MathDB

Problem 2

Part of 2001 Brazil Team Selection Test

Problems(2)

n is power of 2

Source:

10/11/2014
Let f(n)f(n) denote the least positive integer kk such that 1+2+⋯+k1+2+\cdots+k is divisible by nn. Show that f(n)=2n−1f(n)=2n-1 if and only if nn is a power of 22.
number theory unsolvednumber theory
minimization in a floor sequence

Source: Brazil TST 2001 Test 2 P2

4/28/2021
A set SS consists of kk sequences of 0,1,20,1,2 of length nn. For any two sequences (ai),(bi)∈S(a_i),(b_i)\in S we can construct a new sequence (ci)(c_i) such that ci=⌊ai+bi+12⌋c_i=\left\lfloor\frac{a_i+b_i+1}2\right\rfloor and include it in SS. Assume that after performing finitely many such operations we obtain all the 3n3n sequences of 0,1,20,1,2 of length nn. Find the least possible value of kk.
algebrafloor functionoptimization