Mathematical Olympiad - practice problems
MO tasks are not easy, even for adults. At the same time, we believe that the right solution, which is here published almost on one click will serve as the inspiration.Do not be discouraged if you did not discover the right solution. Experiment, sketching, "play" with the problem. Sometimes it helps to look into a book and find out similar problems resolved. Sometimes help three days pause, and then you found the right solution.
Direction: Solve each problem carefully and show your solution in each item.
Number of problems found: 136
- Respectively 81293
The figure shows the squares ABCD, EFCA, CHCE, and IJHE. Points S, B, F, and G are, respectively, the centers of these squares. Line segment AC is 1 cm long. Determine the area of triangle IJS. Please help... - Intersection 81017
There are also two equilateral triangles ABC, and BDE, such that the size of the angle ABD is greater than 120° and less than 180° points C and E lie in the same half-plane defined by the line AD. The intersection of CD and AE is marked F. Determine the s - Respectively 80982
The vertices of the square ABCD are joined by the broken line DEFGHB. The smaller angles at the vertices E, F, G, and H are right angles, and the line segments DE, EF, FG, GH, and HB measure 6 cm, 4 cm, 4 cm, 1 cm, and 2 cm, respectively. Determine the ar - SKMO
Petra had written natural numbers from 1 to 9. She added two of these numbers, deleted them, and wrote the resulting sum instead of the summaries. She thus had eight numbers written down, which she managed to divide into two groups with the same product. - Three-digit 80768
Nikola had one three-digit and one two-digit number written in her notebook. Each of these numbers was made up of different digits. The difference in Nicole's numbers was 976. What was their sum? - Grandchildren 80639
The average age of the grandfather, grandmother, and their five grandchildren is 26 years. The average age of the grandchildren themselves is 7 years. Grandma is a year younger than grandpa. How old is grandma? - Smallest z9 Find the smallest positive numbers a and b for which 7a³ = 11b⁵
- Positive integer integral How many different sets of a positive integer in the form (x, y, z) satisfy the equation xyz=1400?
- Together 70014
Matej and Anton are 44 years old together. Matej is twice as old as Anton was when Matej was half as old as Anton will be when Anton is 3 times older than Matej was when Matej was 3 times as old as Anton. - Different 64304
How many different ways can Anka, Betka, Cyril, Daniel, and Erik line up in the dining room if the boys let the two girls in front of them? - Two-digit 64294
How many natural two-digit numbers can we form from the digits 0, 1, 2, and 3 if the numerals cannot be repeated in these numbers? - Book Store
The Mabini Book Store (MBS) is reducing the prices of Mathematics books for promotion. The store has 6 Algebra books, 6 Geometry books, and 5 Statistics books to be arranged on a shelf. Books of the same kind are to be placed beside each other. How many w - Mr. Product
The product of ages of all of Mr. Product's children is 1408. The age of the youngest child is equal to half the age of the oldest child. How many children does Mr. Product have, and how old are they? - Three-digit 60711
Viera compiled different three-digit numbers from three given digits. When she added up all these numbers, she published 1221. What numbers did Vierka use? Identify five options - Below-average 60271
In a given group of numbers, one number is equal to the average of all, the largest number is 7 larger than the average, the smallest is 7 smaller than the average, and most numbers in the group have below-average values. What is the smallest number of nu - MO Z7–I–6 2021
In triangle ABC, point D lies on the AC side and point E on the BC side. The sizes of the angles ABD, BAE, CAE, and CBD are 30°, 60°, 20°, and 30°, respectively. Find the size of the AED angle. - Mathematical 40213
Sixty-two high school students took part in the Mathematical Olympiad. There were 7 fewer first-year students than sophomores, two-thirds of the number of third-year students, and 5 fourth-year students. How many students from each grade took part in the - Two ports
Between the ports of Mumraj and Zmatek, two ships commute along the same route. They spend negligible time in ports, turn around immediately, and continue sailing. At the same time, a blue ship departs from the port of Mumraj, and a green ship departs fro - Coloured numbers
Mussel wrote four different natural numbers with colored markers: red, blue, green, and yellow. When the red number divides by blue, it gets the green number as an incomplete proportion, and yellow represents the remainder after this division. When it div - Squirrels
The squirrels discovered a bush with hazelnuts. The first squirrel plucked one nut, the second squirrel two nuts, and the third squirrel three nuts. Each new squirrel always tore one nut more than the previous squirrel. When they plucked all the nuts from
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