Difference between revisions of "2023 AIME I Problems"
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==Problem 11== | ==Problem 11== | ||
− | + | Find the number of subsets of <math>{1,2,3,...,10}</math> that contain exactly one pair of consecutive integers. Examples of such subsets are <math>{1,2,5}</math> and <math>{1,3,6,7,10}</math>. | |
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[[2023 AIME I Problems/Problem 11|Solution]] | [[2023 AIME I Problems/Problem 11|Solution]] |
Revision as of 21:10, 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
The sum of all positive integers such that
is a perfect square can be written as
where
and
are positive integers. Find
Problem 5
Let be a point on the circle circumscribing square
that satisfies
and
Find the area of
Problem 6
Alice knows that red cards and
black cards will be revealed to her one at a time in random order. Before each card is revealed, Alice must guess its color. If Alice plays optimally, the expected number of cards she will guess correctly is
where
and
are relatively prime positive integers. Find
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
There is a rhombus in which
. A point
is chosen somewhere on the incircle of
, and the distances from
to sides
,
, and
, are
,
, and
, respectively. Evaluate the perimeter of the rhombus.
Problem 9
Find the number of cubic polynomials , where
,
, and
are integers in
, such that there is a unique integer
with
.
Problem 10
There exists a unique positive integer for which the sum
is an integer strictly between
and
. For that unique
, find
.
(Note that denotes the greatest integer that is less than or equal to
.)
Problem 11
Find the number of subsets of that contain exactly one pair of consecutive integers. Examples of such subsets are
and
.
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
Each face of two noncongruent parallelepipeds is a rhombus whose diagonals have lengths and
. The ratio of the volume of the larger of the two polyhedra to the volume of the smaller is
, where
and
are relatively prime positive integers. Find
. A parallelepiped is a solid with six parallelogram faces such as the one shown below.
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.