[/quote]
,
,
,
at points 
,
,
,
be the diametrically opposite point of 
and 
intersect the incircle again at 
and 
,
,
,
are concurrent, i.e., the lines intersect at the same point.
be positive real numbers . Prove that :
and 
be two similar right-angled triangles (in the same plane) with 
such that 
and 
lie on opposite sides of the line 
.
,
.
is written on the blackboard. Every time, Carmela can erase the number 
is a perfect square. Prove or disprove that all positive integers 
can be written exactly once on the blackboard.
,
be some point lying on 
such that 
lie on 
be some circle passing through 
be its center. Let 
be some point lying on the circumcircle of 
.
is tangent to 
is tangent to 
be an acute triangle, and let 
and 
meet the circumcircle of 
and 
.
,
be the midpoint of 
and 
exist on 
and 
are tangent to 
,
,
,
.
at the orthocenter of 
be its orthocenter, and let 
,
be a point lying on 
such that 
.
.
be its incircle, and let 
be the midpoint of minor arc 
on the circumcircle of 
at points 
,
the intersection of 
and 
.
and 
denote the 
,
,
and 
,
be the intersection of 
and 
.
and 
are tangent to each other.
and 
be circles tangent to 
meets 
.
similarly.
and 
.
and 
.
,
.
and 
touch externally at 
and the radius of 
is any point on 
and 
are points on 
and 
are tangent to 
,
intersect 
,
respectively. 
is the intersection of the tangent at 
.
be non-negative real numbers.Prove that :![\[ \sqrt{x^4+y^4 } +(2+\sqrt{2})xy \geq x^2+y^2 \]](http://latex.artofproblemsolving.com/6/2/3/623d50afffd81b2783ff98568791164b53747567.png)
be real numbers such that 
Prove that
Let 
Prove that
Prove that
Prove that
satisfying![\[\begin{aligned}
\begin{cases}
a+b+c+d=0,\\
a^2+b^2+c^2+d^2=12,\\
abcd=-3.\\
\end{cases}
\end{aligned}\]](http://latex.artofproblemsolving.com/8/b/1/8b1923e0303c0aa79332ee4540ee1885a9cf3f08.png)
intersects line 
.
Y by
Eight circles of radius 
can be placed tangent to side 
of 
such that the first circle is tangent to 
, subsequent circles are externally tangent to each other, and the last is tangent to 
. Similarly, 
circles of radius 
can also be placed along in this manner. The inradius of is 
, where 
and 
are relatively prime positive integers. Find 
.
can be placed tangent to side 
of 
such that the first circle is tangent to 
, subsequent circles are externally tangent to each other, and the last is tangent to 
. Similarly, 
circles of radius 
can also be placed along in this manner. The inradius of is 
, where 
and 
are relatively prime positive integers. Find 
.
,
.
,
is the inradius of 
,
and 
.
.
,
,
.
.![\[\frac{r-34}{r} = \frac{14 \cdot 34}{BC}, \quad \frac{r-1}{r} = \frac{4046 \cdot 1}{BC},\]](http://latex.artofproblemsolving.com/6/c/7/6c7507dae4ca4e6c24ae9d8061162f4c5d398472.png)
.
varies linearly, the distance 
between the centers of circles of two circles of radius 
,
,
,
,
.
,
.![\[ \frac{r - 34}{476} = \frac{r - 1}{4046} \to r = \frac{192}{5} \to \boxed{197}.\]](http://latex.artofproblemsolving.com/1/6/0/1605624c09237b1a2c044a84eeca9eae3e9855b6.png)
![\[\frac{14\cdot 34}{BC}=\frac{r-34}{r}\qquad\text{and}\qquad\frac{4046}{BC}=\frac{r-1}{r},\]](http://latex.artofproblemsolving.com/9/3/a/93abd0a964e50da2e5c030821e0874769dd77018.png)
so![\[\frac{r-34}{r-1}=\frac{2}{17}\implies r=\frac{192}{5}\implies\boxed{197}.\]](http://latex.artofproblemsolving.com/b/3/c/b3cfb51302ed93230d6cbc524d20b81f953763b8.png)
= 
which yields 
where we have used the fact that the ratio of heights in similar triangles is the same scale factor k as the ratio of sides.
