2011 Bogdan Stan

Part of Bogdan Stan

Subcontests

(4)

3

Interesting thing related to a kind of Fourier sum

Let be a natural number

n,

\text{n-tuplets}

of real numbers

a:=\left( a_1,a_2,\ldots, a_n \right) , b:=\left( b_1,b_2,\ldots, b_n \right) ,

and the function

f:\mathbb{R}\longrightarrow\mathbb{R}, f(x)=\sum_{i=1}^na_i\cos \left( b_ix \right)

. Prove that if the numbers of

b

are all positive and pairwise distinct,
a) then,

f\ge 0

implies that the numbers of

a

are all equal. b) if the numbers of

a

are all nonzero and

f

is periodic, then the ratio of any two numbers of

b

is rational.
Marin Tolosi

7 planar vectors; nice problem

Show that among any seven coplanar unit vectors there are at least two of them such that the magnitude of their sum is greater than

\sqrt 3.

Ion Tecu and Teodor Radu

Meticulous problem about lateral derivatives

Let be an open interval

I

and a convex function

f:I\longrightarrow\mathbb{R} .

Prove that the lateral derivatives of

f

are left-continuous on

\mathbb{R}

and also right-continuous on

\mathbb{R} .

4

4

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