MathDB

11.7

Part of III Soros Olympiad 1996 - 97 (Russia)

Problems(2)

4 unequal circles tangent to 2 circles and a line

Source: III Soros Olympiad 1996-97 R1 11.7 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/29/2024
On the plane there are two circles aa and bb and a line ℓ\ell perpendicular to the line passing through the centers of these circles. It is known that there are 44 unequal circles, each of which touches aa, bb and ℓ\ell. Find the radius of the smallest of these four circles if the radii of the other three are 22, 33 and 66. Also find the ratio of the radii of the circles aa and bb.
geometrytangent circles
x^7+x^2+1=0, x^5-x^4 + x^2- x+1.001=0 (III Soros Olympiad 1996-97 R3 11.7)

Source:

5/31/2024
Let us assume that each of the equations x7+x2+1=0x^7 + x^2 + 1= 0 and x5−x4+x2−x+1.001=0x^5- x^4 + x^2- x + 1.001 = 0 has a single root. Which of these roots is larger?
algebrapolynomial