[/quote]
,
such that![\[
f\left(\frac{x^2 - f(x)}{f(x) - 1}\right) = x
\]](http://latex.artofproblemsolving.com/8/0/0/8009a4709ffd6708023d67db2b544e605d1fbe68.png)
for all real numbers 
satisfying 
.
such that![\[ {a^2+b\over b^2-a}\quad\mbox{and}\quad{b^2+a\over a^2-b} \]](http://latex.artofproblemsolving.com/e/3/b/e3bc96af9f635e1c10e21dc91894c8afbf113a6d.png)
.
lie on segments 
and 
,
and 
intersect at point 
.
and 
intersect at a second point 
.
intersect at a second point 
.
lies on segment 
such that 
.
and 
are similar.
| 
with 
are positive integers and 
is prime 
)
,
with integer coefficients and for any prime 
,
integers, such that![\[\bigcup_{s \in S} \{s,\; Q(s),\; Q(Q(s)),\; Q(Q(Q(s))),\; \dots\}\]](http://latex.artofproblemsolving.com/f/9/1/f919f2391121c2b0551d29d4f09ea6cde631a8ba.png)
contains a complete residue system modulo 
such that there exists a prime number 
is a power of 
.
where 
from the non-zero reals to the non-zero reals, such that ![\[f(x+y) \left(f(x) + f(y)\right) = f(xy)\]](http://latex.artofproblemsolving.com/2/f/e/2fe8a09e628f60faedaa07ceb154891375763a83.png)
for all non-zero reals 
such that 
.
and 
such that 
.
players. Every player played every other player once, with no draws. In addition, each player had a numerical rating before the tournament began, with no two players having equal ratings. It turns out there were exactly 
games in which the lower-rated player beat the higher-rated player. Prove that there is some player who won no less than 
and no more than 
games.
be positive integers such that
,
is a permutation of 
.
.
.
be a real numbers such that 
P
Where 
Y by Rounak_iitr
A trapezoid 
with right angles at 
and 
has an inscribed circle with center 
and radius 
. Let the lengths of the parallel sides 
and 
be 
and 
, and the intersection of the diagonals 
and 
be 
.
1. Prove that the perpendicular from to one of the trapezoid sides has the length .
2. Determine the distance between and as a function of and .
with right angles at 
and 
has an inscribed circle with center 
and radius 
. Let the lengths of the parallel sides 
and 
be 
and 
, and the intersection of the diagonals 
and 
be 
.1. Prove that the perpendicular from
to one of the trapezoid sides has the length .2. Determine the distance between
and as a function of and .
.
of the incircle; the line 
is a tangent line if 
.
intersects the line 
in the point 
.
.
