2023 SSMO Accuracy Round Problems/Problem 2

Problem

Suppose that the average of all -digit palindromes is denoted by and the average of all -digit numbers is denoted by Find

Solution

The outermost digits of an -digit palindrome can range from to , each with equal probability (notice that they must be equal due to being a palindrome, so the ones digit cannot be ), so the average is . The inner digits can range from to , again with equal probability, so their average is . Thus .

However, the -digit numbers range in the exact same way except that the ones digit can range from to . Thus .

Then, $$\begin{array}{rl}{P}_n-N_n& =5({10}^{n-1}+1)+4.5(10+{10}^2+\dots +{10}^{n-2})-5({10}^{n-1})-4.5(1+10+{10}^2+\dots +{10}^{n-2})\\ & =5(1)-4.5(1)\\ & =0.5\end{array}$$

However, we must consider one special case: . Here, is an -digit number, so the difference between and is (they are the same set). For all the difference is ; therefore,

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