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are there such that 
and 
?
be real numbers such that 
Prove that
Let be real numbers such that 
Prove that
and 
, then we can deduce 
\le 2 $ ?
satisfying 
. Prove that:![\[
0 \le (x^2 + 1)(y^2 + 1) + 4(x - 1)(y - 1) \le 164.
\]](http://latex.artofproblemsolving.com/4/2/d/42d0b88388462f16dd99d928b94024e689ba7c2e.png)
be a sequence defined by 
. Then the product of four consecutive terms in can be written as the product of two terms in . Find 
if 
.
that satisfy 
with 
, 
and area 
. The circle 
with diameter 
meets 
at 
. Let the orthocenter of the triangle be 
, connect 
and extend to meet at 
such that 
and 
is the diameter of . Draw the circumcircle 
of 
, chord 
of is tangent to and it passes through , compute .
of trapezoid 
, point 
is selected such that 
. It is known that 
is a parallelogram, and its diagonals meet at point 
. On the segment 
, point 
is selected. Prove that if 
, then 
.
of trapezoid , point is selected such that . It is known that is a parallelogram, and its diagonals meet at point . On the segment , point is selected. Prove that if , then .
.
.
.
.
and 
.
.
and natural number 
,
and 
.
