” – said the third, and so on until the \(12\)th, who said: “And now \(12\) times someone has lied.” Then the presenter interrupted the discussion. It turned out that at least one candidate correctly counted how many times someone had lied before him. So how many times have the candidates lied?
Two people play the following game. Each player in turn rubs out 9 numbers (at his choice) from the sequence \(1, 2, \dots , 100, 101\). After eleven such deletions, 2 numbers will remain. The first player is awarded so many points, as is the difference between these remaining numbers. Prove that the first player can always score at least 55 points, no matter how played the second.