MathDB

2

Part of 2008 India Regional Mathematical Olympiad

Problems(2)

Existence of infinite sequences

Source: (Indian) RMO 2008 Problem 2

11/9/2008
Prove that there exist two infinite sequences {an}n≥1 \{a_n\}_{n\ge 1} and {bn}n≥1 \{b_n\}_{n\ge 1} of positive integers such that the following conditions hold simultaneously: (i) (i) 0<a1<a2<a3<⋯ 0 < a_1 < a_2 < a_3 < \cdots; (ii) (ii) an<bn<an2 a_n < b_n < a_n^2, for all n≥1 n\ge 1; (iii) (iii) a_n \minus{} 1 divides b_n \minus{} 1, for all n≥1 n\ge 1 (iv) (iv) a_n^2 \minus{} 1 divides b_n^2 \minus{} 1, for all n≥1 n\ge 1 [19 points out of 100 for the 6 problems]
number theoryleast common multipleinequalitiesalgorithm
A system of equations

Source: An Excursion in Mathematics

6/25/2020
Solve the system of equation x+y+z=2;x+y+z=2;(x+y)(y+z)+(y+z)(z+x)+(z+x)(x+y)=1;(x+y)(y+z)+(y+z)(z+x)+(z+x)(x+y)=1;x2(y+z)+y2(z+x)+z2(x+y)=−6.x^2(y+z)+y^2(z+x)+z^2(x+y)=-6.
system of equationsequationalgebra