MathDB

4

Part of 2008 JBMO Shortlist

Problems(4)

x+y+z=2008, x^2+y^2+z^2=6024^2,1/x+1/y+1/z=1/2008

Source: JBMO 2008 Shortlist A4

10/14/2017
Find all triples (x,y,z)(x,y,z) of real numbers that satisfy the system {x+y+z=2008x2+y2+z2=602421x+1y+1z=12008\begin{cases} x + y + z = 2008 \\ x^2 + y^2 + z^2 = 6024^2 \\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{2008} \end{cases}


JBMOalgebrasystem of equations
bw colorings in a 4x4 table after n steps

Source: JBMO 2008 Shortlist C4

10/14/2017
Every cell of table 4×44 \times 4 is colored into white. It is permitted to place the cross (pictured below) on the table such that its center lies on the table (the whole fi gure does not need to lie on the table) and change colors of every cell which is covered into opposite (white and black). Find all nn such that after nn steps it is possible to get the table with every cell colored black.
JBMOcombinatoricsColoring
2008 JBMO Shortlist G4

Source: 2008 JBMO Shortlist G4

10/10/2017
Let ABCABC be a triangle, (BC<ABBC < AB). The line ll passing trough the vertices CC and orthogonal to the angle bisector BEBE of ∠B\angle B, meets BEBE and the median BDBD of the side ACAC at points FF and GG, respectively. Prove that segment DFDF bisects the segment EGEG.
JBMOgeometry
n^4 + 8n + 11 product of consecutive integers

Source: JBMO 2008 Shortlist N4

10/14/2017
Find all integers nn such that n4+8n+11n^4 + 8n + 11 is a product of two or more consecutive integers.
JBMOnumber theory