Number theory · Palindromic numbers · Decimal representation · Binomial theorem · Binomial coefficients · Carrying
Call a positive integer symmetric if its decimal representation reads the same from left to right as from right to left. Prove that there are infinitely many positive integers \(n\) for which the numbers
\[ n^2, \quad n^3, \quad n^4 \]
are all symmetric, while \(n^5\) is not.
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Serbian National Competition (Drzavno takmicenje) 2011, high school grade I, category A, problem 3. Organized by the Mathematical Society of Serbia (DMS). Source