9

Part of 1996 IMO Shortlist

Problems(2)

Determine maximum and minimum value of a(n) over n <= 1996

Source: IMO Shortlist 1996, A9

Let the sequence a(n), n \equal{} 1,2,3, \ldots be generated as follows with a(1) \equal{} 0, and for

n > 1:

a(n) \equal{} a\left( \left \lfloor \frac{n}{2} \right \rfloor \right) \plus{} (\minus{}1)^{\frac{n(n\plus{}1)}{2}}. 1.) Determine the maximum and minimum value of

a(n)

n \leq 1996

n \leq 1996

for which these extreme values are attained. 2.) How many terms

a(n), n \leq 1996,

are equal to 0?

floor functionmodular arithmeticalgebraSequencerecurrence relationIMO Shortlist

Polygon F and d^2 - h^2 >= p^2 /4

Source: IMO Shortlist 1996, G9

In the plane, consider a point

X

\mathcal{F}

(which is not necessarily convex). Let

p

denote the perimeter of

\mathcal{F}

d

be the sum of the distances from the point

X

to the vertices of

\mathcal{F}

h

be the sum of the distances from the point

X

to the sidelines of

\mathcal{F}

. Prove that d^2 \minus{} h^2\geq\frac {p^2}{4}.

geometryperimeterinequalitiesPythagorean Theoremgeometric inequalityIMO Shortlist