Pythagoras contest - practice for 13 year olds
Number of problems found: 27
- Two-digit 82521
Karel had to multiply two two-digit numbers. Out of care, he changed the order of the digits in one of the factors and got a product that was 4,248 less than the correct result. What is the correct result? How much should Karl have earned? - Kilometers 82408
The longest Brazilian river, the Amazon, is ten times longer than the two Czech rivers, Vltava and Berounka. The ratio of the lengths of the Vltava and Berounka is 9:5. At the same time, the length ratio of the Vltava and Morava rivers is 5:4. How many ki - Rectangle 82087
A 9cm × 15cm rectangle is divided into unit squares. How many paths are there from one rectangle vertex to the opposite vertex if one can only go to the right and up the sides of the squares? - Position 81987
Find a number with six digits. If you put the last digit before the first, you get a new number that is five times larger. The digits between must not change their position. - Nightmares 80568
At the dream market, she offered the Sphinx to a traveler for four dreams, seven illusions, two naps, and one nightmare. Another has seven dreams, four illusions, four naps, and two nightmares. The Sphinx always measures the same for all travelers. How ma - Distribute 70244
We have to distribute the keys to the safe among four people so that no two of them can open the safe but in such a way that any three can open the safe. How many minimum keys do we need? How to divide them? How many minimum locks must be on the safe? All - Grandma
Grandma is 57 years older than her two twin granddaughters. If we add the ages of an older woman with both granddaughters, we get the number 99. How old is her old woman? - Three-digit 58943
The vortex of the three given digits formed different three-digit numbers. When she added up all these numbers, she published 1554. What numbers did Vierka use? - Different 55491
Add the same numbers after the same letters and different numbers after the other letters so that equality applies: KRAVA + KRAVA = MLIEKO, where K is an odd digit. - Four-digit 55481
Find all four-digit abcd numbers to which: abcd = 20. ab + 16. cd, where ab and cd are double digits numbers from digits a, b, c, and d. - Phone number
Ivan's phone number ends with a four-digit number: When we subtract the first from the fourth digit of this four-digit number, we get the same number as when we subtract the second from the third digit. If we write the four-digit number from the back and - Five-digit number
Anna thinks of a five-digit number not divisible by three or four. If he increments each digit by one, it gets a five-digit number divisible by three. If he reduces each digit by one, he gets a five-digit number divisible by four. If it swaps any two digi - All pairs
Find all pairs (m, n) of natural numbers for which is true: m s (n) = n s (m) = 70, where s(a) denotes the digit sum of the natural number a. - Apples and pears
Apples cost 50 cents a piece, pears 60 cents a piece, bananas cheaper than pears. Grandma bought five pieces of fruit. There was only one banana, and I paid 2 euros 75 cents. How many apples and how many pears? - Two cars on ring
There were two cars on the round track (ring) in the adjacent tracks, the first car on the inner track and the second on the outer track. Both cars started at the same time from one starting track. The first toy car drove four laps simultaneously, and the - Self-counting machine
The self-counting machine works exactly like a calculator. The innkeeper wanted to add several three-digit natural numbers on his own. On the first attempt, he got the result in 2224. To check, he added these numbers again, and he got 2198. Therefore, he - Equilateral triangle ABC
In the equilateral triangle ABC, K is the center of the AB side, the L point lies on one-third of the BC side near point C, and point M lies on one-third of the side of the AC side closer to point A. Find what part of the ABC triangle contains the triangl - Special watch
Fero bought a special watch on the market. They have only one (minute) hand and a display showing the angle between the hour and minute hand. How many hours was his watch shown - the minute hand points to number 2; the display shows 125°. - Two rectangles
I cut out two rectangles with 54 cm² and 90 cm². Their sides are expressed in whole centimeters. If I put these rectangles together, I get a rectangle with an area of 144 cm². What dimensions can this large rectangle have? Write all options. Explain your - Triplets
Three years ago, the triplets Pavel, Petr, Luke, and their five-year-old sister Jane had been together for 25 years. How old is Jane today?
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