Mathematical Olympiad - practice problems - page 6 of 11
Number of problems found: 210
- Clubhouse
A clubhouse contained only chairs and tables. Each chair had 4 legs and each table had 3 legs. A group of scouts arrived; each sat on a chair, 2 chairs were left empty, and the total number of legs in the room (people + furniture) was 101. How many chairs - Z7–I–1 MO 2018
A single digit different from zero is written on each of three cards (the digits on different cards are not necessarily different). We know that any three-digit number formed from these cards is divisible by six. Moreover, a three-digit number divisible b - Bombastic number We call a natural number N bombastic if it contains no zero in its notation and if no smaller natural number has the same product of digits as the number N. Charles first became interested in bombastic prime numbers and claimed that there were not many of
- Z7–I–4 2018 MO Betka
Karel was playing with gears assembled into a gear train. When he turned one wheel, all the others turned too. The first wheel had 32 teeth and the second had 24 teeth. When the third wheel (which is in the middle of the gear train) made exactly eight ful - Wall building
The king gave the mason Wesley the task of building a wall 25 cm thick, 50 m long, and 2 m high. If Wesley had worked without a break and at the same pace, he would have built a wall in 26 hours. However, according to the valid royal regulations, Wencesla - MO Z8–I–6 2018
The KLMN trapezium, KL has a 40 cm base and an MN of 16 cm. Point P lies on the KL line so that the NP segment divides the trapezoid into two parts with the same area. Find the length of the KP line. - Positive integers
Find all positive integers x and y for which: 1 / x + 1 / y = 1/4 - Z7-1-3 MO 2018
Grandfather prepared a pile of hazelnuts for his six grandchildren with the instruction that they should divide them somehow. First came Adam, who took half for himself, took one more nut, and left. The same was done by the second Bob, the third Cyril, th - Chair purchase
The hotelier wanted to equip the dining room with new chairs. He chose the type of chair in the catalog. Only when placing an order did he learn from the manufacturer that they offered every fourth chair at half price as part of the discount offer and tha - 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 - Last digit
What is the last number of 2016 power of 2017 - Rectangle perimeter
Adam had three identical rectangles. He put them together and got a rectangle with a circumference of 50 cm. Then, he placed them differently and got a rectangle with a larger circumference. Calculate its perimeter. - Point distances
It is 16 km from point A to B. from point C to B, it is 20 km from point C to D, it is 19 km how many kilometers is it from point D to point A - Square table
Determine the largest integer n for which the square table n×n can be filled with natural numbers from 1 to n² (n squared) so that at least one square power of the integer is written in each of its 3×3 square parts. - Running Track Length
Mike and Rick ran back and forth on the running track. They started towards each other, each from a different end of the track. Both were still running at the same speed, each at a different speed. The first time, they met 800 m from one end of the track, - MO Z6-1-3 2017 chessboard
Veronika has a classical chessboard with 8×8 cells. The rows are marked with digits 1 to 8, the columns with letters A to H. Veronika placed on the cell B1 a knight, which can be moved only as in chess. 1. Is it possible to move the knight in four moves t - Z7-I-5 MO 2017
Prokop constructed a triangle ABC whose interior angle at vertex A was greater than 60° and whose interior angle at vertex B was less than 60°. Juraj drew, in the half-plane determined by line AB and point C, a point D such that triangle ABD was equilater - Z7-1-6 MO 2017
The water sprite Chaluha was pouring fog into various differently sized vessels, which he had carefully arranged on a shelf. When pouring, he proceeded gradually from one side, skipping no vessel. Into each vessel at least a decilitre of fog fits. If he w - 3N on the number axis
The line represents the number axis, and the marked points correspond to the numbers a, - a, and a + 1, but in no particular order. Construct the points that correspond to the numbers 0 and 1. Discuss all the possibilities. - Number line arrangement
On a number line, consider the points corresponding to the values a, 2a, and 3a + 1 arranged in all possible orders. For each arrangement, decide whether it is possible. If yes, give a specific example; if not, explain why.
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