# Mathematical Olympiad + natural numbers - practice problems

#### Number of problems found: 23

- Mr. Product

The product of the ages of all of Mr. Product children's is 1408. The age of the youngest child is equal to half the age of the oldest child. How many children does Mr. Product have and how old are they? - Coloured numbers

Mussel wrote four different natural numbers with coloured markers: red, blue, green and yellow. When the red number divides by blue, it gets the green number as an incomplete proportion, and yellow represents the remainder after this division. When it div - Squirrels

The squirrels discovered a bush with hazelnuts. The first squirrel plucked one nut, the second squirrel two nuts, the third squirrel three nuts. Each new squirrel always tore one nut more than the previous squirrel. When they plucked all the nuts from the - 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 shepherd rewarded him and paid - All pairs

Determine 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

4 dance ensembles were dancing at the festival. None had less than 10 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 danc - Twos

Vojta started writing the number of this year 2019202020192020 into the workbook. .. And so he kept going. When he wrote 2020 digits, he no longer enjoyed it. How many twos did he write? - Self-counting machine

The self-counting machine works exactly like a calculator. The innkeeper wanted to add several three-digit natural numbers on his own. On the first attempt, he got the result in 2224. To check, he added these numbers again and he got 2198. Therefore, he a - 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? - Year 2018

The product of the three positive numbers is 2018. What are the numbers? - MO Z8-I-1 2018

Fero and David meet daily in the elevator. One morning they found that if they multiply their current age, they get 238. If they did the same after four years, this product would be 378. Determine the sum of the current ages of Fero and David. - 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. - Clubhouse

There were only chairs and table in the clubhouse. Each chair had four legs, and the table was triple. Scouts came to the clubhouse. Everyone sat on their chair, two chairs were left unoccupied, and the number of legs in the room was 101. How many chairs - Last digit

What is the last number of 2016 power of 2017 - 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 conduct - Alarm clock

The old watchmaker has a unique digital alarm in its collection that rings whenever the sum of digits of the alarm is equal to 21. Find out when the alarm clock will ring. What is their number? List all options . .. - Z7-I-4 stars 4949

Write instead of stars digits so the next write of product of the two numbers to be valid: ∗ ∗ ∗ · ∗ ∗ ∗ ∗ ∗ ∗ ∗ 4 9 4 9 ∗ ∗ ∗ ∗ ∗ ∗ 4 ∗ ∗ - Star equation

Write digits instead of stars so that the sum of the written digits is odd and is true equality: 42 · ∗8 = 2 ∗∗∗ - Octahedron - sum

On each wall of a regular octahedron is written one of the numbers 1, 2, 3, 4, 5, 6, 7 and 8, wherein on different sides are different numbers. For each wall John make the sum of the numbers written of three adjacent walls. Thus got eight sums, which also - Z9–I–1

In all nine fields of given shape to be filled natural numbers so that: • each of the numbers 2, 4, 6, and 8 is used at least once, • four of the inner square boxes containing the products of the numbers of adjacent cells of the outer square, • in the cir

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Mathematical Olympiad - practice problems. Natural numbers - practice problems.