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The father is 32 years old, the son is 5 years old. In how many years will the father be 10 times older than his son?
Solve this equation: T – W – O = T ÷ W ÷ O = 2 (different letters are different numbers).
There are three milk cans with capacities of 3 litres, 5 litres, and 8 litres. The largest can contains 8 litres of milk, and the rest of the cans are empty. How do you use these cans to split the milk in half?
A horse eats a haystack in 2 days, a cow in 3 days, a sheep in 6 days. How long will it take a horse, a cow and a sheep to eat a haystack together?
🐎🐄 🐑
The number ends in 2.
If this digit is moved to the beginning of the number, it will double.
Find the smallest number that satisfies these conditions.
Numbers from 1 to 10 are arranged in a row.
1 2 3 4 5 6 7 8 9 10 = 0
Is it possible to place '+' and '–' signs between them in such a way that the value of the resulting expression equals zero?
The son of the professor's father is talking to the father of the professor's son, but the professor is not participating in the conversation.
Could this be true?
Can the number 11...11, where there are eighty-one ones, be evenly divided by 81?
There are 9 coins, all except one are the same weight, and the odd one is heavier than the rest. You must determine which is the odd one out using an old-fashioned balance. You may use the balance twice. Explain how this can be done.
All the shapes below are squares. Find the side of the bottom left square if the side of the smallest square is 1.
Prove that no integer in the sequence 11, 111, 1111, 11111 ... is a perfect square.
Express 203 as the sum of distinct natural numbers whose product is also 203.
ABCDEF is a six-digit number. All of the digits are different and arranged from left to right in increasing order. The number is a perfect square. Determine what this number is.
Is there a numeral system in which 3 + 4 = 10 and 3 × 4 = 15?
What is the base of this system?
During the first year, the population of a certain village increased by n people, and for the second year, it increased by 300 people. At the same time, over the first year, the population increased by 300%, and for the second year, it increased by n%. How many inhabitants became part of the village?
The arithmetic mean of ten distinct natural numbers is 15. Find the largest value among these numbers.
A 1992-digit number is written. Each two-digit number formed by neighbouring digits is divisible by 17 or 23. The last digit is 1. What is the first digit?
With a bag of granulated sugar, a cup scale, and a 1 g weight, is it possible to measure 1 kg of sugar in 10 weighings?
There are 28 red, 20 green, 12 yellow, 20 blue, 10 white and 10 black balls in a box.
How many balls need to be randomly drawn from the box, without peeking, to ensure that at least 15 balls of the same color are among those drawn?
Is it possible to arrange the numbers 0, 1 and –1 in a 6x6 table so that all the sums of the numbers along the verticals, horizontals and two main diagonals are different?
Amrita needs to select a new PIN.
She decides it will be made up of four non-zero digits with the following properties:
i) The first two digits and the last two digits each make up a two-digit number which is a multiple of 11.
ii) The sum of all the digits is a multiple of 11.
How many different possibilities are there for Amrita’s PIN?
A three-digit number begins with the number 4. If you move it to the end of the number, you get a number that is 3/4 of the original. Find the original three-digit number.
Are there natural numbers x, y and z that satisfy the condition 28x + 30y + 31z = 365?
The product of five numbers is not equal to zero.
When you decrease each number by one, the product still stays the same.
Can you give an example of such numbers?
N boys and N girls go out to dance. In how many ways can they all dance (boys must dance with girls)? 💃🏻 🕺🏻
Move 3 matchsticks to get 3 squares. Make sure that after you move or remove matchsticks, there are no dangling matchsticks.
It is known that the expressions 4k+5 and 9k+4 are simultaneously perfect squares for certain natural values of k. What values can the expression 7k+4 attain for the same values of k?
In how many ways can we cut a necklace consisting of 30 different beads into 8 parts? (It is allowed to cut only between beads.)
Replace the '*' in the expression ∗1∗2∗4∗8∗16∗32∗64=27 with either '+' or '−' to make the equation correct.
Define an unfortunate number as a number which is 13 times the sum of its digits. How many unfortunate numbers are there?
Prove that the number of USA states with an odd number of neighbours is even.
