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Regional Olympiad - FBH 2018 Grade 12 Problem 1

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2018

September 18, 2018
Setsnumber theorybinomial coefficients

Problem Statement

a)a) Prove that for all positive integers n≥3n \geq 3 holds: (n1)+(n2)+...+(nn−1)=2n−2\binom{n}{1}+\binom{n}{2}+...+\binom{n}{n-1}=2^n-2 where (nk)\binom{n}{k} , with integer kk such that n≥k≥0n \geq k \geq 0, is binomial coefficent
b)b) Let n≥3n \geq 3 be an odd positive integer. Prove that set A={(n1),(n2),...,(nn−12)}A=\left\{ \binom{n}{1},\binom{n}{2},...,\binom{n}{\frac{n-1}{2}} \right\} has odd number of odd numbers
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