The number \(x\) is such that both the sums \(S = \sin 64x + \sin 65x\) and \(C = \cos 64x + \cos 65x\) are rational numbers.
Prove that in both of these sums, both terms are rational.
For all real \(x\) and \(y\), the equality \(f (x^2 + y) = f (x) + f (y^2)\) holds. Find \(f(-1)\).
Prove that for any odd natural number, \(a\), there exists a natural number, \(b\), such that \(2^b - 1\) is divisible by \(a\).
On a lottery ticket, it is necessary for Mary to mark 8 cells from 64. What is the probability that after the draw, in which 8 cells from 64 will also be selected (all such possibilities are equally probable), it turns out that Mary guessed
a) exactly 4 cells? b) exactly 5 cells? c) all 8 cells?
A sequence of natural numbers \(a_1 < a_2 < a_3 < \dots < a_n < \dots\) is such that each natural number is either a term in the sequence, can be expressed as the sum of two terms in the sequence, or perhaps the same term twice. Prove that \(a_n \leq n^2\) for any \(n=1, 2, 3,\dots\)
The triangle visible in the picture is equilateral. The hexagon inside is a regular hexagon. If the area of the whole big triangle is \(18\), find the area of the small blue triangle.
On the left there is a circle inscribed in a square of side 1. On the right there are 16 smaller, identical circles, which all together fit inside a square of side 1. Which area is greater, the yellow or the blue one?
Prove that, for any integer \(n\), among the numbers \(n, n + 1, n + 2, \dots , n + 9\) there is at least one number that is mutually prime with the other nine numbers.
30 pupils in years 7 to 11 took part in the creation of 40 maths problems. Every possible pair of pupils in the same year created the same number of problems. Every possible pair of pupils in different years created a different number of problems. How many pupils created exactly one problem?
You are given \(7\) straight lines on a plane, no two of which are parallel. Prove that there will be two lines such that the angle between them is less than \(26^{\circ}\).