MathDB

3

Part of 2008 JBMO Shortlist

Problems(3)

p(x^2 - y^2) = (p^2- 1)xy , |x - 1|+ |y| = 1

Source: JBMO 2008 Shortlist A3

10/14/2017
Let the real parameter pp be such that the system {p(x2−y2)=(p2−1)xy∣x−1∣+∣y∣=1\begin{cases} p(x^2 - y^2) = (p^2- 1)xy \\ |x - 1|+ |y| = 1 \end{cases} has at least three different real solutions. Find pp and solve the system for that pp.
JBMOsystem of equationsalgebra
2n integers arbitrarily assigned to 2n numbered boxes

Source: JBMO 2008 Shortlist C3

10/14/2017
Integers 1,2,...,2n1,2, ...,2n are arbitrarily assigned to boxes labeled with numbers 1,2,...,2n1, 2,..., 2n. Now, we add the number assigned to the box to the number on the box label. Show that two such sums give the same remainder modulo 2n2n.
JBMOcombinatoricsremainder
a_{n+1} = a_n+s(a_n), whereas s(a)=sum of digits of a

Source: JBMO 2008 Shortlist N3

10/14/2017
Let s(a)s(a) denote the sum of digits of a given positive integer a. The sequence a1,a2,...,an,...a_1, a_2,..., a_n, ... of positive integers is such that an+1=an+s(an)a_{n+1} = a_n+s(a_n) for each positive integer nn. Find the greatest possible n for which it is possible to have an=2008a_n = 2008.
sum of digitsJBMOnumber theory