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Omni-MATH / A group of 101 Dalmathians participate in an election, where they each vote…

Problem

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problem

A group of 101 Dalmathians participate in an election, where they each vote independently on either candidate AA or BB with equal probability. If XX Dalmathians voted for the winning candidate, the expected value of X2X^{2} can be expressed as ab\frac{a}{b} for positive integers a,ba, b with gcd(a,b)=1\operatorname{gcd}(a, b)=1. Find the unique positive integer k≤103k \leq 103 such that 103∣a−bk103 \mid a-bk.
Plain-text mathematical notation (without MathML)
A group of 101 Dalmathians participate in an election, where they each vote independently on either candidate A or B with equal probability. If X Dalmathians voted for the winning candidate, the expected value of X² can be expressed as (a)/(b) for positive integers a,b with gcd(a,b)=1. Find the unique positive integer k≤103 such that 103∣a−bk.
Original LaTeX notation
A group of 101 Dalmathians participate in an election, where they each vote independently on either candidate \(A\) or \(B\) with equal probability. If \(X\) Dalmathians voted for the winning candidate, the expected value of \(X^{2}\) can be expressed as \(\frac{a}{b}\) for positive integers \(a, b\) with \(\operatorname{gcd}(a, b)=1\). Find the unique positive integer \(k \leq 103\) such that \(103 \mid a-bk\).

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Source and history

task_id
73b99aff-45fb-5721-a059-92320fe91f58
task_key
test--73b99aff-45fb-5721-a059-92320fe91f58
task_revision_id
1
config / split
/ test

Official source

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