MathDB

a,b,c

Source: Iran 2005

8/27/2005

Suppose

a,b,c \in \mathbb R^+

and

\frac1{a^2+1}+\frac1{b^2+1}+\frac1{c^2+1}=2

Prove that

ab+ac+bc\leq \frac32

trigonometrygeometryinequalitiesinequalities proposed

Concurrent

Source: Iran 2005

8/27/2005

Suppose

H

and

O

are orthocenter and circumcenter of triangle

ABC

.

\omega

is circumcircle of

ABC

.

AO

intersects with

\omega

at

A_1

.

A_1H

intersects with

\omega

at

A'

and

A''

is the intersection point of

\omega

and

AH

. We define points

B',\ B'',\ C'

and

C''

similiarly. Prove that

A'A'',B'B''

and

C'C''

are concurrent in a point on the Euler line of triangle

ABC

.

geometrycircumcircleEulerconicsprojective geometrygeometry proposed

a^+b^n-c^n

Source: Iran 2005

8/29/2005

Let

a,b,c\in \mathbb N

be such that

a,b\neq c

. Prove that there are infinitely many prime numbers

p

for which there exists

n\in\mathbb N

that

p|a^n+b^n-c^n

.

inductionnumber theoryprime numbersnumber theory proposed

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