[/quote]
,
are given such that 
.
in natural numbers m and n.
,
denote the smallest possible value of 
where 
are sets such that 
and 
whenever 
.
and 
intersect at 
.
of the two circles touches 
at 
respectively. Let 
be the symmetric points of 
intersects circles 
respectively which are distinct from 
and 
are equal.
,
.
intersects 
at 
.
cuts segment 
at 
respectively. If 
and 
,
equals 
.
are points on a circle 
,
,
.
at 
.
is on 
such that 
.
intersects 
.
is the reflection of 
wrt 
.
lies on the same circle.
in its exterio
so that 
,
,
.
and
such that 
and 
.
be the points on 
and 
,
and 
.
and 
is on the circumference of triangle 
has a right angle at 
.
such that 
and 
lies between 
.
is chosen so that 
and 
.
is chosen so that 
and 
is the bisector of 
.
be the point such that 
is a parallelogram. Prove that 
and 
are concurrent.
such that the number of 
which satisfy the following condition is minimal:
has periodic 
and 
are tangent to the circumcircle of 
be the reflection of 
be the reflection of 
.
is the circumcenter of 
,
is perpendicular to 
be a positive integer. Lexi has a dictionary 
consisting of some 
grid so that each column contains a string from 
such that if 
,
,
contained in the interval ![$\left [\frac {1}{2},1 \right ]$](http://latex.artofproblemsolving.com/f/9/f/f9ff3f08482967aa0c76d307674d0f7479ac3b89.png)
.
grid. They start in one of the grid's squares and can move from one square to another adjacent square via a street (for each pair of neighboring squares on the grid, there is a street connecting them). Some streets are dangerous. If Davi or George pass through a dangerous street, they get scared and swap who is leading the city tour. Their goal is to pass through every block of Fortaleza exactly once. However, if the city tour ends with George in command, the entire world becomes unemployed and everyone starves to death. Given that there is at least one street that is not dangerous, prove that Davi and George can achieve their goal without everyone dying of hunger.
Y by Adventure10, Mango247, and 1 other user
Consider 
a semicircle of diameter 
. A parallel to cuts the semicircle at 
such that 
separates 
. The parallel at through 
intersects the semicircle the second time at 
. Let 
be the intersection point of the lines 
and 
. The parallel through at cuts in 
. Prove that 
is tangent to .
Author: Cosmin Pohoata
a semicircle of diameter 
. A parallel to cuts the semicircle at 
such that 
separates 
. The parallel at through 
intersects the semicircle the second time at 
. Let 
be the intersection point of the lines 
and 
. The parallel through at cuts in 
. Prove that 
is tangent to .Author: Cosmin Pohoata
is a parallellogram, we have that 
.
such that 
.![\[\left|PO\right|=\left|PA\right|+\left|AO\right|=\left|FD\right|+\left|DU\right|=\left|FU\right| \hspace{1cm}(1)\]](http://latex.artofproblemsolving.com/f/4/7/f471aece004d10625b5fe0c45f58a046fcd70b6c.png)
![\[\left|BU\right|=\left|OD\right|=\left|OC\right| \hspace{1cm}(2)\]](http://latex.artofproblemsolving.com/c/3/f/c3f2075ece3f1be803671760f8b5810787f2b99c.png)
![\[\angle DUB = \angle DOB = \angle COP \hspace{1cm}(3)\]](http://latex.artofproblemsolving.com/6/8/4/684b91c934f76f13bf1de6c65ef0c9a333d56ded.png)
and 
follows that 
,
.
.
,
,
is an issoscel trapezoid.
.
,
is cyclic, and 
.
,
