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?

There is a $5\times 9$ rectangle drawn on squared paper. In the lower left corner of the rectangle is a button. Kevin and Sophie take turns moving the button any number of squares either to the right or up. Kevin goes first. The winner is the one who places the button in upper right corner. Who would win, Kevin or Sophie, by using the right strategy?

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

There are $12$ aliens in the High Council of the planet of liars and truth tellers. “There is no-one honest here,” said the first member of the council. “There is at most one honest person here,” said the second person. The third person said that there are at most $2$ honest members, the fourth person said there are at most $3$ honest aliens, and so on until the twelfth person, who said there are at most $11$ honest aliens. How many honest members are in the High Council?

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 $a$.

All of the sweets of different sorts in stock are arranged in $n$ boxes, for which prices are set at $1,2,\dots ,n$, respectively. It is required to buy such $k$ of these boxes of the least total value, which contain at least $k/n$ of the mass of all of the sweets. It is known that the mass of sweets in each box does not exceed the mass of sweets in any more expensive box.

a) What boxes should I buy when $n=10$ and $k=3$?

b) The same question for arbitrary natural numbers $n\ge k$.

In a class there are 50 children. Some of the children know all the letters except “h” and they miss this letter out when writing. The rest know all the letters except “c” which they also miss out. One day the teacher asked 10 of the pupils to write the word “cat”, 18 other pupils to write “hat” and the rest to write the word “chat”. The words “cat” and “hat” each ended up being written 15 times. How many of the pupils wrote their word correctly?

The student did not notice the multiplication sign between two three-digit numbers and wrote one six-digit number, which turned out to be seven times bigger than their product. Determine these numbers.

The student did not notice the multiplication sign between two seven-digit numbers and wrote one fourteen-digit number, which turned out to be three times bigger than their product. Determine these numbers.

A cherry which is a ball of radius r is dropped into a round glass whose axial section is the graph of the function $y=x^4$. At what maximum r will the ball touch the most bottom point of the bottom of the glass? (In other words, what is the maximum radius r of a circle lying in the region $y\ge x^4$ and containing the origin?).