Problems

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?

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.

Given a square trinomial \(f (x) = x^2 + ax + b\). It is known that for any real \(x\) there exists a real number \(y\) such that \(f (y) = f (x) + y\). Find the greatest possible value of \(a\).

Hannah has a calculator that allows you to multiply a number by 3, add 3 to the number or (4 if the number is divisible by 3 to make a whole number) divide by 3. How can the number 11 be made on this calculator from the number 1?

In the \(4 \times 4\) square, the cells in the left half are painted black, and the rest – in white. In one go, it is allowed to repaint all cells inside any rectangle in the opposite colour. How, in three goes, can one repaint the cells to get the board to look like a chessboard?

In the king’s prison, there are five cells numbered from 1 to 5. In each cell, there is one prisoner. Kristen persuaded the king to conduct an experiment: on the wall of each cell she writes at one point a number and at midnight, each prisoner will go to the cell with the indicated number (if the number on the wall coincides with the cell number, the prisoner does not go anywhere). On the following night at midnight, the prisoners again must move from their cell to another cell according to the instructions on the wall, and they do this for five nights. If the location of prisoners in the cells for all six days (including the first) is never repeated, then Kristen will be given the title of Wisdom, and the prisoners will be released. Help Kristen write numbers in the cells.

Which numbers can stand in place of the letters in the equality \(AB \times C = DE\), if different letters denote different numbers and from left to right the numbers are written in ascending order?

At the vertices of the hexagon \(ABCDEF\) (see Fig.) There were 6 identical balls: at \(A\) – one with mass 1 g, at \(B\) – 2 g, ..., at \(F\) – 6 g. Callum changed the places of two balls in opposite vertices. A set of weighing scales with 2 plates is available, which let you know which plate contains the balls of greater mass. How, in one weighing, can it be determined which balls were rearranged?

There are 40 weights of weights of 1 g, 2 g, ..., 40 grams. Of these, 10 weights of even weight were chosen and placed on the left hand side of the scales. Then we selected 10 weights of odd weight and put it on the right hand side of the scales. The scales were balanced. Prove that on one of the bowls of the scales there are two weights with a mass difference of 20 g.

Seven coins are arranged in a circle. It is known that some four of them, lying in succession, are fake and that every counterfeit coin is lighter than a real one. Explain how to find two counterfeit coins from one weighing on scales without any weights. (All counterfeit coins weigh the same.)