Natural numbers + Mathematical Olympiad - practice problems - page 3 of 4
Number of problems found: 65
- Together 7114
Michaella has five crayons. Victor has fewer of them than Michaella. Vendelín has as many as Michaella and Vojto have together. All three have seven times more crayons than Victor. How many crayons does Vendelín have? - MO C–I–1 2018
An unknown number is divisible by just four numbers from the set {6, 15, 20, 21, 70}. Determine which ones. - Star equation
Write digits instead of stars so that the sum of the written digits is odd and is true equality: 42 · ∗8 = 2 ∗∗∗ - Solutions 8481
For which integers x is the ratio (x + 11) / (x + 7) an integer? Find all solutions.
- 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. - Z7-I-4 stars 4949
Write instead of stars digits, so the next write of the product of the two numbers is valid: ∗ ∗ ∗ · ∗ ∗ ∗ ∗ ∗ ∗ ∗ 4 9 4 9 ∗ ∗ ∗ ∗ ∗ ∗ 4 ∗ ∗ - Six-digit primes
Find all six-digit prime numbers that contain each one of digits 1,2,4,5,7 and 8 just once. How many are they? - Last digit
What is the last number of 2016 power of 2017 - Year 2018
The product of the three positive numbers is 2018. What are the numbers?
- Expression 4451
Find the largest natural number d that has that property for any natural number the number n is the value of the expression V (n) = n ^ 4 + 11n²−12 is divisible by d. - Determine 8611
Determine all natural numbers A and B pairs for which the sum of twice the least common multiple and three times the greatest common divisor of natural numbers A and B is equal to their product. - Positive integer integral
How many different sets of a positive integer in the form (x, y, z) satisfy the equation xyz=1400? - Equations: 80499
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.) - Justification 8468
The natural number n has at least 73 two-digit divisors. Prove that one of them is the number 60. Also, give an example of the number n, which has exactly 73 double-digit divisors, including a proper justification.
- Differences 80551
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 - Determine 5893
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. - Alarm clock
The old watchmaker has a unique digital alarm in its collection that rings whenever the sum of the alarm's digits equals 21. Find out when the alarm clock will ring. What is their number? List all options. - Circumference 9811
Kristýna chose a certain odd natural number divisible by three. Jakub and David then examined triangles with a circumference in millimeters equal to the number selected by Kristýna and whose sides have lengths in millimeters expressed by different integer - 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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