Mathematical Olympiad - practice problems - page 3 of 11
Number of problems found: 210
- Nine-digit numbers
Determine the number of nine-digit numbers in which each of the digits 0 through 9 occurs at most once and in which the sums of the digits 1 through 3, 3 through 5, 5 through 7, and 7 to the 9th place are always equal to 10. Find the smallest and largest - MO Z7-II-1 2020
Dragons and cyclops live on a fairy-tale island. All dragons are red, three-headed, and two-legged. All cyclops are brown, one-headed, and two-legged. Cyclops have one eye in the middle of their forehead; dragons have two eyes on each head. The cyclops an - 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. - Triangle
In triangle ABC, point S is the incentre (centre of the inscribed circle). The area of quadrilateral ABCS equals four-fifths of the area of triangle ABC. The side lengths of triangle ABC are all integers expressed in centimetres, and the perimeter of tria - Olympiad student
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
Two ships commute along the same route between the ports of Mumraj and Zmatek. 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 from - Coloured numbers
Mussel wrote four different natural numbers with colored markers: red, blue, green, and yellow. When the red number is divided by blue, it gets the green number is an incomplete proportion, and yellow represents the remainder after this division. When it - 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 - Shepherd
Kuba makes a deal with a shepherd to take care of his sheep. Shepherd said to Kuba that he would receive twenty gold coins and one sheep after a year of service. But Kuba resigned just after the seventh month of service. But the shepherd rewarded him and - 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. - Dance ensembles
Four dance ensembles were dancing at the festival. None had less than ten and more than 20 members. All dancers from some of the two ensembles were represented in each dance. First, 31 participants were on the stage, then 32, 34, 35, 37, and 38. How many - Z6–I–5 MO 2019
The shape in the picture was created by cutting a small cross out of a large cross. Each of these crosses can be composed of five identical squares, with the sides of the small squares being half the length of the sides of the large squares. The area of t - Snow White 2019 MO Z7
Snow White with her seven dwarfs gathered pinecones for a bonfire. Snow White said that the number of all pinecones is a number divisible by two. The first dwarf stated that it is a number divisible by three, the second dwarf said that it is a number divi - Pages MO Z6-I-4
Once the king called all his pages and placed them in a line. To the first page he gave a certain number of ducats, to the second he gave two ducats less, to the third again two ducats less, and so on. When he reached the last page, he gave him the approp - Z6 – I – 6 MO 2019
A mother was studying multi-digit numbers in which odd and even digits alternate regularly. She called those starting with an odd digit 'comical' and those starting with an even digit 'cheerful'. (For example, the number 32387 is comical, and the number 4 - Grandpa MO Z5–I–5 2019
Grandpa has three apple trees in his garden with a total of 39 apples on them. There are apples on only eight branches: one apple tree has two branches, and the other two apple trees have three branches each. Different branches bear different numbers of a - Construct rhombus - MO
Construct the diamond ABCD so that its diagonal BD is 8 cm and the distance of apex B from the line AD is 5 cm. Specify all possibilities. How long is a side of a rhombus? - Z8–I–5 MO 2019
For eight different points as shown in the figure: points C, D, and E lie on a line parallel to line AB; F is the midpoint of segment AD; G is the midpoint of segment AC; and H is the intersection of lines AC and BE. The area of triangle BCG is 12 cm² and - Magic number conjuring
Roman likes magic and math. Last time he conjured three- or four-digit numbers like this: • created two new numbers from the given number by dividing it between digits in the place of hundreds and tens (e.g., from the number 581, he would get 5 and 81), • - Five number operations For five whole numbers, if we add one to the first, multiply the second by the second, subtract three from the third, multiply the fourth by four, and divide the fifth by five, we get the same result each time. Find all five of the numbers that add up to
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