A $6$ meter ladder rests against a vertical wall. The midpoint of the ladder is twice as far from the ground as it is from the wall. At what height on the wall does the ladder reach?

Two players take turns to write natural numbers on a board. The rules forbid writing numbers greater than $p$ and also divisors of previously written numbers. The player who has no move loses. Determine which player has a winning strategy for $p = 10$ and describe this strategy.

Three equal circles of radius $r$ each pass through the centres of the other two. What is the area of intersection that is common to all the three circles?

The positive integers from $ 1$ to $n$ inclusive are written on a whiteboard. After one number is erased, the average (arithmetic mean) of the remaining $n - 1$ numbers is $22$. Knowing that $n$ is odd, determine $n$ and the number that was erased. Explain your reasoning.

A rectangle $ABCD$ is given. On the side $AB$, n different points are chosen strictly between $A$ and $B$. Similarly, $m$ different points are chosen on the side $AD$ between $A$ and $D$. Lines are drawn from the points parallel to the sides. How many rectangles are formed in this way? An example of a particular rectangle $ABCD$ is shown with a shaded one rectangle that may be formed in this way.

In an apartment block there live only couples of parents with children. It is known that every couple has at least one child, that every child has exactly two parents, that every little boy in this building has a sister, and that among the children there are more boys than girls. You may also assume that there are no grandparents living in the building. Is it possible that there are more parents than children in the building? Explain your reasoning.

The sum of the three nonnegative real numbers $ x_1, x_2, x_3$ is not greater than $\frac12$. Prove that $(1 - x_1)(1 - x_2)(1 - x_3) \ge \frac12$

The positive integer $N = 11...11$, whose decimal representation contains only ones, is divisible by $7$. Prove that this positive integer is also divisible by $13$.

There are $11$ empty boxes and a pile of stones. Two players play the following game by alternating moves: In one move a player takes $10$ stones from the pile and places them into boxes, taking care to place no more than one stone in any box. The winner is the player after whose move there appear $21$ stones in one of the boxes for the first time. If a player wants to guarantee that they win the game, should they go first or second? Explain your reasoning.

The altitudes of triangle $ABC$ intersect at a point $H$.Find $\angle ACB$ if it is known that $AB = CH$.