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

1. About vases with apples and peaches

60 apples and 60 peaches were laid out in vases so that all the vases contained an equal number of apples, but any two vases contained a different number of peaches. What is the largest number of vases that could be used?

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 divisible 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. In the first vase we put 1 peach, in the second - 2 peaches, in the third - 3 peaches, 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 amount of peaches in some vase: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 15 = 60. Everything fits. The answer is 10 vases.

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2. About portions of ice cream

While Cheburashka eats two servings of ice cream, Winnie the Pooh manages to eat five of the same servings, and while Winnie the Pooh eats three servings, Carlson eats seven. Working together, Cheburashka and Carlson ate 82 servings. How many servings did Winnie the Pooh eat during this time?

Let's pay attention to Winnie the Pooh: it is through him that the speed of eating ice cream is correlated by all the heroes. Find the least common multiple of 3 (through which Winnie the Pooh is related to Carlson) and 5 (through which Winnie the Pooh is related to Cheburashka) - 15.

This means that when Vinnie eats 15 servings of ice cream, Cheburashka will eat 2 × 3 = 6 servings, and Carlson will eat 7 × 5 = 35 servings. While Vinnie is eating 15 servings of ice cream, Cheburashka and Carlson together destroy 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 twice as many servings in the same time: 15 × 2 = 30.

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3. About the Australian Zoo

At the Australian Zoo, 35% of all kangaroos are gray, and 13% of all zoo animals are kangaroos, but not gray. What percentage of all animals in the zoo are kangaroos?

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 resulting 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.

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4. About gnome-liars

Several gnomes have gathered in the room and they always lie. They are all of different heights and different weights. Each of them said: "Everyone else is lighter than me, and some of them are lower than me." Which of the statements A - D is necessarily true?

A. The heaviest gnome - the lowest

B. The lightest gnome - the lowest

B. The heaviest gnome is the tallest

D. The lightest gnome is the tallest

E. None of the statements A through D are required to be fulfilled.

For the heaviest gnome, the phrase “Everyone else is lighter than me” is true, 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.

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5. About the invention of the Mad Hatter

The Mad Hatter made a strange clock. Their minute hand is stationary, and the dial and hour hand rotate so that the watch always shows the correct time. How many revolutions per day does the hour hand of such a clock make?

The minute hand is motionless. In order for it to show the correct time, the dial must move in the opposite direction (counterclockwise) at the same speed as the minute hand moves in an ordinary watch, that is, make a full revolution in 1 hour, and 24 revolutions in a day.

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.

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The selection used problems from the international mathematical competition-game "Kangaroo" for and years.