On a globe, there are 17 parallels of latitude and 24 meridians of longitude. How many regions does the surface of the globe get divided into by the intersection of these parallels and meridians?
A mother has two apples, three pears, and four oranges. For nine consecutive days, she gives her son one piece of the remaining fruit each day. In how many ways can she give the fruits to her son over the nine days?
Kate has a calculator with three operations: multiply a number by 3, add 3 to a number, or (if the number is divisible by 3) divide it by 3. Using these operations, how can you transform the number 1 into the number 11?
There are two hourglasses, one measuring 7 minutes and the other 11 minutes. You need to boil an egg for exactly 15 minutes. How can you measure this time using only the given hourglasses?
A quadrilateral has all integer side lengths, a perimeter of 26, and one side of length � n. What is the greatest possible length of one side of this quadrilateral?
A number is chosen at random from among the first 100 positive integers, and a positive integer divisor of that number is then chosen at random. What is the probability that the chosen divisor is divisible by 11?
How many different ways are there to read the word BANANA in the following table if we can only cross to a field that shares an edge with the current field, and we can use fields several times?
Santa Claus gave out 47 chocolates to a group of kids. Every girl got one more chocolate than every boy. After that, Santa shared 74 jelly beans, ensuring each boy got one more jelly bean than each girl. How many kids were there in total?
On December 31st, Michael made a statement: "After the New Year, everything I said before the New Year will become a lie." Was he telling the truth?
Except for the first term, each term in the following sequence is found by multiplying the same number to the previous term:3, a, b, c, 48, …Find the value of c.
What is the largest product that can be formed from using the digits 1, 2, 3, and 4, and one multiplication sign? You are only allowed to combine the digits to form two numbers.
Find the missing number in the grid below.
Hint: Four-digit perfect squares.
The diagram shows a point E inside a rectangle ABCD such that AE = 16 cm, DE = 20 cm and CE = 13 cm. Find the length of BE.
Fred has five cards labeled 1, 2, 3, 4, and 5. Create a three-digit number and a two-digit number using these cards so that the larger number is divisible by the smaller one.
Let a be an integer such that a+1 is divisible by 3. Prove that 4+7a is divisible by 3.
How many three-digit numbers are there that are equal to five times the product of their digits?
In a biology lab, there are people, mice, and snakes. The total count of heads is 40, legs amount to 100, and tails sum up to 36. Find the number of snakes in the lab.
The diagram shows four semicircles, one with radius 2 cm, touching the other three, which have radius 1 cm. What is the total area of the shaded regions?
Do there exist positive integers x, and y, such that x+y, 2x+y and x+2y are perfect squares?
The areas of the two rectangles in the diagram are 25 cm² and 13 cm². What is the value of x?
Two brothers and three sisters form a single line for a photograph. The two boys refuse to stand next to each other.
How many different line-ups are possible?
In the Maths Premier League, teams get 3 points for a win, 1 point for a draw and 0 points for a loss. Last year, my team played 38 games and got 80 points. We won more than twice the number of games we drew and more than five times the number of games we lost. How many games did we draw?
Alice refuses to sit next to either Bob or Carla. Derek refuses to sit next to Eric. How many ways are there for the five of them to sit in a row of 5 chairs under these conditions?
Prove that, for real numbers x, y, z: |x| + |y| + |z| ≤ |x + y − z| + |x − y + z| + |−x + y + z|.
Two straight lines have equations y = px + 4 and py = qx − 7, where p and q are constants.
The two lines meet at the point (3, 1).
What is the value of q?
Factorial n, written n!, is defined by: n! = 1 × 2 × 3 × · · · × n.
What is the remainder when 1! + 2! + 3! + 4! + 5! + 6! + 7! + 8! + 9! + 10! is divided by 5?
The number 21! = 51,090,942,171,709,440,000 has over 60,000 positive integer divisors. One of them is chosen at random.
What is the probability that it is odd?
Find all positive integers n such that n × 2n + 1 is a square.
Source: United Kingdom - BMO 2023 - Round 1 - 4
Do there exist 99 consecutive natural numbers such that the smallest one is divisible by 100, the next by 99, the third by 98, ..., and the last one by 2?
