Difference between revisions of "2022 USAJMO Problems"
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==Day 2== | ==Day 2== | ||
===Problem 4=== | ===Problem 4=== | ||
− | + | Let <math>ABCD</math> be a rhombus, and let <math>K</math> and <math>L</math> be points such that <math>K</math> lies inside the rhombus, <math>L</math> lies outside the rhombus, and <math>KA=KB=LC=LD</math>. Prove that there exist points <math>X</math> and <math>Y</math> on lines <math>AC</math> and <math>BD</math> such that <math>KXLY</math> is also a rhombus. | |
[[2022 USAJMO Problems/Problem 4|Solution]] | [[2022 USAJMO Problems/Problem 4|Solution]] | ||
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===Problem 5=== | ===Problem 5=== | ||
− | + | Find all pairs of primes <math>(p,q)</math> for which <math>p-q</math> and <math>pq-q</math> are both perfect squares. | |
[[2022 USAJMO Problems/Problem 5|Solution]] | [[2022 USAJMO Problems/Problem 5|Solution]] | ||
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===Problem 6=== | ===Problem 6=== |
Revision as of 21:19, 19 April 2022
Contents
Day 1
For any geometry problem whose statement begins with an asterisk
, the first page of the solution must be a large, in-scale, clearly labeled diagram. Failure to meet this requirement will result in an automatic 1-point deduction.
Problem 1
For which positive integers does there exist an infinite arithmetic sequence of integers
and an infinite geometric sequence of integers
satisfying the following properties?
is divisible by
for all integers
;
is not divisible by
.
Problem 2
Let and
be positive integers. The cells of an
grid are colored amber and bronze such that there are at least
amber cells and at least
bronze cells. Prove that it is possible to choose
amber cells and
bronze cells such that no two of the
chosen cells lie in the same row or column.
Problem 3
Solution
Let and
be fixed integers, and
. Given are
identical black rods and
identical white rods, each of side length 1.
We assemble a regular -gon using these rods so that parallel sides are the same color. Then, a convex
-gon
is formed by translating the black rods, and a convex
-gon
is formed by translating the white rods. An example of one way of doing the assembly when
and
is shown below, as well as the resulting polygons
and
.
[image here]
Prove that the difference of the areas of and
depends only on the numbers
and
, and not on how the
-gon was assembled.
Day 2
Problem 4
Let be a rhombus, and let
and
be points such that
lies inside the rhombus,
lies outside the rhombus, and
. Prove that there exist points
and
on lines
and
such that
is also a rhombus.
Problem 5
Find all pairs of primes for which
and
are both perfect squares.
Problem 6
Let be complex numbers, and define
for all nonnegative integers
.
Suppose that for all
. Prove that
2021 USAJMO (Problems • Resources) | ||
Preceded by 2021 USAJMO |
Followed by 2023 USAJMO | |
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The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.