Problems

In a burrow there is a family of 24 mice. Every night exactly four of them are sent to the warehouse for cheese.

Could it occur that at some point in time each mouse went to the warehouse with every other mouse exactly one time?

Let $x$ be a 2 digit number. Let $A$, $B$ be the first (tens) and second (units) digits of $x$, respectively. Suppose $A$ is twice as large as $B$. If we add the square of $A$ to $x$ then we get the square of a certain whole number. Find the value of $x$.

In a physics club, the teacher created the following experiment. He spread out 16 weights of weight 1, 2, 3, ..., 16 grams onto weighing scales, so that one of the bowls outweighed the other. Fifteen students in turn left the classroom and took with them one weight each, and after each student’s departure, the scales changed their position and outweighed the opposite bowl of the scales. What weight could remain on the scales?

Solve the equation: $|x-2005|+|2005-x|=2006$.

Solve the equation: $$x+\frac{x}{x}+\frac{x}{x+\frac{x}{x}}=1$$

Between them, Jennifer and Alex shared the money they made from running a lemonade stand. Jennifer thought: “If I took $40\mathrm{\%}$ more money then Alex’s share would decrease by $60\mathrm{\%}$”. How would Alex’s share of the profits change if Jennifer took $50\mathrm{\%}$ more money for herself?

Find all functions $f(x)$ such that $f(2x+1)=4x^2+14x+7$.

In a board, 20 pins are placed (see the picture). The distance between any adjacent pins is 1 inch. Pull a string of length 19 inches from the first pin to the second one, so that it goes through all the pins.

Henry did not manage to get into the elevator on the first floor of the building and decided to go up the stairs. It takes 2 minutes to rise to the third floor. How long does it take to rise to the ninth floor?

The grasshopper jumps on the interval $[0,1]$. On one jump, he can get from the point $x$ either to the point $x/3^{1/2}$, or to the point $x/3^{1/2}+(1-(1/3^{1/2}))$. On the interval $[0,1]$ the point $a$ is chosen.

Prove that starting from any point, the grasshopper can be, after a few jumps, at a distance less than $1/100$ from point $a$.