2008 Alexandru Myller

Part of Alexandru Myller

Subcontests

(4)

3

independent sum of squares involving two concentric circles

C_1,C_2

be two distinct concentric circles, and

BA

be a diameter of

C_1.

Choose the points

M,N

C_1,C_2,

respectively, but not on the line

BA.

a) Show that there are unique points

P,Q

MA,MB,

respectively, so that

N

is the middle of

PQ.

b) Prove that the value

AP^2+BQ^2

does not depend on

M,N.

Mihai Piticari, Mihail Bălună

leveraging the continuity of a function

Let be a function

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

satisfying the following properties:

\text{(i)}

is continuous on the rational numbers.

\text{(ii)} f(x)<f\left( x+\frac{1}{n}\right) ,

x

n.

f

is increasing.
Gabriel Mârşanu, Mihai Piticari

units as many as nilpotent elements implies there are a power of 2 of elements

In a certain ring there are as many units as there are nilpotent elements. Prove that the order of the ring is a power of

2.

Dinu Şerbănescu

4

4

4