Problems

Prove that the following inequalities hold for the Brockard angle $\phi$:

a) ${\phi }^3\le (\alpha -\phi )(\beta -\phi )(\gamma -\phi )$ ;

b) $8{\phi }^3\le \alpha \beta \gamma$ (the Jiff inequality).

Suppose that $n\ge 3$. Are there n points that do not lie on one line, whose pairwise distances are irrational, and the areas of all of the triangles with vertices in them are rational?

Do there exist three points $A$, $B$ and $C$ on the plane such that for any point $X$ the length of at least one of the segments $XA$, $XB$ and $XC$ is irrational?

How many rational terms are contained in the expansion of

a) $(\sqrt{2}+\sqrt[4]{3}{)}^{100}$;

b) $(\sqrt{2}+\sqrt[3]{3}{)}^{300}$?

Which term in the expansion $(1+\sqrt{3}{)}^{100}$ will be the largest by the Newton binomial formula?

Find the sums of the following series:

a) $\frac{1}{1\times 2}+\frac{1}{2\times 3}+\frac{1}{3\times 4}+\frac{1}{4\times 5}+\dots$;

b) $\frac{1}{1\times 2\times 3}+\frac{1}{2\times 3\times 4}+\frac{1}{3\times 4\times 5}+\frac{1}{4\times 5\times 6}+\dots$;

c) $\frac{0!}{r!}+\frac{1!}{(r-1)!}+\frac{2!}{(r-2)!}+\frac{3!}{(r-3)!}+\dots$ for $r\ge 2$.

Suppose that there are 15 prime numbers forming an arithmetic progression with a difference of $d$. Prove that $d>30,000$.

Could it be that a) $\sigma (n)>3n$; b) $\sigma (n)>100n$?

Prove that in any infinite decimal fraction you can rearrange the numbers so that the resulting fraction becomes a rational number.

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.