Year 2018

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1. 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.
2. Sum of two primes
   Christian Goldbach, a mathematician, found out that every even number greater than 2 can be expressed as a sum of two prime numbers. Write or express 2018 as a sum of two prime numbers.
3. Amazing number
   An amazing number is name for such even number, the decomposition product of prime numbers has exactly three not necessarily different factors and the sum of all its divisors is equal to twice that number. Find all amazing numbers.
4. MO Z6-6-1
   Write integers greater than 1 to the blanks in the following figure, so that each darker box was product of the numbers in the neighboring lighter boxes. What number is in the middle box?
5. Two-digit number
   I am a two-digit number less than 20. When I divided by three, then you get the rest 1 and when you divide me by four you get also rest 1. What number am I?
6. Class
   When Pythagoras asked how many students attend the school, said: "Half of the students studying mathematics, 1/4 music, seventh silent and there are three girls at school." How many students had Pythagoras at school?
7. Z9–I–4 MO 2017
   Numbers 1, 2, 3, 4, 5, 6, 7, 8 and 9 were prepared for a train journey with three wagons. They wanted to sit out so that three numbers were seated in each carriage and the largest of each of the three was equal to the sum of the remaining two. The conducto
8. Toy cars
   Pavel has a collection of toy cars. He wanted to regroup them. But in the division of three, four, six, and eight, he was always one left. Only when he formed groups of seven, he divided everyone. How many toy cars have in the collection?
9. Cakes Z8-I-5
   Mom brought 10 cakes of three types: kokosek was less than laskonek and most were caramel cubes. John chose two different kinds of cakes, Stephan did the same and for Margerith leave only the cakes of the same type. How many kokosek, laskonek and caramel c
10. One hundred stamps
    A hundred letter stamps cost a hundred crowns. Its costs are four levels - twenty tenths , one crown, two-crown and five-crown. How many are each type of stamps? How many does the problem have solutions?
11. Divisibility 2
    How many divisors has integer number 13?
12. School
    Less than 500 pupils attend school. When it is sorted into pairs, one pupil remains. Similarly, when sorted into 3, 4, 5 and 6 members team one remains. Sorted to seven members teams, no left behind. How many pupils are attending this school?
13. Package
    The package has no more than 67 m of cloth. If we just cut it all on the blouses or all on dresses, no cloth left remain. On the one blouse consumes 3.8 m of cloth and on one dress 1.7 m. Determine the amount of the cloth in the package.
14. Together
    If 8 men, 10 women, 16 children collects ₹1024 in 4 days, how many days will be required for 6 men, 5 women and 4 boys to collect ₹768? (₹ is Indian Rupee)
15. Digit sum
    Determine for how many integers greater than 900 and less than 1,001 has digit sum digit of the digit sum number 1.
16. 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?
17. Primes 2
    Which prime numbers is number 2025 divisible?