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The numbers x, y and z are such that all three numbers x+yz, y+zx and z+xy are rational, and x2+y2=1. Prove that the number xyz2 is also rational.

In the Republic of mathematicians, the number α>2 was chosen and coins were issued with denominations of 1 pound, as well as in αk pounds for every natural k. In this case α 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.

The number x is such that both the sums S=sin64x+sin65x and C=cos64x+cos65x are rational numbers.

Prove that in both of these sums, both terms are rational.

Author: A.K. Tolpygo

An irrational number α, where 0<α<12, is given. It defines a new number α1 as the smaller of the two numbers 2α and 12α. For this number, α2 is determined similarly, and so on.

a) Prove that for some n the inequality αn<3/16 holds.

b) Can it be that αn>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.