11.5

Part of V Soros Olympiad 1998 - 99 (Russia)

Problems(3)

min (x -y)^2 + (z - u)^2 (V Soros Olympiad 1998-99 Round 1 11.5)

Source:

Find the smallest value of the expression

(x -y)^2 + (z - u)^2,

(x -1)^2 + (y -4)^2 + (z-3)^2 + (u-2)^2 = 1.

algebrainequalities

edge of a cube, distances from a plane (V Soros Olympiad 1998-99 Round 2 11.5)

Source:

It is known that the distances from all the vertices of a cube and the centers of its faces to a certain plane (

14

values in total) take two different values. The smallest is

1

. What can the edge of a cube be equal to?

geometry3D geometry

x^4 - 5x + a = 0 (V Soros Olympiad 1998-99 Round 3 11.5)

Source:

Find all values of the parameter

a

for which the sum of all solutions (meaning real solutions) of the equation

x^4 - 5x + a = 0

a