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Six chess players participated in a tournament. Each two participants of the tournament played one game against each other. How many games were played? How many games did each participant play? How many points did the chess players collect all together?

Before you is a lock “with a secret” $($see the picture$)$.

If you put the arrows on the desired letters, you will get the keyword and the lock will open. What is this word?

Try to read the word in the first figure, using the key $($see the second figure$)$.

In a room, there are 85 red and blue balloons. It is known that: 1) at least one of the balloons is red; 2) from each arbitrarily chosen pair of balloons at least one blue. How many red balloons are there in the room?

An entire set of dominoes, except for 0-0, was laid out as shown in the figure. Different letters correspond to different numbers, the same – the same. The sum of the points in each line is 24. Try to restore the numbers.

Twenty-eight dominoes can be laid out in various ways in the form of a rectangle of $8 \times 7$ cells. In Fig. 1-4 four variants of the arrangement of the figures in the rectangles are shown. Can you arrange the dominoes in the same arrangements as each of these options?

A page of a calendar is partially covered by the previous torn sheet $($see the figure$)$. The vertices A and B of the upper sheet lie on the sides of the bottom sheet. The fourth vertex of the lower leaf is not visible – it is covered by the top sheet. The upper and lower pages, of course, are identical in size to each other. Which part of the lower page is greater, that which is covered or that which is not?

Can the following equality be true:

$K \times O \times T$ = $A \times B \times C \times D \times E \times F$

if you substitute the letters with the numbers from 1 to 9? Different letters correspond to different numbers.

In one move, it is permitted to either double a number or to erase its last digit. Is it possible to get the number 14 from the number 458 in a few moves?

Is it possible to cut a square into four parts so that each part touches each of the other three $($ie has common parts of a border$)$?

Is it possible to arrange 44 marbles into 9 piles, so that the number of marbles in each pile is different?

Write the first 10 prime numbers in a line. How can you remove 6 digits to get the largest possible number?

What word is encrypted: 22212221265121? Each letter is replaced by its number in the English alphabet.

Before the start of the Olympics, the price of hockey pucks went up by 10%, and after the end of the Olympics they fell by 10%.

When were the pucks more expensive – before the price rise or after the fall?

Two classes with the same number of students took a test. Having checked the test, the strict teacher Mr Jones said that he gave out 13 more twos than other marks $($where the marks range from 2 to 5 and 5 is the highest$)$. Was Mr Jones right?

In the gymnasium, all students know at least one of the ancient languages – Greek or Latin, some – both languages. 85% of all children know the Greek language and 75% know Latin. How many students know both languages?

a) Prove that within any 6 whole numbers there will be two that have a difference between them that is a multiple of 5.

b) Will this statement remain true if instead of the difference we considered the total?

Find all of the natural numbers that, when divided by 7, have the same remainder and quotient.