MathDB

1

Part of 2008 JBMO Shortlist

Problems(4)

x^3 +y^3 +k(x^2 +y^2) = y^3 +z^3 +k(y^2 +z^2) = ...=2008

Source: JBMO 2008 Shortlist A1

10/14/2017
If for the real numbers x,y,z,kx, y,z, k the following conditions are valid, x≠y≠z≠xx \ne y \ne z \ne x and x3+y3+k(x2+y2)=y3+z3+k(y2+z2)=z3+x3+k(z2+x2)=2008x^3 +y^3 +k(x^2 +y^2) = y^3 +z^3 +k(y^2 +z^2) = z^3 +x^3 +k(z^2 +x^2) = 2008, fi nd the product xyzxyz.
JBMOalgebra
n white markers in n x n board

Source: JBMO 2008 Shortlist C1

10/14/2017
On a 5×55 \times 5 board, nn white markers are positioned, each marker in a distinct 1×11 \times 1 square. A smart child got an assignment to recolor in black as many markers as possible, in the following manner: a white marker is taken from the board, it is colored in black, and then put back on the board on an empty square such that none of the neighboring squares contains a white marker (two squares are called neighboring if they share a common side). If it is possible for the child to succeed in coloring all the markers black, we say that the initial positioning of the markers was good. a) Prove that if n=20n = 20, then a good initial positioning exists. b) Prove that if n=21n = 21, then a good initial positioning does not exist.
JBMOcombinatorics
2008 JBMO Shortlist G1

Source: 2008 JBMO Shortlist G1

10/10/2017
Two perpendicular chords of a circle, AM,BNAM, BN , which intersect at point KK, define on the circle four arcs with pairwise different length, with ABAB being the smallest of them. We draw the chords AD,BCAD, BC with AD//BCAD // BC and C,DC, D different from N,MN, M . If LL is the intersection point of DN,MCDN, M C and TT the intersection point of DC,KL,DC, KL, prove that ∠KTC=∠KNL\angle KTC = \angle KNL.
JBMOgeometry
x(x - y) = 8y - 7 in NxN

Source: JBMO 2008 Shortlist N1

10/14/2017
Find all the positive integers xx and yy that satisfy the equation x(x−y)=8y−7x(x - y) = 8y - 7
JBMOnumber theory