Source: Tournament of Towns - 2017 - O Level - 2
Let ℕ = {1, 2, 3, . . .} be the set of all positive integers.
Find all functions f : ℕ → ℕ such that for any positive integers a and b, the following two conditions hold:
(1) f(ab) = f(a)f(b)
(2) at least two of the numbers f(a), f(b) and f(a + b) are equal.
Source: European Girls’ Mathematical Olympiad - 2022 - Day 1 - Problem 2
Margie's winning art design is shown. The smallest circle has radius 2 cm, with each successive circle's radius increasing by 2 cm.
Which of the following is closest to the percent of the design that is black?
The diagram shows six identical squares arranged symmetrically. What fraction of the diagram is shaded?
Source: UK JMO - 2017 - A6
A gold coin is worth x% more than a silver coin. The silver coin is worth y% less than the gold coin.
Both x and y are positive integers.
How many possible values for x are there?
Source: UK - IMC - 2024 - 25
In one glass, there was milk, and in the other, an equal amount of coffee.
A spoonful of milk was transferred from the milk glass to the coffee glass and then stirred.
Subsequently, the same spoonful of the resulting mixture was returned to the milk glass.
What is more now: the coffee in the milk glass or the milk in the coffee glass?
What is the maximum number of pieces that a round pancake can be divided into using three staright cuts?
Two squares of side 1 have a common centre. Show that the area of their intersection is greater than 3/4.
In many parts of the world, including the USA and Europe, dates are written differently. In the USA, it's customary to write the month first, followed by the day and year (MM/DD/YYYY), while in Europe, it's common to write the day first, followed by the month and year (DD/MM/YYYY). This can sometimes lead to ambiguity, where a date could be interpreted differently depending on the format used.
How many days in a year have dates that cannot be interpreted unambiguously without knowing the specific date format being used?
A student made an error while multiplying two three-digit numbers. Instead of noticing the multiplication sign between them, the student wrote the numbers together, forming a six-digit number.
Surprisingly, this mistake resulted in a product that was three times greater than the actual product of the two numbers.
Can you find these two original three-digit numbers?
Arrange the digits 1 to 9 in a row so that each pair of consecutive numbers forms a two-digit number divisible by either 7 or 13.
Each number from 1 to 9 should appear exactly once.
A positive integer is called friendly if it is divisible by the sum of its digits.
For example, 111 is friendly but 123 is not.
Find the number of all two-digit friendly numbers.
Seven standard dice are glued together to make the solid shown.
The pairs of faces of the dice that are glued together have the same number of dots on them.
How many dots are on the surface of the solid?
Punam puts counters onto some of the cells of a 5 × 5 board.
She can put more than one counter on each cell, and she can leave some cells empty.
She tells Quinn how many counters there are in each row and column. These ten numbers are all different.
Can Quinn always work out which cells, if any, are empty?
PT is the tangent to a circle O, and PB is the angle bisector of the angle TPA (see diagram). How big is the angle TBP?
Given a circle with radius 1 and 100 points marked along its circumference. Prove that there exists at least one point on the circle such that the sum of the distances from this point to all 100 marked points is at least 100.
A semicircle is inscribed in a quarter circle as shown. What fraction of the quarter circle is shaded?
For every natural number n, prove that the expression n5 - 5n3 + 4n is always divisible by 120.
A piece fell out of a book, where the first page is numbered 439 and the last page is numbered with the same digits in a different order. Determine the number of pages in the missing piece.
In the center of a square field, a hare sits, surrounded by four wolves positioned at each corner.
The hare can move freely within the field, while the wolves can only move along the sides of the square.
Given that the maximum speed of each wolf is at most 1.4 times the maximum speed of the hare, is it possible for the hare to escape the square without being caught by the wolves?
Find all prime numbers p for which the number p2+11 has exactly 6 different divisors (including 1 and the number itself).
My 24-hour digital clock displays hours and minutes only. How many displayed times in a 24-hour period contain at least one occurrence of the digit 5?
Some of the digits in the following correct addition have been replaced by the letters P, Q, R, and S, as shown. What is the value of P + Q + R + S?
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