2023 IMO Problems
Problems of the 2023 IMO.
Contents
Day I
Problem 1
Determine all composite integers that satisfy the following property: if are all the positive divisors of with , then divides for every .
Problem 2
Let be an acute-angled triangle with . Let be the circumcircle of . Let be the midpoint of the arc of containing . The perpendicular from to meets at and meets again at . The line through parallel to meets line at . Denote the circumcircle of triangle by . Let meet again at . Prove that the line tangent to at meets line on the internal angle bisector of .
Problem 3
For each integer , determine all infinite sequences of positive integers for which there exists a polynomial of the form , where are non-negative integers, such that for every integer .
Day II
Problem 4
Let be pairwise different positive real numbers such that is an integer for every . Prove that .
Problem 5
Let be a positive integer. A Japanese triangle consists of circles arranged in an equilateral triangular shape such that for each , , , , the row contains exactly circles, exactly one of which is coloured red. A ninja path in a Japanese triangle is a sequence of circles obtained by starting in the top row, then repeatedly going from a circle to one of the two circles immediately below it and finishing in the bottom row. Here is an example of a Japanese triangle with , along with a ninja path in that triangle containing two red circles.
In terms of , find the greatest such that in each Japanese triangle there is a ninja path containing at least red circles.
Problem 6
Let be an equilateral triangle. Let be interior points of such that , , , and Let and meet at let and meet at and let and meet at
Prove that if triangle is scalene, then the three circumcircles of triangles and all pass through two common points.
See Also
2023 IMO (Problems) • Resources | ||
Preceded by 2022 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 2024 IMO |
All IMO Problems and Solutions |