Author: A.V. Shapovalov
We call a triangle rational if all of its angles are measured by a rational number of degrees. We call a point inside the triangle rational if, when we join it by segments with vertices, we get three rational triangles. Prove that within any acute-angled rational triangle there are at least three distinct rational points.
What is the largest number of horses that can be placed on an \(8\times8\) chessboard so that no horse touches more than seven of the others?
Harry thought of two positive numbers \(x\) and \(y\). He wrote down the numbers \(x + y\), \(x - y\), \(xy\) and \(x/y\) on a board and showed them to Sam, but did not say which number corresponded to which operation.
Prove that Sam can uniquely figure out \(x\) and \(y\).
The functions \(f\) and \(g\) are defined on the entire number line and are reciprocal. It is known that \(f\) is represented as a sum of a linear and a periodic function: \(f (x) = kx + h (x)\), where \(k\) is a number, and \(h\) is a periodic function. Prove that \(g\) is also represented in this form.
Two play the following game. There is a pile of stones. The first takes either 1 stone or 10 stones with each turn. The second takes either m or n stones with every turn. They take turns, beginning with the first player. He who can not make a move, loses. It is known that for any initial quantity of stones, the first one can always play in such a way as to win (for any strategy of the second player). What values can m and n take?
It is known that \(a > 1\). Is it always true that \(\lfloor \sqrt{\lfloor \sqrt{a}\rfloor }\rfloor = \lfloor \sqrt{4}{a}\rfloor\)?
A robot came up with a cipher for writing words: he replaced some letters of the alphabet with single-digit or two-digit numbers, using only the digits 1, 2 and 3 (different letters it replaces with different numbers). First, he wrote down, using the cipher: \(ROBOT = 3112131233\). Having encrypted the words \(CROCODIL\) and \(BEGEMOT\), he was surprised to note that the numbers were completely identical! Then the Robot ciphered the word \(MATHEMATICS\). Write down the number that he got.
Is there a positive integer \(n\) such that \[\sqrt{n}{17\sqrt{5} + 38} + \sqrt{n}{17\sqrt{5} - 38} = 2\sqrt{5}\,?\]
Does there exist a function \(f (x)\) defined for all real numbers such that \(f(\sin x) + f (\cos x) = \sin x\)?
On the left bank of the river, there were 5 physicists and 5 chemists. All of them need to cross to the right bank. There is a two-seater boat. On the right bank at any time there can not be exactly three chemists or exactly three physicists. How do they all cross over by making 9 trips to the right side?