Problems

Prove that if the irreducible rational fraction $p/q$ is a root of the polynomial $P(x)$ with integer coefficients, then $P(x)=(qx-p)Q(x)$, where the polynomial $Q(x)$ also has integer coefficients.

Prove the irrationality of the following numbers:

a) $\sqrt{3}17$

b) $\sqrt{2}+\sqrt{3}$

c) $\sqrt{2}+\sqrt{3}+\sqrt{5}$

d) $\sqrt{3}3-\sqrt{2}$

e) $\cos {10}^{\circ }$

f) $\tan {10}^{\circ }$

g) $\sin 1^{\circ }$

h) ${\log }_{2}3$

Is it possible for

a) the sum of two rational numbers irrational?

b) the sum of two irrational numbers rational?

c) an irrational number with an irrational degree to be rational?

One of the roots of the equation $x^2+ax+b=0$ is $1+\sqrt{3}$. Find $a$ and $b$ if you know that they are rational.

Prove that the number $\sqrt{2}+\sqrt{3}+\sqrt{5}+\sqrt{7}+\sqrt{11}+\sqrt{13}+\sqrt{17}$ is irrational.

Prove that there is at most one point of an integer lattice on a circle with centre at $(\sqrt{2},\sqrt{3})$.

Prove that if $(p,q)=1$ and $p/q$ is a rational root of the polynomial $P(x)=a_n{x}^n+\cdots +a_{1}x+a_0$ with integer coefficients, then

a) $a_0$ is divisible by $p$;

b) $a_n$ is divisible by $q$.

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.

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.