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f(x+2^y)=f(2^x)+f(y) (I Soros Olympiad 1994-95 R1 10.6 11.6)

Source:

7/31/2021

Find all functions

f:R\to R

such that for any real

x, y

,

f(x+2^y)=f(2^x)+f(y)

algebrafunctional equation

max of inradii and exradii in different circles

Source: I Soros Olympiad 1994-95 Round 2 10.6 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

5/25/2024

The radius of the circle inscribed in triangle

ABC

is equal to

r

, and the radius of the circle tangent to the segment

BC

and the extensions of sides

AB

and

AC

(the exscribed circle corresponding to angle

A

) is equal to

R

. A circle with radius

x < r

is inscribed in angle

\angle BAC

. Tangents to this circles passing through points

B

and

C

and different from

BA

and

AC

intersect at point

A'

. Let

y

be the radius of the circle inscribed in triangle

BCK

. Find the greatest value of the sum

x + y

as x changes from

0

to

r

. (In this case, it is necessary to prove that this largest value is the same in any triangle with given

r

and

R

).

geometryexradiusgeometric inequality

all the turtles will be at the vertices of some convex polygon.

Source: I Soros Olympiad 1994-95 Ukraine R2 10.6 https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics

6/6/2024

Several (at least three) turtles are crawling along the plane, the velocities of which are constant in magnitude and direction (all are equal in magnitude, but pairwise different in direction). Prove that regardless of the initial location, after some time all the turtles will be at the vertices of some convex polygon.

combinatoricscombinatorial geometry

10.6

Problems(3)