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.)

Natural numbers from 1 to 200 are divided into 50 sets. Prove that in one of the sets there are three numbers that are the lengths of the sides of a triangle.

On a plane, there are 1983 points and a circle of unit radius. Prove that there is a point on the circle, from which the sum of the distances to these points is no less than 1983.

In an isosceles triangle, the sides are equal to either 3 or 7. Which side length is the base?

$a$, $b$ 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 $a^2+b^2+c^2<2(ab+bc+ca)$.

In a triangle, the lengths of two of the sides are 3.14 and 0.67. Find the length of the third side if it is known that it is an integer.