Problems

The dragon locked six dwarves in the cave and said, "I have seven caps of the seven colors of the rainbow. Tomorrow morning I will blindfold you and put a cap on each of you, and hide one cap. Then I’ll take off the blindfolds, and you can see the caps on the heads of others, but not your own and I won’t let you talk any more. After that, everyone will secretly tell me the color of the hidden cap. If at least three of you guess right, I’ll let you all go. If less than three guess correctly, I’ll eat you all for lunch." How can dwarves agree in advance to act in order to be saved?

Jason has $20$ red balls and $14$ bags to store them. Prove that there is a bag, which contains at least two balls.

One of the most useful tools for proving mathematical statements is the Pigeonhole principle. Here is one example: suppose that a flock of $10$ pigeons flies into a set of $9$ pigeonholes to roost. Prove that at least one of these $9$ pigeonholes must have at least two pigeons in it.

Show the following: Pigeonhole principle strong form: Let $q_1,q_2,...,q_n$ be positive integers. If $q_1+q_2+...+q_n-n+1$ objects are put into $n$ boxes, then either the $1$st box contains at least $q_1$ objects, or the $2$nd box contains at least $q_2$ objects, . . ., or the $n$th box contains at least $q_n$ objects.
How can you deduce the usual Pigeonhole principle from this statement?

Let $r$ be a rational number and $x$ be an irrational number (i.e. not a rational one). Prove that the number $r+x$ is irrational.
If $r$ and $s$ are both irrational, then must $r+s$ be irrational as well?

Definition: We call a number $x$ rational if there exist two integers $p$ and $q$ such that $x=\frac{p}{q}$. We assume that $p$ and $q$ are coprime.
Prove that $\sqrt{2}$ is not rational.

Let $n$ be an integer such that $n^2$ is divisible by $2$. Prove that $n$ is divisible by $2$.

Let $n$ be an integer. Prove that if $n^3$ is divisible by $3$, then $n$ is divisible by $3$.

The numbers $x$ and $y$ satisfy $x+3=y+5$. Prove that $x>y$.

The numbers $x$ and $y$ satisfy $x+7\ge y+8$. Prove that $x>y$.