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Integer for all n ≥ k

Source: IMO LongList 1982 - P1

March 16, 2011
inductionleast common multiplenumber theory proposednumber theory

Problem Statement

(a) Prove that 1n+1⋅(2nn)\frac{1}{n+1} \cdot \binom{2n}{n} is an integer for n≥0.n \geq 0.
(b) Given a positive integer kk, determine the smallest integer CkC_k with the property that Ckn+k+1⋅(2nn)\frac{C_k}{n+k+1} \cdot \binom{2n}{n} is an integer for all n≥k.n \geq k.
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