1999 Italy TST

Part of Italy TST

Subcontests

(3)

1

A subset of X with at least root 2n elements

X

n

-element set and let

A_1,\ldots ,A_m

X

|A_i|=3

i=1,\ldots ,m

|A_i\cap A_j|\le 1

for any two distinct indices

i,j

.
Show that there exists a subset of

X

\lfloor\sqrt{2n}\rfloor

elements which does not contain any of the

A_i

1

Function doesn't exist when power of y is > 1

(a) Find all strictly monotone functions

f:\mathbb{R}\rightarrow\mathbb{R}

such that f(x+f(y))=f(x)+y \text{for all real}\ x,y. (b) If

n>1

is an integer, prove that there is no strictly monotone function

f:\mathbb{R}\rightarrow\mathbb{R}

such that f(x+f(y))=f(x)+y^n \text{for all real}\ x, y.

1

Geometry inequality involving tangent to incircle

D

E

be points on sides

AB

AC

respectively of a triangle

ABC

DE

BC

and tangent to the incircle of

ABC

DE\le\frac{1}{8}(AB+BC+CA)