8 tasks from the school math olympiad that are not so easy to solve

Miscellaneous / by admin / April 04, 2023

To find the answer, you will have to not only remember the rules from textbooks, but also use logic and imagination.

We advise you to take a piece of paper and a pen, they will be useful in solving some problems.

– 1 –

By what do you need to multiply a third of a quarter of a number to get three times half of the same number?

You can solve the problem with a simple equation. To do this, the hidden number should be designated as x, and the number by which it needs to be multiplied - n. Then a third of a quarter of a number is x / (3 × 4), and three times half is 3x / 2.

It turns out the equation nx / 12 = 3x / 2. According to the rules, both parts of it can be multiplied by the same number, and the equality remains true. Multiply by 12 to get rid of fractions. Then it turns out: nx \u003d 18x, and n \u003d 18.

Answer: 18.

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– 2 –

Zhenya daily writes down the current date and calculates the product of the digits. For example, on March 19, she wrote down 19.03 and made the calculations: 1 × 9 × 0 × 3 = 0. What is the largest product of numbers she can get?

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There must be no zeros in the date entry. That is, of the months, only November (11) and December (12) are suitable. The number 2 is greater than one, so you need to select the last month.

The day of the month must contain the number 9. The largest number that can come before it is 2, because there are no more than 31 days in a month. It turns out the date 29.12. Let's do the calculations: 2 × 9 × 1 × 2 = 36.

Answer: 36.

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– 3 –

How many decigrams are in a kilogram?

The prefix deci- means one tenth. That is, in one decigram, there are 0.1 grams or 0.0001 kilograms. So, in a kilogram 1,000/0.1 = 10,000 decigrams.

Answer: 10 000.

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– 4 –

When four children were asked how many of them went to ice rink, Sasha replied that no one, Kolya - that one person, Tonya - that two, and Zhenya - that three. It is known that only those who went to the skating rink told the truth. How many kids were there yesterday?

Sasha replied that none of the guys were at the rink. But she wouldn't have been able to find out if she hadn't gone there herself. It turns out that Sasha is lying.

If Tony's statement were true, then Zhenya or Kolya should have also said that two people went to the skating rink. Zhenya's answer also does not coincide with other statements of the guys. So both Zhenya and Tonya are lying. From this we can conclude that Kolya was at the rink alone and only his answer was true.

Answer: one.

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– 5 –

Fedya writes down natural numbers: 1, 2, 3... After he wrote down the 2015th figure, he ran out of ink. What was the last number he wrote?

Natural numbers are those used in counting. There are only 9 single-digit natural numbers (from 1 to 9), two-digit natural numbers - 90 (from 10 to 99).

This means that before starting to write three-digit numbers, Fedya had 2,015 − 9 × 1 − 90 × 2 = 1,826 digits before the end of the ink. To find out how many three-digit numbers he wrote, divide 1,826 by 3. In the answer, you need to get an integer value, since Fedya could not write fraction of a number. But 1,826 is not a multiple of 3, so when dividing, there will be a remainder: 1,826 / 3 = 608 three-digit integers + two more digits.

To find the desired three-digit number, add 99 to its serial number, so many numbers are up to 100 - the first three-digit number: 608 + 99 = 707. This is the last number that Fedya managed to write in full. The next number is 708, it had two characters left before the end of the ink. So the last number that Fedya wrote is 0.

Answer: 0.

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– 6 –

There are 10 vases on the table, in any two vases together there are no more than five nuts. What is the maximum number of nuts that can be in all the vases together?

According to the terms, two vases should not contain more than five nuts, that is, one can contain a maximum of four nuts. Then in the remaining nine vases there will be only one nut each. In total, 10 vases will contain: 4 + 9 = 13 nuts.

If the maximum number of nuts in one vase is three, then in order to meet the conditions, the remaining nine vases must contain two nuts. In this case, in 10 vases there will be: 3 + 9 × 2 = 21 nuts. This is the largest possible number of nuts.

Answer: 21.

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– 7 –

Mad Hatter made strange watch. Their minute hand is stationary, while the dial and the 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?

In order for such a watch to always show the correct time, the dial must rotate at the speed of the minute hand, but in the opposite direction. This means that the dial makes 24 turns counterclockwise per day.

The hour hand in a conventional watch rotates twice a day in a clockwise direction. But in the Hatter's watch, it must additionally turn along with the dial in the other direction 24 times. That is, in a day the hour hand makes: 24 − 2 = 22 revolutions.

Answer: 22.

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– 8 –

Turtle Chapa is 180 years old, and turtle Pashka is 173 years old. In how many years will Pashka's age be 99% of Chapa's age?

Turtle age difference: 180 − 173 = 7 years. She will always be the same. According to the condition of the problem, after a certain number of years Pashka should be 99% of Chapa's age. The age of Chapa himself is 100%. 100 − 99 = 1% is their age difference.

As we wrote above, this value will not change over time, so 1% will be equal to 7 years. That is, the full age of Chapa by the time when Pashka's age will be 99% of his age is: 7 × 100 = 700 years. This means that Pashka's age will correspond to the condition of the problem in: 700 − 180 = 520 years.

Answer: 520.

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The selection used tasks from the mathematical competition "Kangaroo" for 2015, 2017, 2018 and 2019.