MathDB

0<a,b,c,d<1 => 1 + ab + bc + cd + da + ac + bd > a+b+c+d

Source: JBMO 2008 Shortlist A6

10/14/2017

If the real numbers

a, b, c, d

are such that

0 < a,b,c,d < 1

, show that

1 + ab + bc + cd + da + ac + bd > a + b + c + d

.

JBMOalgebrainequalities

2008 JBMO Shortlist G6

Source: 2008 JBMO Shortlist G6

10/10/2017

Let

ABC

be a triangle with

\angle A<{{90}^{o}}

. Outside of a triangle we consider isosceles triangles

ABE

and

ACZ

with bases

AB

and

AC

, respectively. If the midpoint

D

of the side

BC

is such that

DE \perp DZ

and

EZ = 2 \cdot ED

, prove that

\angle AEB = 2 \cdot \angle AZC

.

JBMOgeometry

for n, exists p so that p|n and f(n) = f(n/p)-f(p)

Source: JBMO 2008 Shortlist N6

10/14/2017

Let

f : N \to R

be a function, satisfying the following condition: for every integer

n > 1

, there exists a prime divisor

p

of

n

such that

f(n) = f \big(\frac{n}{p}\big)-f(p)

. If

f(2^{2007}) + f(3^{2008}) + f(5^{2009}) = 2006

, determine the value of

f(2007^2) + f(2008^3) + f(2009^5)

JBMOnumber theory

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