Filter Problems

Showing 1 to 20 of 285 entries

20 birds fly into a photographer’s studio – 8 starlings, 7 wagtails and 5 woodpeckers. Each time the photographer presses the shutter to take a photograph, one of the birds flies away and doesn’t come back. How many photographs can the photographer take to be sure that at the end there will be no fewer than 4 birds of one species and no less than 3 of another species remaining in the studio.

100 cars are parked along the right hand side of a road. Among them there are 30 red, 20 yellow, and 20 pink Mercedes. It is known that no two Mercedes of different colours are parked next to one another. Prove that there must be three Mercedes cars parked next to one another of the same colour somewhere along the road.

2003 dollars were placed into some wallets and the wallets were placed in some pockets. It is known that there are more wallets in total than there are dollars in any pocket. Is it true that there are more pockets than there are dollars in one of the wallets? You are not allowed to place wallets one inside the other.

17 squares are marked on an 8×8 chessboard. In chess a knight can move horizontally or vertically, one space then two or two spaces then one – eg: two down and one across, or one down and two across. Prove that it is always possible to pick two of these squares so that a knight would need no less than three moves to get from one to the other.

A village infant school has 20 pupils. If we pick any two pupils they will have a shared granddad.

Prove that one of the granddads has no fewer than 14 grandchildren who are pupils at this school.

An after school club was attended by 60 pupils. It turns out that in any group of 10 there will always be 3 classmates. Prove that within the group of 60 who attended there will always be at least 15 pupils from the same class.

Ben noticed that all 25 of his classmates have a different number of friends in this class. How many friends does Ben have?

A gang contains 101 bandits. The whole gang has never taken part in a raid together, but every possible pair of bandits have taken part in a raid together exactly once. Prove that one of the bandits has taken part in no less than 11 different raids.

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 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?

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 $A \subset B \subset C$.

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.

a) We are given two cogs, each with 14 teeth. They are placed on top of one another, so that their teeth are in line with one another and their projection looks like a single cog. After this 4 teeth are removed from each cog, the same 4 teeth on each one. Is it always then possible to rotate one of the cogs with respect to the other so that the projection of the two partially toothless cogs appears as a single complete cog? The cogs can be rotated in the same plane, but cannot be flipped over.

b) The same question, but this time two cogs of 13 teeth each from which 4 are again removed?

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.

The function F is given on the whole real axis, and for each x the equality holds: F $(x + 1)$ F $(x)$ + F $(x + 1)$ + 1 = 0.

Prove that the function F can not be continuous.

A group of numbers $A_1, A_2, …, A_{100}$ is created by somehow re-arranging the numbers $1, 2, …, 100$.

100 numbers are created as follows:

$$B_1=A_1, B_2=A_1+A_2, B_3=A_1+A_2+A_3, …, B_{100} = A_1+A_2+A_3…+A_{100}$$

Prove that there will always be at least 11 different remainders when dividing the numbers $B_1, B_2, …, B_{100}$ by 100.

In the race of six athletes, Andrei lagged behind Boris and two more athletes. Victor finished after Dmitry, but before Gennady. Dmitry beat Boris, but still came after Eugene. What place did each athlete take?

A class contains 38 pupils. Prove that within the class there will be at least 4 pupils born in the same month.

Is it always the case that in any 25 GBP banknotes – that is £5, £10, £20, and £50 notes – there will always be 7 banknotes of the same denomination?