Problems

A point $P$ is somewhere inside the triangle $\mathrm{△}ABC$. Show that $AP+BP<AC+BC$.

The distance between London and Warsaw equals $1450$ km, between Warsaw and Kyiv is $680$ km. The distance from London to New Delhi, is $6700$ km and the distance from Kyiv to New Delhi is $4570$ km. What is the distance from London to Kyiv?

Show that if all sides of a triangle have integer lengths and one of them is equal to $1$, then the other two have lengths equal to each other.

A billiard ball lies on a table in the shape of an acute angle. How should you hit the ball so that it returns to its starting location after hitting each of the two banks once? Is it always possible to do so?
(When the ball hits the bank, it bounces. The way it bounces is determined by the shortest path rule – if it begins at some point $D$ and ends at some point $D^{\prime }$ after bouncing, the path it takes is the shortest possible path that includes the bounce.)

Prove that $S_{ABCD}\le (AB\times BC+AD\times DC)/2$.

Prove that $\mathrm{\angle }ABC>{90}^{\circ }$ if and only if the point $B$ lies inside a circle with diameter $AC$.

The radii of two circles are $R$ and $r$, and the distance between their centres is equal $d$. Prove that these circles intersect if and only if $|R-r|<d<R+r$.

Prove that $(a+b-c)/2<m_c<(a+b)/2$, where $a$, $b$ and $c$ are the lengths of the sides of an arbitrary triangle and $m_c$ is the median to side $c$.

$a$, $b$ and $c$ are the lengths of the sides of an arbitrary triangle. Prove that $a=y+z$, $b=x+z$ and $c=x+y$, where $x$, $y$ and $z$ are positive numbers.

a, b and c are the lengths of the sides of an arbitrary triangle. Prove that $a^2+b^2+c^2<2(ab+bc+ca)$.