Problems

Prove that $\mathrm{\angle }ABC<\mathrm{\angle }BAC$ if and only if $AC<BC$, that is, the larger side lies opposite the larger angle of the triangle, and opposite the larger side lies the larger angle.

$ABC$ is a right angled triangle with a right angle at $C$. Prove that $c^n>a^n+b^n$ for $n>2$.

Prove that the area $S_{ABC}$ of a triangle is equal to $abc/4R$.

The point $D$ lies on the base $AC$ of the isosceles triangle $ABC$. Prove that the radii of the circumscribed circles of the triangles $ABD$ and $CBD$ are equal.

Express the area of the triangle $ABC$ through the length of the side $BC$ and the angles $B$ and $C$.

Two intersecting circles of radius $R$ are given, and the distance between their centers is greater than $R$. Prove that $\beta =3\alpha$ (Fig.).

Find all triangles in which the angles form an arithmetic progression, and the sides form: a) an arithmetic progression; b) a geometric progression.

Prove that the point $X$ lies on the line $AB$ if and only if $\overrightarrow{OX}=t\overrightarrow{OA}+(1-t)\overrightarrow{OB}$ for some $t$ and any point $O$.

Several points are given and for some pairs $(A,B)$ of these points the vectors $\overrightarrow{AB}$ are taken, and at each point the same number of vectors begin and end. Prove that the sum of all the chosen vectors is $\overrightarrow{0}$.

Prove that, when a circle is translated it becomes a circle.