MathDB

1

Function: f(f(x) + y) = 2x + f(f(y) − x)

f : \mathbb{R} \to \mathbb{R}

that satisfy the condition f(f(x) + y) = 2x + f(f(y) - x) \text{ for all } x, y \in\mathbb{R}.

4

1

If the resulting quad is cyclic, then ABC must be isosceles

Point

P

lies on the median from vertex

C

of a triangle

ABC

. Line

AP

meets

BC

at

X

, and line

BP

meets

AC

at

Y

. Prove that if quadrilateral

ABXY

is cyclic, then triangle

ABC

is isosceles.

2

1

Czech-Polish-Slovak 2003 Geometry

In an acute-angled triangle

ABC

the angle at

B

is greater than

45^\circ

. Points

D,E, F

are the feet of the altitudes from

A,B,C

respectively, and

K

is the point on segment

AF

such that

\angle DKF = \angle KEF

. (a) Show that such a point

K

always exists. (b) Prove that

KD^2 = FD^2 + AF \cdot BF

.

1

System of equations: max(x_i, i)=x_{i+1}

Given an integer

n \ge 2

, solve in real numbers the system of equations \begin{align*} \max\{1, x_1\} &= x_2 \\ \max\{2, x_2\} &= 2x_3 \\ &\cdots \\ \max\{n, x_n\} &= nx_1. \\ \end{align*}

3

1

a,b,c and pa,qb,rc triangles with equal areas

Numbers

p,q,r

lies in the interval

(\frac{2}{5},\frac{5}{2})

nad satisfy

pqr=1

. Prove that there exist two triangles of the same area, one with the sides

a,b,c

and the other with the sides

pa,qb,rc

.

2003 Czech-Polish-Slovak Match

Subcontests

Function: f(f(x) + y) = 2x + f(f(y) − x)

If the resulting quad is cyclic, then ABC must be isosceles

Czech-Polish-Slovak 2003 Geometry

System of equations: max(x_i, i)=x_{i+1}

a,b,c and pa,qb,rc triangles with equal areas