2021 IMO Problems
Problem 1
Let be an integer. Ivan writes the numbers
each on different cards. He then shuffles these
cards, and divides them into two piles. Prove that at least one of the piles contains two cards such that the sum of their numbers is a perfect square.
Problem 2
Show that the inequality
holds for all real numbers
.
Problem 3
Let be an interior point of the acute triangle
with
so that
. The point
on the segment
satisfies
, the point
on the segment
satisfies
, and the point
on the line
satisfies
. Let
and
be the circumcenters of the triangles
and
respectively. Prove that the lines
,
, and
are concurrent.
Problem 4
Let be a circle with centre
, and
a convex quadrilateral such that each of
the segments
and
is tangent to
. Let
be the circumcircle of the triangle
.
The extension of
beyond
meets
at
, and the extension of
beyond
meets
at
.
The extensions of
and
beyond
meet
at
and
, respectively. Prove that
Problem 5
Two squirrels, Bushy and Jumpy, have collected 2021 walnuts for the winter. Jumpy numbers the walnuts from 1 through 2021, and digs 2021 little holes in a circular pattern in the ground around their favourite tree. The next morning Jumpy notices that Bushy had placed one walnut into each hole, but had paid no attention to the numbering. Unhappy, Jumpy decides to reorder the walnuts by performing a sequence of 2021 moves. In the -th move, Jumpy swaps the positions of the two walnuts adjacent to walnut
.
Prove that there exists a value of such that, on the
-th move, Jumpy swaps some walnuts
and
such that
.
Problem 6
Let be an integer,
be a finite set of (not necessarily positive) integers, and
be subsets of
. Assume that for each
the sum of the elements of
is
. Prove that
contains at least
elements.
Resources
2021 IMO (Problems) • Resources | ||
Preceded by 2020 IMO Problems |
1 • 2 • 3 • 4 • 5 • 6 | Followed by 2022 IMO Problems |
All IMO Problems and Solutions |