10.1

Part of VI Soros Olympiad 1999 - 2000 (Russia)

Problems(4)

a + b + ab -> 1999, 2000 starting with1,4 - VI Soros Olympiad 1999-00 Round 1 1

Source:

Let's call the "Soros product" of two different numbers,

a

b

a + b + ab

. Is it possible, based on numbers

1

4

, after repeated application of this operation to the already obtained products, to obtain: a) the number

1999

? b) the number

2000

3(x + y)>= 2xy + 4 (VI Soros Olympiad 1990-00 R1 10.1)

Source:

For real numbers

x,y, \in [1,2]

, prove the inequality

3(x + y)\ge 2xy + 4

algebrainequalities

(sin x]^2 + (cos x]^2 =|tg x| +|ctg x| (VI Soros Olympiad 1990-00 R2 10.1)

Source:

For a real number

a

(a]

the smallest integer that is not less than

a

. Find all real values of

x

for which holds the equality

(\sin x]^2 + (\cos x]^2 =|tg x| +|ctg x|.

algebranumber theorytrigonometry

f(x)f(y)f(z)-f(xyz)=xy+yz+xz+x+y+z (VI Soros Olympiad 1990-00 R3 10.1)

Source:

Find all real functions of a real numbers, such that for any

x

y

z

holds the equality

f(x)f(y)f(z)-f(xyz)=xy+yz+xz+x+y+z.

algebrafunctionalfunctional equation