MathDB

1

circumcenter of a triangle lies on a given circle

A

that lies on circle

c(o,R)

(with center

O

and radius

R

). Let

(e)

be the tangent of the circle

c

at point

A

and a line

(d)

that passes through point

O

and intersects

(e)

at point

M

and the circle at points

B,C

(let

B

lie between

O

and

A

). If

AM = R\sqrt3

, prove that a) triangle

AMC

is isosceles. b) circumcenter of triangle

AMC

lies on circle

c

.

3

1

among d_1=x_2-x_1, d_2=x_3-x_2, ..., d_{101}=x_{102}-x_{101}, 26 are equal

Let

x_1,x_2,x_3,...,x_{102}

be natural numbers such that

x_1<x_2<x_3<...<x_{102}<255

. Prove that among the numbers

d_1=x_2-x_1, d_2=x_3-x_2, ..., d_{101}=x_{102}-x_{101}

there are at least

26

equal.

2

1

(a^2b^2)/(a+b)+(b^2c^2)/(b+c)+(c^2a^2)/(c+a) <=(a^3+b^3+c^3)/2

If

a,b,c

are positive real numbers, prove that

\frac{a^2b^2}{a+b}+\frac{b^2c^2}{b+c}+\frac{c^2a^2}{c+a}\le \frac{a^3+b^3+c^3}{2}

4

1

solve in Z: xy = 3 (x+y)-1

Product of two integers is

1

less than three times of their sum. Find those integers.

2008 Greece JBMO TST

Subcontests

circumcenter of a triangle lies on a given circle

among d_1=x_2-x_1, d_2=x_3-x_2, ..., d_{101}=x_{102}-x_{101}, 26 are equal

(a^2b^2)/(a+b)+(b^2c^2)/(b+c)+(c^2a^2)/(c+a) <=(a^3+b^3+c^3)/2

solve in Z: xy = 3 (x+y)-1

2008 Greece JBMO TST

Subcontests

circumcenter of a triangle lies on a given circle

among d_1=x_2-x_1, d_2=x_3-x_2, ..., d_{101}=x_{102}-x_{101}, 26 are equal

(a^2b^2)/(a+b)+(b^2c^2)/(b+c)+(c^2a^2)/(c+a) &lt;=(a^3+b^3+c^3)/2

solve in Z: xy = 3 (x+y)-1

(a^2b^2)/(a+b)+(b^2c^2)/(b+c)+(c^2a^2)/(c+a) <=(a^3+b^3+c^3)/2