The smell of a flowering lavender plant diffuses through a radius of 20 m around it. How many lavender plants must be planted along a straight 400m path so that the smell of the lavender reaches every point on the path.
The angle at the top of a crane is \(20^{\circ}\). How will the magnitude of this angle change when looking at the crane with binoculars which triple the size of everything?
Prove that in any group of 7 natural numbers – not necessarily consecutive – it is possible to choose three numbers such that their sum is divisible by 3.
What is the minimum number of squares that need to be marked on a chessboard, so that:
1) There are no horizontally, vertically, or diagonally adjacent marked squares.
2) Adding any single new marked square breaks rule 1.
We are given 101 rectangles with integer-length sides that do not exceed 100.
Prove that amongst them there will be three rectangles \(A, B, C\), which will fit completely inside one another so that \(A \subset B \subset C\).
A staircase has 100 steps. Vivian wants to go down the stairs, starting from the top, and she can only do so by jumping down and then up, down and then up, and so on. The jumps can be of three types – six steps (jumping over five to land on the sixth), seven steps or eight steps. Note that Vivian does not jump onto the same step twice. Will she be able to go down the stairs?
10 friends sent one another greetings cards; each sent 5 cards. Prove that there will be two friends who sent cards to one another.
A gang contains 101 gangsters. The whole gang has never taken part in a raid together, but every possible pair of gangsters have taken part in a raid together exactly once. Prove that one of the gangsters has taken part in no less than 11 different raids.
Ben noticed that all 25 of his classmates have a different number of friends in this class. How many friends does Ben have?
A village infant school has \(20\) pupils. If we pick any two pupils, then they will have a shared granddad.
Prove that one of the granddads has at least \(14\) grandchildren who are pupils at this school.