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.
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?
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?
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.
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?
Is the number \(10^{2002} + 8\) divisible by 9?
Is the sum of the numbers \(1 + 2 + 3 + \dots + 1999\) divisible by 1999?
Try to read the word in the first figure, using the key (see the second figure).
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?
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?