Mathematical Olympiad - math problems

MO tasks are not easy, even for adults. At the same time, we believe that the right solution, which is here published almost on one click will serve as the inspiration.

Do not be discouraged if you did not discover the right solution. Experiment, sketching, "play" with the problem. Sometimes it helps to look into a book and find out similar problems resolved. Sometimes help three days pause, and then you found the right solution.

Number of problems found: 67

Year 2018
The product of the three positive numbers is 2018. What are the numbers?
Pytagoriade
Two fifth-graders teams competing in math competitions - in Mathematical Olympiad and Pytagoriade. Of the 33 students competed in at least one of the contest 22 students. Students who competed only in Pytagoriade were twice more than those who just compet
Christmas Day
In leap years was 53 Sundays. On what day of the week fell to Christmas Day?
Pentagon
Within a regular pentagon ABCDE point, P is such that the triangle is equilateral ABP. How big is the angle BCP? Make a sketch.
One million
Write the million number (1000000) using only nine numbers and algebraic operations plus, minus, times, divided, powers, and squares. Find at least three different solutions.
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.
Twos
Vojta started writing the number of this year 2019202020192020 into the workbook. .. And so he kept going. When he wrote 2020 digits, no longer enjoyed it. How many twos did he write?
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.
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?
Star equation
Write digits instead of stars so that the sum of the written digits is odd and is true equality: 42 · ∗8 = 2 ∗∗∗
Hexagon - MO
The picture shows the ABCD square, the EFGD square and the HIJD rectangle. Points J and G lie on the side CD and is true |DJ|
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?
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.
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 ∗ ∗
Bedrich and Adam
When Bedrich is as old as Adam today, Adam will be 14 years old. When Adam will be as old as Bedrich, Bedrich was two years old today. How old are Adam and Bedrich today?
Pyramid Z8–I–6
Each brick of pyramid contains one number. Whenever possible, the number in each brick is lowest common multiple of two numbers of bricks lying directly above it. That number may be in the lowest brick? Determine all possibilities.
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
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 . ..
Last digit
What is the last number of 2016 power of 2017
Bicycles
You're the owner of the transport 's learning playground. Buy bicycles of two colors but you've got to spend accurately 120000 Kč. Blue bike costs 3600Kč and red bicycle 3200Kč.

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