Prove that the equation \[a_1 \sin x + b_1 \cos x + a_2 \sin 2x + b_2 \cos 2x + \dots + a_n \sin nx + b_n \cos nx = 0\] has at least one root for any values of \(a_1 , b_1, a_2, b_2, \dots, a_n, b_n\).
Solve the equation \(2x^x = \sqrt {2}\) for positive numbers.
Two players play the following game. They take turns. One names two numbers that are at the ends of a line segment. The next then names two other numbers, which are at the ends of a segment nested in the previous one. The game goes on indefinitely. The first aims to have at least one rational number within the intersection of all of these segments, and the second aims to prevent such occurring. Who wins in this game?
The positive irrational numbers \(a\) and \(b\) are such that \(1/a + 1/b = 1\). Prove that among the numbers \(\lfloor ma\rfloor , \lfloor nb\rfloor\) each natural number occurs exactly once.
A rectangular billiard with sides 1 and \(\sqrt {2}\) is given. From its angle at an angle of \(45 ^\circ\) to the side a ball is released. Will it ever get into one of the pockets? (The pockets are in the corners of the billiard table).
Suppose that \(n \geq 3\). Are there n points that do not lie on one line, whose pairwise distances are irrational, and the areas of all of the triangles with vertices in them are rational?
Do there exist three points \(A\), \(B\) and \(C\) on the plane such that for any point \(X\) the length of at least one of the segments \(XA\), \(XB\) and \(XC\) is irrational?
Find the sums of the following series:
a) \({\frac {1} {1 \times 2}} + {\frac {1} {2 \times 3}} + {\frac {1} {3 \times 4}} + {\frac {1} {4 \times 5}} + \dots\);
b) \({\frac {1} {1 \times 2 \times 3}} + {\frac {1} {2 \times 3 \times 4}} + {\frac {1} {3 \times 4 \times 5}} + {\frac {1} {4 \times 5 \times 6}} + \dots\);
c) \({\frac {0!} {r!}} + {\frac {1!} {(r-1)!}} + {\frac {2!} {(r-2) !}} + {\frac {3!} {(r-3)!}} + \dots\) for \(r \geq 2\).
Could it be that a) \(\sigma(n) > 3n\); b) \(\sigma(n) > 100n\)?
Prove that there are infinitely many composite numbers among the numbers \(\lfloor 2^k \sqrt{2}\rfloor\) (\(k = 0, 1, \dots\)).