5 Olympiad problems in mathematics with which not every adult will cope with

60 apples were laid on the vases and 60 peaches so that in all the vases it turned out to be equal, but in any two vases – a different number of peaches. What the largest number of VAZ could be used?

60 apples are equal to all vases. It means that the possible amount of VAZ should be chosen from the numbers to which 60 is divided without a residue.

It is also known that in each vase there should be a different number of peaches. Let’s try to decompose the fruits into each vase and understand when they become more than 60. In the first vase, place 1 peach, in the second – 2 peach, in the third – 3 peach and so on: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 = 66. This exceeds the number of peaches that we have, therefore, to decompose them in 11 VAZ will not work.

So you need to take less than the terms (and less than VAZ): 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 55. It is less than 60. So we can add the missing number of peaches in some vase: 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 15 = 60. Everything converges. Answer – 10 VAZ.

2. On the portions of ice cream

While Cheburashka eats two portions of ice cream, Winnie Pooh time to eat five of the same portions, but for now Winnie the Pooh eats three servings, Carlson eats seven. "Working" together, Cheburashka and Carlson ate 82 servings. How many portions during this time ate Winnie the Pooh?

Pay attention to Winnie the Pooh: It is through it that the rate of eating ice cream is associated with all the characters. We will find the smallest common multiple numbers 3 (through it Winnie the Pooh correlates with Carlson) and 5 (through it Winnie the Pooh correlates with Cheburashka) – 15.

This means that when Winnie will eat 15 portions of ice cream, Cheburashka will eat 2 × 3 = 6 servings, and Carlson 7 × 5 = 35 servings. While Winnie eats 15 portions of ice cream, Cheburashka and Carlson together destroy 6 + 35 = 41 portion. 82 Portions of ice cream they will eat twice as long because 82 ÷ 41 = 2. So, and Winnie the Pooh will have time to eat during the same time 2 times more portions: 15 × 2 = 30.

3. About the Australian Zoo

In the Australian Zoo, 35% of all Kangaroo gray, and 13% of all animals Zoo – Kangaroo, but not gray. How many percent of all animals in the zoo constitute a kangaroo?

Let N be the total number of animals in the zoo, C – the amount of gray kangaroo, and k – the number of all kangaroo.

35% of the total kangaroo – gray. We write it: 0.35K = C.
13% of all animals are not gray kangaroo. We write down and this: 0.13n = k – 0,35K.

We simplify the resulting expression: 0.13N = 0.65K; n = 5k; k = 1 / 5n = 20 / 100n = 20%. So, the kangaroo makes up 20% of all animals in the zoo.

4. About gnomes-lguns

There are several gnomes in the room, which are always lyout. All of them different growth and different weight. Each of them said: "All the others are easier for me, and some of them below me". Which of the statements a – r is necessarily true?

A. The most severe dwarf – the lowest
B. The easiest gnome is the lowest
V. The most severe dwarf is the highest
G. The easiest gnome – the highest
D. None of the statements of A – G is not obliged to.

For the very heavy gnome phrase "all the rest lightering me" – truth, and its continuation – "… And one of them below me" – should be a lie. So all the other gnomes above it. "The most difficult gnome is the lowest" is a true statement. For all the other dwarves, the phrase "all the rest is easier for me" is already a lie, so you can not say anything about them.

5. About the invention of a mad hat

Mad Hatter made a strange watch. The minute arrow is fixed, and the dial and the hour arrow rotate so that the clock always shows the right time. How many revolutions per day makes the hour arrow of such hours?

Minute arrow is stationary. So that it showed the right time, the dial must move in the opposite direction (counterclockwise) with the speed with which a minute arrow moves in ordinary hours, that is, to make a complete turn in 1 hour, and during the day – 24 turns.

Hour arrow should also show right time. Together with the dial, it will make one turnover in an hour, that is, 24 turns per day. She also goes to the usual direction – one full revolution in 12 hours and two full turnover in 24 hours in a clockwise direction. Therefore, as a result, it will make 24 – 2 = 22 turns per day.

In the selection used tasks from the International Mathematical Competition-game "Kangaroo" for the years 2018 and 2019.

As you task? How many decided without tips? Share in the comments!