P1. Do there exist two real monic polynomials 
and 
of degree 3, such that the roots of 
are nine pairwise distinct nonnegative integers that add up to 
?
and of degree 3, such that the roots of are nine pairwise distinct nonnegative integers that add up to ?
(In a monic polynomial of degree 3, the coefficient of 
is 
.)
is .)
P2. Determine the least integer 
for which the following story could hold true: In a chess tournament with 
players, every pair of players plays at least 
and at most games against each other. At the end of the tournament, it turns out that every player has played a different number of games.
for which the following story could hold true: In a chess tournament with players, every pair of players plays at least and at most games against each other. At the end of the tournament, it turns out that every player has played a different number of games.
P3. Let 
be positive reals for which:
be positive reals for which:
.
Prove that:
.
P4. 
is quadrilateral inscribed in a circle 
.Lines 
and 
intersect at 
and lines
and 
intersect at 
. Prove that the circle with diameter 
and circle are orthogonal.
is quadrilateral inscribed in a circle .Lines and intersect at and lines and intersect at . Prove that the circle with diameter and circle are orthogonal.
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