10.5

Part of III Soros Olympiad 1996 - 97 (Russia)

Problems(3)

\frac{x+y}{1+xy}=\frac{1-2y}{2-y} (III Soros Olympiad 1996-97 R1 10.5)

Source:

Solve the system of equations

\begin{cases} \dfrac{x+y}{1+xy}=\dfrac{1-2y}{2-y} \\ \dfrac{x-y}{1-xy}=\dfrac{1-3x}{3-x} \end{cases}

algebrasystem of equations

circumcenter constrcuction using at most 5 circles, given more circle

Source: III Soros Olympiad 1996-97 R2 10.5 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

A circle is drawn on a plane, the center of which is not indicated. On this circle, point

A

is marked and a second circle with center at

A

is constructed. The second circle has a radius greater than the radius of the first and intersects the first at two points. Construct the center of the first circle using only a compass, drawing no more than five more circles.

geometryconstructioncircles

area of quad. wanted, starting with intersecting circles

Source: III Soros Olympiad 1996-97 R3 10.5 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

Two circles intersect at two points

A

B

. The radii of these circles are equal to

R

r

, respectively; the angle between the radii going to the points of intersection is equal to

a

KM

b

is taken in a circle of radius

r

. Straight lines

KA

KB

MA

MB

intersect the other circle for second time at four points. Find the area of the quadrilateral with vertices at these points.