Find the minimum possible value of 
.
.
2. For 
, define
, define
Find the set 
3. Let 
satisfy
satisfy
For all 
Show that 
where 
is a constant.
Show that where is a constant.
4. In a badminton tournament, each of 
players play all the other 
players. Each game results in either a win, or a loss. The players then write down the names of those whom they defeated, and also of those who they defeated. For example, if 
beats 
and beats 
then writes the names of both and 
. Show that there will be one person, who has written down the names of all the other players.
players play all the other players. Each game results in either a win, or a loss. The players then write down the names of those whom they defeated, and also of those who they defeated. For example, if beats and beats then writes the names of both and . Show that there will be one person, who has written down the names of all the other players.
and 
Find the ratio 
Show that
(i) There exists an integer 
and a polynomial 
with 
such that
and a polynomial with such that
(ii) Show that 
where 
is described as above.
where is described as above.
7. Find all natural numbers 
for which
for which
is a perfect square.
8. Let 
be a square such that 
lies along the line 
and and 
lie on the parabola 
Find all possible values of sidelength of the square.
be a square such that lies along the line and and lie on the parabola Find all possible values of sidelength of the square.
Πηγή: artofproblemsolving
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