MathDB

2.1

Part of 2011 Saudi Arabia Pre-TST

Problems(2)

17 soldiers patrol in equilateral military base

Source: 2011 Saudi Arabia Pre-TST November 2.1

1/1/2022
The shape of a military base is an equilateral triangle of side 1010 kilometers. Security constraints make cellular phone com­munication possible only within 2.52.5 kilometers. Each of 1717 soldiers patrols the base randomly and tries to contact all others. Prove that at each moment at least two soldiers can communicate.
combinatoricscombinatorial geometryEquilateral
an interval does not contain any integer

Source: 2011 Saudi Arabia Pre-TST February 2.1

1/1/2022
Let nn be a positive integer. Prove that the interval In=(1+8n+12,1+8n+92)I_n= \left( \frac{1+\sqrt{8n+1}}{2}, \frac{1+\sqrt{8n+9}}{2}\right) does not contain any integer.
number theoryInteger