9.4

Part of III Soros Olympiad 1996 - 97 (Russia)

Problems(3)

2x2 x^4-2x^3+x=y^2-y , y^4-2y^3+y=x^2-x (III Soros Olympiad 1996-97 R1 9.4)

Source:

Solve the system of equations

\begin{cases} x^4-2x^3+x=y^2-y \\ y^4-2y^3+y=x^2-x \end{cases}

algebrasystem of equations

x^2+ax+1996=0, x^2+1996x+a=0 (III Soros Olympiad 1996-97 R2 9.4)

Source:

At what values of

a

does each of the equations

x^2 + ax + 1996 = 0

x^2 + 1996x + a = 0

have two integer roots?

computational geo with 2 circles

Source: III Soros Olympiad 1996-97 R3 9.4 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

AB = a

is drawn in a circle of radius

B

. A circle with center on line

AB

A

and intersects this circle a second time at point

C

M

be an arbitrary point of the second circle. Straight lines

MA

MC

intersect the first circle a second time at points

P

Q

PQ