10.1

Part of II Soros Olympiad 1995 - 96 (Russia)

Problems(3)

a^2x^2 + y^2 + z^2 >= ayz+xy+xz (II Soros Olympiad 1995-96 R1 10.1)

Source:

Find all values of

a

for which the inequality

a^2x^2 + y^2 + z^2 \ge ayz+xy+xz

x

y

z

inequalitiesalgebra

min max y=\sqrt{7+5cosx}-cosx (II Soros Olympiad 1995-96 R2 10.1)

Source:

Find the largest and smallest value of the function

y=\sqrt{7+5\cos x}-\cos x.

trigonometryalgebrainequalities

x^2-ax +21 = 0 (II Soros Olympiad 1997-98 R3 10.1)

Source:

Find the smallest

a

for which the equation

x^2-ax +21 = 0

has a root that is a natural number.