2021 AIME II Problems/Problem 1

Problem

Find the arithmetic mean of all the three-digit palindromes. (Recall that a palindrome is a number that reads the same forward and backward, such as $777$ or $383$ .)

Solution 1

Recall the the arithmetic mean of all the $n$ digit palindromes is just the average of the largest and smallest $n$ digit palindromes, and in this case the $2$ palindromes are $101$ and $999$ and $\frac{101+999}{2}=550$ and $\boxed{550}$ is the final answer.

~ math31415926535

Solution 2 (Symmetry)

For any palindrome $ABA,$ note that $ABA,$ is 100A + 10b + A which is also 101A + 10B.

Will finish up soon. No edit please. Thanks.

~MRENTHUSIASM

See also

2021 AIME II (Problems • Answer Key • Resources)

Preceded by
First Problem Followed by
Problem 2

1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

All AIME Problems and Solutions
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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2021 AIME II Problems/Problem 1

Contents

Problem

Solution 1

Solution 2 (Symmetry)

See also

2021 AIME II (Problems • Answer Key • Resources)
Preceded by First Problem	Followed by Problem 2
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions