Mathematical Olympiad - practice problems - page 2 of 11
Number of problems found: 210
- SKMO
Patricia 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 produc - Nikola number difference
Nicole 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? - Sheep shepherd legs
There were 45 sheep and several shepherds in the meadow. After half of the shepherds and a third of the sheep left the meadow, the remaining shepherds and sheep had a total of 126 legs. All sheep and all shepherds usually had leg counts. how many shepherd - Grandparent age average
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?
- Sequence difference ratio
Bolek and Lolek each had their own arithmetic sequence. Both Lolek and Bolek's sequence started with the number 2023 and ended with the number 3023. The two sequences had 26 numbers in common. The ratio of Bolek's and Lolka's difference was 5:2. What is t - Integer part system
In the field of real numbers, solve the system of equations: 2x + ⌊y⌋ = 2022, 3y + ⌊2x⌋ = 2023. (⌊a⌋ denotes the (lower) integer part of the real number a, i.e., the largest integer not greater than a., E.g., ⌊1.9⌋ = 1 and ⌊−1.1⌋ = −2.) - Matej Anton age
Matthew and Anthony are 44 years old together. Matthew is twice as old as Anthony was when Matthew was half as old as Anthony will be when Anthony is 3 times older than Matthew was when Matthew was 3 times as old as Anthony. - Gathering of solvers
The best solvers of the mathematical olympiad and their leaders came to a gathering. They all stayed in a pension with two-bed and three-bed rooms, such that all beds in these rooms were fully occupied. The two-bed rooms were exactly a third of all rooms - Kuba four-digit number
Jake wrote down a four-digit number containing two even digits and two odd digits. If he crossed out both even digits, he would get a number four times smaller than if he crossed out both odd digits. What is the largest number with these characteristics t - Dining room lineup
If the boys let the two girls in front of them, how many different ways can Annie, Betty, Cyril, Daniel, and Eric line up in the dining room? - Two-digit number creation
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 - Son birth year
My only son was born when I was 37 years old. That was exactly 32 years after the death of my grandfather, and he died at the age of 64. Dedecek was 12 years older than Grandma; they got married in 1947, just when Grandma was 18 years old. What year was m - Aquaman
Aquaman Kebule shopped at Captain Nemo's fishmonger, where the prices of all goods were listed in whole scales. If Kebule bought 2 crayfish, 3 clams, and 1 pike, he would pay 49 scales. If he bought 5 more crayfish, 11 clams, and 1 pike, he would pay 154 - Mr. Product
The product of the ages of all of Mr. Product's children is 1,408. The youngest child's age is half the oldest child's age. How many children does Mr. Product have, and how old are they? - Three-digit unknown int
Vera compiled different three-digit numbers from three given digits. When she added up all these numbers, the sum was 1221. What numbers did Vera use? Identify five options - Group number average
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 - Three-leggers
On a newly discovered planet live animals that astronauts named after the number of legs one-leggers, two-leggers, three-leggers, and so on (there were no animals without legs there). Animals with an odd number of legs have two heads, animals with an even
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