## Maths Problems

Problems from the first maths battle. November 06, 2015. The Royal Grammar School for Boys in High Wycombe.

## Problem 1

Milk, lemonade, cola and water are poured into a cup, a glass, a bowl and a jar. Determine which liquid is in which vessel if

• water and milk are not in the cup
• the vessel with lemonade is standing between the bowl and the vessel with cola
• the jar contains neither lemonade nor water
• the glass is standing near the jar and the vessel with milk

## Problem 2

A stadium is filled with Harlequins and Wasps fans (every person in the stadium is supporting either Harlequins or Wasps.) During the game every person at the stadium bought at least one drink. The total amount of money spent on drinks by Wasps supporters is greater than the total amount of money spent on drinks by Harlequins supporters. Also, the total amount of money spent by women is larger than the total amount of money spent by men. Prove that there is at least one female supporter of Wasps present at the stadium.

## Problem 3

Olivia, Zara, James and Alex have being playing darts. Alex scored most points, James’ score is not the worst and Zara performed better than Olivia. Have the boys outplayed the girls in this game?

## Problem 4

A number is called “cool” if it has only even numbers as digits. (For example, 480 is a “cool” number.) How many different “cool” 4-digit numbers exist?

## Problem 5

A sloppy sandwich is made from a squared piece of bread and a circular piece of cheese (see the picture below). Can you divide the sandwich into two equal parts with one single cut?!

## Problem 6

Ten friends posted funny messages on each other’s timeline (on Facebook) in such a way, that every person wrote exactly on five timelines (nobody wrote funny messages on his own timeline). Show that there are two friends who wrote on each other’s timelines.