Olympiad problems

2007 Sharygin Geometry Olympiad, 11

A boy and his father are standing on a seashore. If the boy stands on his tiptoes, his eyes are at a height of $1$ m above sea-level, and if he seats on father’s shoulders, they are at a height of $2$ m. What is the ratio of distances visible for him in two eases? (Find the answer to $0,1$, assuming that the radius of Earth equals $6000$ km.)

2001 Kazakhstan National Olympiad, 8

Tags: geometry , combinatorial geometry , distance

There are $ n \geq4 $ points on the plane, the distance between any two of which is an integer. Prove that there are at least $ \frac {1} {6} $ distances, each of which is divisible by $3$.

1987 Polish MO Finals, 1

Tags: geometry , combinatorics , combinatorial geometry , square , distance , point

There are $n \ge 2$ points in a square side $1$. Show that one can label the points $P_1, P_2, ... , P_n$ such that $\sum_{i=1}^n |P_{i-1} - P_i|^2 \le 4$, where we use cyclic subscripts, so that $P_0$ means $P_n$.

2015 JBMO Shortlist, C2

Tags: combinatorics , point , combinatorial geometry , distance

$2015$ points are given in a plane such that from any five points we can choose two points with distance less than $1$ unit. Prove that $504$ of the given points lie on a unit disc.

2018 Adygea Teachers' Geometry Olympiad, 1

Tags: geometry , distance , square

Can the distances from a certain point on the plane to the vertices of a certain square be equal to $1, 4, 7$, and $8$ ?