MathDB

3

Part of 2016 IMC

Problems(2)

IMC 2016, Problem 3

Source: IMC 2016

7/27/2016
Let nn be a positive integer. Also let a1,a2,…,ana_1, a_2, \dots, a_n and b1,b2,…,bnb_1,b_2,\dots, b_n be real numbers such that ai+bi>0a_i+b_i>0 for i=1,2,…,ni=1,2,\dots, n. Prove that ∑i=1naibi−bi2ai+bi≤∑i=1nai⋅∑i=1nbi−(∑i=1nbi)2∑i=1n(ai+bi)\sum_{i=1}^n \frac{a_ib_i-b_i^2}{a_i+b_i}\le\frac{\displaystyle \sum_{i=1}^n a_i\cdot \sum_{i=1}^n b_i - \left( \sum_{i=1}^n b_i\right) ^2}{\displaystyle\sum_{i=1}^n (a_i+b_i)}.
(Proposed by Daniel Strzelecki, Nicolaus Copernicus University in Toruń, Poland)
IMCIMC 2016Summationcollege contests
IMC 2016, Problem 8

Source: IMC 2016

7/28/2016
Let nn be a positive integer, and denote by Zn\mathbb{Z}_n the ring of integers modulo nn. Suppose that there exists a function f:Zn→Znf:\mathbb{Z}_n\to\mathbb{Z}_n satisfying the following three properties:
(i) f(x)≠xf(x)\neq x,
(ii) f(f(x))=xf(f(x))=x,
(iii) f(f(f(x+1)+1)+1)=xf(f(f(x+1)+1)+1)=x for all x∈Znx\in\mathbb{Z}_n.
Prove that n≡2(mod4)n\equiv 2 \pmod4.
(Proposed by Ander Lamaison Vidarte, Berlin Mathematical School, Germany)
IMCIMC 2016college contestsabstract algebrafunction