Problems

Jane’s birthday cake is square-shaped and has side length 25 cm. Suppose she makes 4 horizontal cuts perpendicular to the vertical edges of the cake and 4 vertical cuts perpendicular to the horizontal edges of the cake. Show that at least one of the pieces has an area of at most 25 cm${}^2$.

You may have seen the pigeonhole principle before, sometimes called Dirichlet’s box principle. It says that if you have more pigeons than pigeonholes, and you put all of the pigeons into some pigeonhole, then there exists at least one pigeonhole with at least two pigeons. While it sounds quite simple, it’s a powerful technique. The difficult thing is often choosing the appropriate pigeons and pigeonholes.
It has multiple applications in various situations.
Today we will see how to use it in geometric problems.

Show that given any nine points on a sphere, there is a closed hemisphere that contains six of them. A closed hemiphere is one that contains the equator with respect to the division.

I’m thinking of a positive number less than $100$. This number has remainder $1$ when divided by $3$, it has remainder $2$ when divided by $4$, and finally, it leaves remainder $3$ when divided by $5$. What number am I thinking of?

I’m thinking of two prime numbers. The first prime number squared is thirty-six more than the second prime number. What’s the second prime number?

How many integers less than $2025$ are divisible by $18$ or $21$, but not both?

Determine all prime numbers $p$ such that $p^2-6$ and $p^2+6$ are both prime numbers.

Let $ABCD$ be a square and let $X$ be any point on side $BC$ between $B$ and $C$. Let $Y$ be the point on line $CD$ such that $BX=YD$ and $D$ is between $C$ and $Y$. Prove that the midpoint of $XY$ lies on diagonal $BD$.

Let $ABCD$ be a trapezium such that $AB$ is parallel to $CD$. Let $E$ be the intersection of diagonals $AC$ and $BD$. Suppose that $AB=BE$ and $AC=DE$. Prove that the internal angle bisector of $\mathrm{\angle }BAC$ is perpendicular to $AD$.

Let $ABC$ be an isosceles triangle with $AB=AC$. Point $D$ lies on side $AC$ such that $BD$ is the angle bisector of $\mathrm{\angle }ABC$. Point $E$ lies on side $BC$ between $B$ and $C$ such that $BE=CD$. Prove that $DE$ is parallel to $AB$.