0
votes
1answer
39 views

If 51 mosquitoes are sitting on a square with side 1m, are at least 3 of them within a disk of radius 1/7?

There are 51 mosquitoes on a square-shaped window with side 1 m. Can Stephen kill 3 mosquitoes with a circular plastic disk of radius 1/7 m in a single strike? I know this can be solved by ...
4
votes
1answer
99 views

10 points inside a square - minimum distance between any of them

A square of side 1 is given, and 10 points are inside the square. If we divide the square into 9 smaller squares, and apply Dirichlet principle, we can prove that there are 2 of these 10 points whose ...
4
votes
1answer
256 views

Issue concerning enumerating vertices in a prism (number of two adjacent vertices can only differ by a certain amount)

There are 100 vertices in a prism with a 50-gon as its base. Those vertices are assigned integers 1 to 100 (inclusive) in a random order. Each number can only be assigned once. The objective is to ...
20
votes
1answer
1k views

if there are 5 points on a sphere then 4 of them belong to a half-sphere.

If there are 5 points on the surface of a sphere then there is a closed half sphere containing at least 4 of them. It's in a pidgeonhole list of problems, but I think I have to use rotations in more ...
-1
votes
1answer
195 views

Area of the triangle [closed]

Choose any 9 points on or within a unit square. Prove that there always exists 3 points such that triangle formed by them has area 1/3
9
votes
1answer
187 views

Pigeonhole principle for a triangle

Consider a equilateral triangle of total area 1. Suppose 7 points are chosen inside. Show that some 3 points form a triangle of area $\leq\frac 14$.
6
votes
0answers
206 views

How small parallelograms are we guaranteed to get, when we select the two sides from different plane lattices?

This a shortened version (motivation from telecommunications stripped away) of a question I asked in MO in late May (no answers). I am mostly checking, if somebody has seen this or a related question ...
4
votes
1answer
271 views

If any x points are elected out of a unit square, then some two of them are no farther than how many units apart?

If 5 points are randomly positioned in a unit square, no two points can be greater than square root of 2 divided by 2 apart; divide up the unit square into four squares, and, based on the pigeonhole ...