Problems

Prove that for each $x$ such that $\sin x\ne 0$, there is a positive integer $n$ such that $|\sin nx|\ge \sqrt{3}/2$.

For what natural numbers $n$ are there positive rational but not whole numbers $a$ and $b$, such that both $a+b$ and $a^n+b^n$ are integers?

The base of the pyramid is a square. The height of the pyramid crosses the diagonal of the base. Find the largest volume of such a pyramid if the perimeter of the diagonal section containing the height of the pyramid is 5.

One day all the truth tellers on the planet decided to carry a clearly visible mark of truth in order to be distinguished from liars. Two truth tellers and two liars met and looked at each other. Which of them could say the phrase:

• “All of us are truth tellers.”

• “Only one of you is a truth teller.”

• “Exactly two of you are truth tellers.”

A New Year’s garland, hanging along the school corridor, consists of red and blue light bulbs. Next to each red light bulb there must necessarily be a blue one. What is the largest number of red light bulbs in this garland, if it consists of only 50 light bulbs?

A continuous function $f(x)$ is such that for all real $x$ the following inequality holds: $f(x^2)-(f(x){)}^2\ge 1/4$. Is it true that the function $f(x)$ necessarily has an extreme point?

A magician with a blindfold gives a spectator five cards with the numbers from 1 to 5 written on them. The spectator hides two cards, and gives the other three to the assistant magician. The assistant indicates to the spectator two of them, and the spectator then calls out the numbers of these cards to the magician (in the order in which he wants). After that, the magician guesses the numbers of the cards hidden by the spectator. How can the magician and the assistant make sure that the trick always works?

The volume of the regular quadrangular pyramid $SABCD$ is equal to $V$. The height $SP$ of the pyramid is the edge of the regular tetrahedron $SPQR$, the plane of the face $PQR$ which is perpendicular to the edge $SC$. Find the volume of the common part of these pyramids.

The height $SO$ of a regular quadrilateral pyramid $SABCD$ forms an angle $\alpha$ with a side edge and the volume of this pyramid is equal to $V$. The vertex of the second regular quadrangular pyramid is at the point $S$, the centre of the base is at the point $C$, and one of the vertices of the base lies on the line $SO$. Find the volume of the common part of these pyramids.

In 10 boxes there are pencils (there are no empty boxes). It is known that in different boxes there is a different number of pencils, and in each box, all pencils are of different colors. Prove that from each box you can choose a pencil so that they will all be of different colors.