There are 6 locked suitcases and 6 keys for them. It is not known which keys are for which suitcase. What is the smallest number of attempts do you need in order to open all the suitcases? How many attempts would you need if there are 10 suitcases and keys instead of 6?
On the table four figures lie in a row: a triangle, a circle, a rectangle and a rhombus. They are painted in different colors: red, blue, yellow, green. It is known that the red figure lies between the blue and green figures; to the right of the yellow figure lies the rhombus; the circle lies to the right of both the triangle and the rhombus; the triangle does not lie on the edge; the blue and yellow figures are not next to each other. Determine in which order the figures lie and what colors they are.
One day a strange notebook was found on the stairs. It contained one hundred statements:
“There is exactly one incorrect statement in this notebook”;
“There are exactly two incorrect statements in this notebook”;
“There are exactly three incorrect statements in this notebook”;
...
“There are exactly one hundred incorrect statements in this notebook.”
Are any of these statements true, and if so, which ones?
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?
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?
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?
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?