Problems

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 $f(-1)$.

It is known that $a=x+y+\sqrt{xy}$, $b=y+z+\sqrt{yz}$, $c=x+z+\sqrt{xz}$. where $x>0$, $y>0$, $z>0$. Prove that $a+b+\sqrt{ab}>c$.

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?

Prove that for any positive integer $n$, it is always possible to find a number, consisting of the digits $1$ and $2,$ that is divisible by $2^n$. (For example, $2$ is divisible by $2$, $12$ is divisible by $4,$ $112$ is divisible by $8,$ $2112$ is divisible by $16$ and so on...).

A sequence of natural numbers $a_1<a_2<a_3<\cdots <a_n<\dots$ 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 $a_n\le n^2$ for any $n=1,2,3,\dots$

Out of the given numbers 1, 2, 3, ..., 1000, find the largest number $m$ that has this property: no matter which $m$ of these numbers you delete, among the remaining $1000-m$ numbers there are two, of which one is divisible by the other.

Let’s call a natural number good if in its decimal record we have the numbers 1, 9, 7, 3 in succession, and bad if otherwise. (For example, the number 197,639,917 is bad and the number 116,519,732 is good.) Prove that there exists a positive integer $n$ such that among all $n$-digit numbers (from ${10}^{n-1}$ to ${10}^{n-1}$) there are more good than bad numbers.

Try to find the smallest possible $n$.

An infinite sequence of digits is given. Prove that for any natural number $n$ that is relatively prime with a number 10, you can choose a group of consecutive digits, which when written as a sequence of digits, gives a resulting number written by these digits which is divisible by $n$.