MathDB

IMO Shortlist 2011, Algebra 3

Source: IMO Shortlist 2011, Algebra 3

7/11/2012

Determine all pairs

(f,g)

of functions from the set of real numbers to itself that satisfy

g(f(x+y)) = f(x) + (2x + y)g(y)

for all real numbers

x

and

y

.
Proposed by Japan

functionalgebrafunctional equationIMO Shortlist

IMO Shortlist 2011, G3

Source: IMO Shortlist 2011, G3

7/13/2012

Let

ABCD

be a convex quadrilateral whose sides

AD

and

BC

are not parallel. Suppose that the circles with diameters

AB

and

CD

meet at points

E

and

F

inside the quadrilateral. Let

\omega_E

be the circle through the feet of the perpendiculars from

E

to the lines

AB,BC

and

CD

. Let

\omega_F

be the circle through the feet of the perpendiculars from

F

to the lines

CD,DA

and

AB

. Prove that the midpoint of the segment

EF

lies on the line through the two intersections of

\omega_E

and

\omega_F

.
Proposed by Carlos Yuzo Shine, Brazil

geometryparallelogramcircumcircleperpendicular bisectorpower of a pointIMO Shortlist

IMO Shortlist 2011, Number Theory 3

Source: IMO Shortlist 2011, Number Theory 3

7/11/2012

Let

n \geq 1

be an odd integer. Determine all functions

f

from the set of integers to itself, such that for all integers

x

and

y

the difference

f(x)-f(y)

divides

x^n-y^n.

Proposed by Mihai Baluna, Romania

functionalgebranumber theoryDivisibilityIMO ShortlistHi

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