Problems

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 $AB$.

Is it possible to place a positive integer in every cell of a $10\times 10$ array in such a way that both the following conditions are satisfied?

1. Each number (not in the bottom row) is a proper divisor of the number immediately below.

2. The numbers in each row, rearrange if necessary, form a sequence of 10 consecutive numbers.

A round-robin tournament is one where each team plays every other team exactly once. Five teams take part in such a tournament getting: $3$ points for a win, $1$ point for a draw and $0$ points for a loss. At the end of the tournament the teams are ranked from first to last according to the number of points.
Is it possible that at the end of the tournament, each team has a different number of points, and each team except for the team ranked last has exactly two more points than the next-ranked team?

$ABCD$ is a rectangle with side lengths $AB=CD=1$ and $BC=DA=2$. Let $M$ be the midpoint of $AD$. Point $P$ lies on the opposite side of line $MB$ to $A$, such that triangle $MBP$ is equilateral. Find the value of $\mathrm{\angle }PCB$.