Hundredth Decimal Digit

This hard combinatorics-flavored question asks for a specific decimal digit deep in the expansion of a large power built from a simple algebraic expression. The setup involves raising a sum of a rational number and an irrational square-root term to a high exponent, then examining the fine structure of its decimal representation far past the decimal point. The twist is that directly computing or approximating such a huge power numerically would be infeasible in an interview setting, so the candidate must rely on structural properties of the expression rather than brute force or heavy computation.

Answering it well draws on understanding of binomial expansions, conjugate pairs, and how irrational terms can cancel to leave an integer. The problem leans on bounding arguments, size comparisons, and reasoning about how close a number is to an integer without explicitly calculating it. An interviewer is watching for the ability to spot and exploit symmetry, use inequalities cleanly, control error terms, and convert an algebraic observation into precise conclusions about decimal behavior.

What it tests

When an expression of the form $(a + b)^n$ is paired with its conjugate $(a - b)^n$, their sum often yields an integer because the irrational parts cancel due to symmetry in the binomial expansion. For large $n$, if $|a - b| < 1$, then $(a - b)^n$ becomes extremely small, so $(a + b)^n$ is just less than an integer by a tiny amount. This structure is a consequence of how binomial coefficients and powers distribute over conjugate pairs, causing most non-integer terms to cancel. The key is that the sum is forced to be an integer, so the original term must be just below this integer if the conjugate is positive and tiny. This principle allows us to deduce the decimal behavior of such expressions without explicit computation.