MathDB

3

Part of 2009 JBMO Shortlist

Problems(4)

(|x| + |y| - 2)^2 = 1 , y = ax + 5

Source: JBMO 2009 Shortlist A3

10/14/2017
Find all values of the real parameter aa, for which the system (∣x∣+∣y∣−2)2=1(|x| + |y| - 2)^2 = 1 y=ax+5y = ax + 5 has exactly three solutions
JBMOalgebrasystem of equations
SARAJEVO in a table

Source: JBMO 2009 Shortlist C3

10/14/2017
a) In how many ways can we read the word SARAJEVO from the table below, if it is allowed to jump from cell to an adjacent cell (by vertex or a side) cell? b) After the letter in one cell was deleted, only 525525 ways to read the word SARAJEVO remained. Find all possible positions of that cell.
JBMOcombinatorics
2009 JBMO Shortlist G3

Source: 2009 JBMO Shortlist G3

10/8/2017
Parallelogram ABCD{ABCD}with obtuse angle ∠ABC\angle ABC is given. After rotation of the triangle ACD{ACD} around the vertex C{C}, we get a triangle CD′A′{CD'A'}, such that points B,CB,C and D′{D'}are collinear. Extensions of median of triangle CD′A′{CD'A'} that passes through D′{D'}intersects the straight line BD{BD} at point P{P}. Prove that PC{PC}is the bisector of the angle ∠BPD′\angle BP{D}'.
geometryJBMO
(x + y)^2(x^2 + y^2) = 2009^2

Source: JBMO 2009 Shortlist N3

10/14/2017
Find all pairs (x,y)(x,y) of integers which satisfy the equation (x+y)2(x2+y2)=20092(x + y)^2(x^2 + y^2) = 2009^2
JBMOnumber theory