Problems

Derive from the theorem in question 61013 that $\sqrt{17}$ is an irrational number.

The numbers $x$, $y$ and $z$ are such that all three numbers $x+yz$, $y+zx$ and $z+xy$ are rational, and $x^2+y^2=1$. Prove that the number $xy{z}^2$ is also rational.

In the Republic of mathematicians, the number $\alpha >2$ was chosen and coins were issued with denominations of 1 pound, as well as in ${\alpha }^k$ pounds for every natural $k$. In this case $\alpha$ was chosen so that the value of all the coins, except for the smallest, was irrational. Could it be that any amount of a natural number of pounds can be made with these coins, using coins of each denomination no more than 6 times?

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.

Is there a positive integer $n$ such that $$\sqrt{n}17\sqrt{5}+38+\sqrt{n}17\sqrt{5}-38=2\sqrt{5}?$$

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.

Author: A.K. Tolpygo

An irrational number $\alpha$, where $0<\alpha <\frac{1}{2}$, is given. It defines a new number ${\alpha }_1$ as the smaller of the two numbers $2\alpha$ and $1-2\alpha$. For this number, ${\alpha }_2$ is determined similarly, and so on.

a) Prove that for some $n$ the inequality ${\alpha }_n<3/16$ holds.

b) Can it be that ${\alpha }_n>7/40$ for all positive integers $n$?

Let $n$ numbers are given together with their product $p$. The difference between $p$ and each of these numbers is an odd number.

Prove that all $n$ numbers are irrational.