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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
The diagram shows six identical squares arranged symmetrically. What fraction of the diagram is shaded?
Source: UK JMO - 2017 - A6
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?
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
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
Find all positive integers n such that n × 2n + 1 is a square.
Source: United Kingdom - BMO 2023 - Round 1 - 4
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?
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?
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?
Prove that, for real numbers x, y, z: |x| + |y| + |z| ≤ |x + y − z| + |x − y + z| + |−x + y + z|.
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?
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?
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?
The areas of the two rectangles in the diagram are 25 cm² and 13 cm². What is the value of x?
Do there exist positive integers x, and y, such that x+y, 2x+y and x+2y are perfect squares?
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?
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.
How many three-digit numbers are there that are equal to five times the product of their digits?
Let a be an integer such that a+1 is divisible by 3. Prove that 4+7a is divisible by 3.
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.
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.
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