5 olympiad problems in mathematics that not every adult can cope with

In all vases 60 apples are equally distributed. This means that the possible number of vases should be chosen from the numbers by which 60 is divided without a remainder.

It is also known that each vase must have a different number of peaches. Let's try to put the fruits in each vase and understand when there will be more than 60 of them. Place 1 peach in the first vase, 2 peaches in the second, 3 peaches in the third, and so on: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 = 66. This exceeds the number of peaches that we have, so it will not work to arrange them in 11 vases.

This means that you need to take fewer terms (and less vases): 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55. This is less than 60. This means that we can add the missing number of peaches to another vase: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 15 = 60. It all fits together. The answer is 10 vases.

Let's imagine that Winnie the Pooh eats 15 servings of ice cream, then Cheburashka will eat 2 × 3 = 6 servings, and Carlson will eat 7 × 5 = 35 servings. Together, Cheburashka and Carlson will eat 6 + 35 = 41 servings. They will eat 82 servings of ice cream twice as long, because 82 ÷ 41 = 2. This means that Winnie the Pooh will have time to eat 2 times more portions during the same time: 15 × 2 = 30.

Let n be the total number of animals in the zoo, c the number of gray kangaroos, and k the number of all kangaroos.

35% of the total number of kangaroos are gray. Let's write this: 0.35k = c.
13% of all animals are not gray kangaroos. We also write this: 0.13n = k - 0.35k.

Let's simplify the obtained expression: 0.13n = 0.65k; n = 5k; k = 1 / 5n = 20 / 100n = 20%. This means that kangaroos make up 20% of all animals in the zoo.

For the heaviest gnome, the phrase “Everyone else is lighter than me” is the truth, and its continuation - “… and one of them is lower than me” - must be a lie. So all the other dwarves are taller than him. “The heaviest gnome is the lowest” is a true statement. For all the other gnomes, the phrase "Everyone else is lighter than me" is already a lie, so nothing can be said about them.

The minute hand is motionless. For it to show the correct time, the dial must move in the opposite direction (counterclockwise) from the same the speed at which the minute hand moves in an ordinary watch, that is, make a full revolution in 1 hour, and in 24 hours turnover.

The hour hand must also show the correct time. Together with the dial, it will make one revolution per hour, that is, 24 revolutions per day. It also goes in its usual direction - one full revolution in 12 hours and two full revolutions in 24 hours in a clockwise direction. Therefore, in the end, it will make 24 - 2 = 22 revolutions per day.