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==Problem 5== | ==Problem 5== | ||
| − | Let <math>P</math> be a point on the circle circumscribing square <math>ABCD</math> that satisfies <math>PA\cdot PC=56</math> and <math>PB\cdot PD=90</math> | + | Let <math>P</math> be a point on the circle circumscribing square <math>ABCD</math> that satisfies <math>PA \cdot PC = 56</math> and <math>PB \cdot PD = 90.</math> Find the area of <math>ABCD.</math> |
[[2023 AIME I Problems/Problem 5|Solution]] | [[2023 AIME I Problems/Problem 5|Solution]] | ||
Revision as of 14:49, 8 February 2023
| 2023 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
|
Instructions
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| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
to 
, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.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 ![\[\sqrt{\log_b n} = \log_b \sqrt{n} \qquad \text{and} \qquad b \cdot \log_b n = \log_b (bn).\]](http://latex.artofproblemsolving.com/e/1/8/e188ed1d68418cfaed35d91cc7b0f333a84f6b1b.png)
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
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
There exists a unique positive integer such that the sum
![\[U=\sum_{n=1}^{2023}\left\lfloor\frac{n^2-n a}{5}\right\rfloor\]](http://latex.artofproblemsolving.com/a/d/2/ad257828cdba5520d49781266097aa37e262baf8.png)
is an integer strictly between 
and 1000 . For that unique , find 
. Here 
is the greatest integer less than or equal to 
.
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.
Unofficial problem statement has been posted.
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. 
