Difference between revisions of "2023 AIME II Problems"
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==Problem 5== | ==Problem 5== | ||
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| + | Let <math>S</math> be the set of all positive rational numbers <math>r</math> such that when the two numbers <math>r</math> and <math>55r</math> are written as fractions in lowest terms, the sum of the numerator and denominator of one fraction is the same as the sum of the numerator and denominator of the other fraction. The sum of all the elements of <math>S</math> can be expressed in the form <math>\frac{p}{q},</math> where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q.</math> | ||
[[2023 AIME II Problems/Problem 5|Solution]] | [[2023 AIME II Problems/Problem 5|Solution]] | ||
Revision as of 14:50, 16 February 2023
| 2023 AIME II (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
The numbers of apples growing on each of six apple trees form an arithmetic sequence where the greatest number of apples growing on any of the six trees is double the least number of apples growing on any of the six trees. The total number of apples growing on all six trees is 
Find the greatest number of apples growing on any of the six trees.
Problem 2
Recall that a palindrome is a number that reads the same forward and backward. Find the greatest integer less than 
that is a palindrome both when written in base ten and when written in base eight, such as 
Problem 3
Let 
be an isosceles triangle with 
There exists a point 
inside such that 
and 
Find the area of 
Problem 4
Let 
and 
be real numbers satisfying the system of equations
\begin{align*}
xy + 4z &= 60 \\
yz + 4x &= 60 \\
zx + 4y &= 60.
\end{align*}
Let 
be the set of possible values of 
Find the sum of the squares of the elements of 
Problem 5
Let be the set of all positive rational numbers 
such that when the two numbers and 
are written as fractions in lowest terms, the sum of the numerator and denominator of one fraction is the same as the sum of the numerator and denominator of the other fraction. The sum of all the elements of can be expressed in the form 
where 
and 
are relatively prime positive integers. Find 
Problem 6
These problems will not be available until the 2023 AIME II is released in February 2023.
Problem 7
These problems will not be available until the 2023 AIME II is released in February 2023.
Problem 8
These problems will not be available until the 2023 AIME II is released in February 2023.
Problem 9
These problems will not be available until the 2023 AIME II is released in February 2023.
Problem 10
These problems will not be available until the 2023 AIME II is released in February 2023.
Problem 11
These problems will not be available until the 2023 AIME II is released in February 2023.
Problem 12
These problems will not be available until the 2023 AIME II is released in February 2023.
Problem 13
These problems will not be available until the 2023 AIME II is released in February 2023.
Problem 14
These problems will not be available until the 2023 AIME II is released in February 2023.
Problem 15
These problems will not be available until the 2023 AIME II is released in February 2023.
See also
| 2023 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by 2023 AIME I |
Followed by 2024 AIME I | |
| 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. 
