Pigeonhole problem (weight and height)

Question

There are 33 people in a room. We ask them two questions each:

1-How many people in this room have the same height as you?
2-How many people in this room have the same weight as you?

The answers are all between 0 to 10, and all the numbers between 0 to 10 are heard. Prove there are two people of same weight and height in this room.

Answer 1

First of all we know that 66 answers were given: 10 was said 11 times, 9 10 times and so on. Due that we kbow now, all the possible answers are the numbers from 0 to 10.

An other fact that we should recognise is that, if a number once was said as weight(or hight) then all the other times it was said as weight(or height).

So take a look at the case when 10 was said. If two people said 10 and x, then they height and weight is the same. If not then there need to be 11 (10, y) different pairs, where y is a number from 0 to 9. From 0 to 9 there are 10 numbers, so there will be two pairs that are the same, so there will be at least two people with the same height and weight.