11.3

Part of III Soros Olympiad 1996 - 97 (Russia)

Problems(3)

y=x^4+a , x=\1/y^4+b (III Soros Olympiad 1996-97 R1 11.3)

Source:

Find the greatest

a

for which there is

b

such that the system

\begin{cases} y=x^4+a \\ x=\dfrac{1}{y^4}+b \end{cases}

has exactly two solutions.

algebrasystem of equations

root of x^3- x- 3 = 0 vs \sqrt[5]{13} (III Soros Olympiad 1996-97 R2 11.3)

Source:

Prove that the equation x^3- x- 3 = 0 has a unique root. Which is greater, the root of this equation or

\sqrt[5]{13}

? (Use of a calculator is prohibited.)

Comparealgebrainequalities

2 circles tangent to an arc and a line

Source: III Soros Olympiad 1996-97 R3 11.3 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

AB

is drawn in a certain circle. The smaller of the two arcs

AB

corresponds to a central angle of

120^o

p

to this arc is drawn. Two circles with radii

R

r

are constructed, touching this smaller arc

AB

and straight lines

AB

p

. Find the radius of the original circle.

geometrytangent circles