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3

Part of 1997 China National Olympiad

Problems(2)

Split 1,2,..,3n into 3 sequences of n terms

Source: Chinese Mathematical Olympiad 1997 Problem 3

8/26/2013
Prove that there are infinitely many natural numbers nn such that we can divide 1,2,…,3n1,2,\ldots ,3n into three sequences (an),(bn)(a_n),(b_n) and (cn)(c_n), with nn terms in each, satisfying the following conditions: i) a1+b1+c1=a2+b2+c2=…=an+bn+cna_1+b_1+c_1= a_2+b_2+c_2=\ldots =a_n+b_n+c_n and a1+b1+c1a_1+b_1+c_1 is divisible by 66; ii) a1+a2+…+an=b1+b2+…+bn=c1+c2+…+cn,a_1+a_2+\ldots +a_n= b_1+b_2+\ldots +b_n=c_1+c_2+\ldots +c_n, and a1+a2+…+ana_1+a_2+\ldots +a_n is divisible by 66.
number theory unsolvednumber theory
Sequence satisfying the condition a_(m+n) < a_n + a_m

Source: Chinese Mathematical Olympiad 1997 Problem 6

8/26/2013
Let (an)(a_n) be a sequence of non-negative real numbers satisfying an+m≤an+ama_{n+m}\le a_n+a_m for all non-negative integers m,nm,n. Prove that if n≥mn\ge m then an≤ma1+(nm−1)ama_n\le ma_1+\left(\dfrac{n}{m}-1\right)a_m holds.
inductioninequalities