MathDB

2

Part of 2006 Bulgaria Team Selection Test

Problems(4)

Polynomial

Source: Bulgarian TST1/2006 Problem 2

5/31/2006
Find all couples of polynomials (P,Q)(P,Q) with real coefficients, such that for infinitely many x∈Rx\in\mathbb R the condition P(x)Q(x)−P(x+1)Q(x+1)=1x(x+2) \frac{P(x)}{Q(x)}-\frac{P(x+1)}{Q(x+1)}=\frac{1}{x(x+2)} Holds. Nikolai Nikolov, Oleg Mushkarov
algebrapolynomialalgebra unsolved
Tough ineq

Source: Bulgarian TST1/2006 Problem 5

5/31/2006
Prove that if a,b,c>0,a,b,c>0, then ab3a+4b+5c+bc3b+4c+5a+ca3c+4a+5b≤a+b+c12. \frac{ab}{3a+4b+5c}+\frac{bc}{3b+4c+5a}+\frac{ca}{3c+4a+5b}\le \frac{a+b+c}{12}.
Nikolai Nikolov
inequalitiesinequalities unsolvedalgebra
Sequence

Source: Bulgarian TST2/2006 Problem 2

5/31/2006
a) Let {an}n=1∞\{a_n\}_{n=1}^\infty is sequence of integers bigger than 1. Proove that if x>0x>0 is irrational, then \ds x_n>\frac{1}{a_{n+1}} for infinitely many nn, where xnx_n is fractional part of anan−1…a1xa_na_{n-1}\dots a_1x.
b)Find all sequences {an}n=1∞\{a_n\}_{n=1}^\infty of positive integers, for which exist infinitely many x∈(0,1)x\in(0,1) such that \ds x_n>\frac{1}{a_{n+1}} for all nn.
Nikolai Nikolov, Emil Kolev
algebra unsolvedalgebra
Divisors

Source: Bulgarian TST1/2006 Problem 5

5/31/2006
Problem 5. Denote with d(a,b)d(a,b) the numbers of the divisors of natural aa, which are greater or equal to bb. Find all natural nn, for which d(3n+1,1)+d(3n+2,2)+…+d(4n,n)=2006.d(3n+1,1)+d(3n+2,2)+\ldots+d(4n,n)=2006. Ivan Landgev
number theory unsolvednumber theory