Question

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 $383.$ )

Answer

Answer: 550

Concepts:palindrome place value mean

Difficulty rating: 1750

Solution:

A three-digit palindrome has the form $\overline{aba} = 101a + 10b$ with $a \in \{1, \ldots, 9\}$ and $b \in \{0, \ldots, 9\},$ and every such pair of digits occurs exactly once, so the two digits vary independently over the $90$ palindromes.

By linearity, the mean is $101$ times the average of $a$ plus $10$ times the average of $b,$ namely $\begin{aligned} &101 \cdot 5 \\ &\quad {}+ 10 \cdot \frac{9}{2} = 505 + 45 = 550. \end{aligned}$

2021 AIME II Problem 1

Answer: 550

Difficulty rating: 1750

Solution:

Problem 1 in Other Years