MathDB

11.2

Part of III Soros Olympiad 1996 - 97 (Russia)

Problems(3)

f(f(x)) = 1 + 2x, f(f(x)) = 1 - 2x (III Soros Olympiad 1996-97 R1 11.2)

Source:

5/29/2024
Is there a function f(x)f(x) defined and continuous on RR such that:
a) f(f(x))=1+2xf(f(x)) = 1 + 2x ?
b) f(f(x))=1−2xf(f(x)) = 1 - 2x ?
algebrafunctional equation
min y= x^2/8+x cosx +cos2x (III Soros Olympiad 1996-97 R2 11.2)

Source:

5/31/2024
Find the smallest value of the expression: y=x28+xcos⁡x+cos⁡2xy=\frac{x^2}{8}+x \cos x +\cos 2x
algebrainequalitiestrigonometry
45^o rotation gives y=x^3+ax^2+19x+97 (III Soros Olympiad 1996-97 R3 11.2)

Source:

5/31/2024
It is known that the graph of the function y=f(x)y = f(x) after a rotation of 45o45^o around a certain point turns into the graph of the function y=x3+ax2+19x+97y = x^3 + ax^2 + 19x + 97. At what aa is this possible?
geometrygeometric transformationrotation