2024 IMO Problems
Problems of the 2024 IMO.
Contents
[hide]Day I
Problem 1
Determine all real numbers such that, for every positive integer
, the integer
is a multiple of . (Note that
denotes the greatest integer less than or equal to
. For example,
and
.)
Problem 2
Find all positive integer pairs such that there exists positive integers
such that
holds for all integer
.
Problem 3
Let be an infinite sequence of positive integers, and let
be a positive integer. Suppose that, for each
,
is equal to the number of times
appears in the list
.
Prove that at least one of the sequence and
is eventually periodic.
(An infinite sequence is eventually periodic if there exist positive integers
and
such that
for all
.)
Day II
Problem 4
Let be a triangle with
. Let the incentre and incircle of triangle
be
and
, respectively. Let
be the point on line
different from
such that the line
through
parallel to
is tangent to
. Similarly, let
be the point on line
different from
such that the line through
parallel to
is tangent to
. Let
intersect the circumcircle of
triangle
again at
. Let
and
be the midpoints of
and
, respectively.
Prove that
.
Problem 5
Turbo the snail plays a game on a board with 2024 rows and 2023 columns. There are hidden monsters in 2022 of the cells. Initially, Turbo does not know where any of the monsters are, but he knows that there is exactly one monster in each row except the first row and the last row, and that each column contains at most one monster.
Turbo makes a series of attempts to go from the first row to the last row. On each attempt, he chooses to start on any cell in the first row, then repeatedly moves to an adjacent cell sharing a common side. (He is allowed to return to a previously visited cell.) If he reaches a cell with a monster, his attempt ends and he is transported back to the first row to start a new attempt. The monsters do not move, and Turbo remembers whether or not each cell he has visited contains a monster. If he reaches any cell in the last row, his attempt ends and the game is over.
Determine the minimum value of for which Turbo has a strategy that guarantees reaching
the last row on the
attempt or earlier, regardless of the locations of the monsters.
Problem 6
Let be the set of rational numbers. A function
is called
if the following property holds: for every
,
Show that there exists an integer
such that for any aquaesulian function
there are at most
different rational numbers of the form
for some rational number
, and find the smallest possible value of
.
See Also
2024 IMO (Problems) • Resources | ||
Preceded by 2023 IMO |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 2025 IMO |
All IMO Problems and Solutions |