Pythagoras contest - practice problems
The Pythagoras contest refers to mathematics competitions, often named after the ancient Greek mathematician Pythagoras, designed to challenge students with creative problem-solving beyond standard curriculum. Problems typically emphasize logical reasoning, pattern recognition, and elegant solutions rather than computational complexity. Contests may be local, national, or international, targeting various age groups and skill levels. They encourage mathematical thinking, healthy competition, and interest in mathematics as a discipline. Participants develop persistence, analytical skills, and confidence in tackling unfamiliar problems. Such competitions often serve as talent identification for advanced mathematics programs and olympiads. The Pythagoras contest tradition honors mathematical excellence and fosters a community of young mathematicians.Remember: Solve each problem thoughtfully and make sure to show your complete solution for every question.
Number of problems found: 59
- Wands - Wizard
The number promised the wizard Rhododendron that she would teach him a lot about numbers, but he must first solve her riddle: My two magic wands measure 63 cm together. One is 7 cm longer than the other. How long are the wands? - All zeros
How many zeros does the product of the numbers 10 × 11 × 12 × 13 × 14 × 15 × 20 × 21 × 22 × 23 × 24 × 25 end with? - In a football
In a football tournament of eight teams, where each team played each other exactly once, points were awarded as follows: the winner received 3 points, the loser received 0 points, and in the event of a draw, each team received 1 point. At the end of the t - On the board
A division problem with two positive numbers was written on the board. David noticed that if he increased the dividend by 2 and the divisor by 7, the quotient would not change. By how much should the divisor be increased so that when the dividend is incre - Factorization and primes
How many times does the digit 2 appear in the prime factorisation of 2,024? - Divisible PIN
Mark's mother helped him with his shopping. She wanted to pay by card when collecting the goods in person, but she was not sure what her PIN was. Could her PIN be a number divisible by six? - Four-digit numbers
Find all four-digit numbers whose decimal notation begins with a 6 and ends with a 2 and is divisible by 24. - Two cubes 3
Two cubes made of plasticine have single-digit integer edge lengths differing by 1 cm. Can a single larger cube with an integer edge length be made from the same total amount of plasticine? - Four-digit - sum A four-digit number has a digit sum of 20. The sum of its last two digits equals the second digit increased by 5. The sum of the first and last digits equals the second digit decreased by 3. If we write the digits of this number in reverse order, the numb
- Number line distance
Consecutive natural numbers on the number line are always 1 cm apart. Write the sum of the numbers 9 cm away from 517 on the number line. - Lucka candy count
Ten different packages contain 1 to 10 candies respectively. Each child took two packages. They received the following number of candies: Annie: 5 Suzy: 7 Katie: 9 Monica: 15 How many candies does Lucy have? - Remaining
Mum baked buns for dinner. Dad ate 1/3 of them. Then George came and ate a quarter of the remaining buns. After George, Martin came and ate a third of what was left. Annie ate half of the remaining buns, and Mum was left with four buns. How many buns did - Karel digit error Carl 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?
- Amazon river length
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 and squares
A 9 cm × 15 cm 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 of digits 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.
- Classroom clock
On a classroom clock, the large (minute) hand on the clock will travel through an angle of 120 degrees in some time. What angle does the small (hour) hand pass in this time? - Average of average Three different numbers are given. The average of the average of two smaller numbers and the average of the two larger numbers is equal to the average of all three numbers. The average of the smallest and largest number is 2022. Determine the sum of the t
- Dream market equivalence 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
- Water container measurement
Happy Mom needs to measure exactly 6 liters of water. It only has a five-liter and a seven-liter container. How can a mother measure exactly 6 liters of water by gradually pouring? He doesn't care about other containers.
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