Difference between revisions of "2024 USAJMO Problems"
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[[2024 USAJMO Problems/Problem 2|Solution]] | [[2024 USAJMO Problems/Problem 2|Solution]] | ||
===Problem 3=== | ===Problem 3=== | ||
− | Let <math>a(n)</math> be the sequence defined by | + | Let <math>a(n)</math> be the sequence defined by <math>a(n+1)=(a(n))^{n+1}-1</math> for each integer <math>n\geq1</math>. Suppose that <math>p>2</math> is prime and <math>k</math> is a positive integer. Prove that some term of the sequence <math>a(n)</math> is divisible by <math>p^k</math>. |
[[2024 USAJMO Problems/Problem 3|Solution]] | [[2024 USAJMO Problems/Problem 3|Solution]] | ||
+ | |||
==Day 2== | ==Day 2== | ||
===Problem 4=== | ===Problem 4=== | ||
Line 20: | Line 21: | ||
*no matter how Rowan permutes the rows of the coloring, Colin can then permute the columns to restore the original grid coloring; and | *no matter how Rowan permutes the rows of the coloring, Colin can then permute the columns to restore the original grid coloring; and | ||
− | *no matter how Colin permutes the | + | *no matter how Colin permutes the columns of the coloring, Rowan can then permute the rows to restore the original grid coloring; |
In terms of <math>n</math>, how many orderly colorings are there? | In terms of <math>n</math>, how many orderly colorings are there? | ||
[[2024 USAJMO Problems/Problem 4|Solution]] | [[2024 USAJMO Problems/Problem 4|Solution]] | ||
+ | |||
===Problem 5=== | ===Problem 5=== | ||
Find all functions <math>f:\mathbb{R}\rightarrow\mathbb{R}</math> that satisfy | Find all functions <math>f:\mathbb{R}\rightarrow\mathbb{R}</math> that satisfy |
Latest revision as of 17:11, 12 June 2024
Contents
Day 1
Problem 1
Let be a cyclic quadrilateral with
and
. Points
and
are selected on segment
such that
. Points
and
are selected on segment
such that
. Prove that
is a cyclic quadrilateral.
Problem 2
Let and
be positive integers. Let
be the set of integer points
with
and
. A configuration of
rectangles is called happy if each point in
is a vertex of exactly one rectangle, and all rectangles have sides parallel to the coordinate axes. Prove that the number of happy configurations is odd.
Problem 3
Let be the sequence defined by
for each integer
. Suppose that
is prime and
is a positive integer. Prove that some term of the sequence
is divisible by
.
Day 2
Problem 4
Let be an integer. Rowan and Colin play a game on an
grid of squares, where each square is colored either red or blue. Rowan is allowed to permute the rows of the grid, and Colin is allowed to permute the columns of the grid. A grid coloring is
if:
- no matter how Rowan permutes the rows of the coloring, Colin can then permute the columns to restore the original grid coloring; and
- no matter how Colin permutes the columns of the coloring, Rowan can then permute the rows to restore the original grid coloring;
In terms of , how many orderly colorings are there?
Problem 5
Find all functions that satisfy
for all
.
Problem 6
Point is selected inside acute triangle
so that
and
. Point
is chosen on ray
so that
. Let
be the midpoint of
. Show that line
is tangent to the circumcircle of triangle
.
See also
2024 USAJMO (Problems • Resources) | ||
Preceded by 2023 USAJMO Problems |
Followed by 2025 USAJMO Problems | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.