Difference between revisions of "2023 AIME I Problems"
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==Problem 7== | ==Problem 7== | ||
− | + | Call a positive integer <math>n</math> extra-distinct if the remainders when <math>n</math> is divided by <math>2, 3, 4, 5,</math> and <math>6</math> are distinct. Find the number of extra-distinct positive integers less than <math>1000</math>. | |
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[[2023 AIME I Problems/Problem 7|Solution]] | [[2023 AIME I Problems/Problem 7|Solution]] |
Revision as of 14:52, 8 February 2023
2023 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
Problem 1
Five men and nine women stand equally spaced around a circle in random order. The probability that every man stands diametrically opposite a woman is where
and
are relatively prime positive integers. Find
Problem 2
Positive real numbers and
satisfy the equations
The value of
is
where
and
are relatively prime positive integers. Find
Problem 3
A plane contains lines, no
of which are parallel. Suppose that there are
points where exactly
lines intersect,
points where exactly
lines intersect,
points where exactly
lines intersect,
points where exactly
lines intersect, and no points where more than
lines intersect. Find the number of points where exactly
lines intersect.
Problem 4
These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.
Problem 5
Let be a point on the circle circumscribing square
that satisfies
and
. Find the area of
.
Problem 6
These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.
Problem 7
Call a positive integer extra-distinct if the remainders when
is divided by
and
are distinct. Find the number of extra-distinct positive integers less than
.
Problem 8
These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.
Problem 9
Find the number of cubic polynomials , where
,
, and
are integers in
, such that there is a unique integer
with
.
Problem 10
These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.
Problem 11
These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM. Unofficial problem statement has been posted.
Problem 12
These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.
Problem 13
These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.
Problem 14
These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.
Problem 15
These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.
See also
2023 AIME I (Problems • Answer Key • Resources) | ||
Preceded by 2022 AIME II |
Followed by 2023 AIME II | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
- American Invitational Mathematics Examination
- AIME Problems and Solutions
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.