AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
are given such that 
. Prove the inequality 
+1 w
, let 
denote the smallest possible value of 
where 
are sets such that 
and 
whenever 
. Determine for each positive integer .
and 
intersect at 
. The common tangent line nearer to 
of the two circles touches 
at 
respectively. Let 
be the symmetric points of with respect to respectively. The circumcircle of triangle 
intersects circles and at points 
respectively which are distinct from . Prove that the circumradii of the triangles 
and 
are equal.
, 
. 
intersects 
at 
. A line 
cuts segment 
at 
respectively. If 
and 
, prove that the diameter of the circumcircle of 
equals 
.
are points on a circle 
, satisfying 
, 
. meets 
at 
. 
is on 
such that 
. Suppose 
intersects again at 
. 
is the reflection of 
wrt 
. Prove that 
lies on the same circle.
in its exterior
we draw the triangles 
so that 
, 
, 
.
and
and be on segment of an acute triangle 
such that 
and 
. Let and 
be the points on 
and 
, respectively, such that is the midpoint of 
and is the midpoint of 
. Prove that the intersection of 
and 
is on the circumference of triangle .
has a right angle at 
. Let be the point on line 
such that 
and 
lies between and 
. Point 
is chosen so that 
and is the bisector of 
. Point 
is chosen so that 
and 
is the bisector of 
. Let be the midpoint of . Let 
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 a point such that 
and 
are tangent to the circumcircle of . Let 
be the reflection of over and 
be the reflection of over 
. If 
is the circumcenter of 
, help Oscar prove that 
is perpendicular to .
be a positive integer. Lexi has a dictionary 
consisting of some -letter strings containing only the letters and . Lexi would like to write either the letter or the letter in each cell of a 
grid so that each column contains a string from when read from top-to-bottom and each row contains a string from when read from left-to-right.
such that if contains at least different strings, then Lexi can fill her grid in this manner, no matter what strings are in ?
, 
, 
contained in the interval ![$\left [\frac {1}{2},1 \right ]$](http://latex.artofproblemsolving.com/f/9/f/f9ff3f08482967aa0c76d307674d0f7479ac3b89.png)
. Prove that 
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.
of the set 
such that 
, we have that 
and $\left\lfloor{\sqrt{n}\right\rfloor}+1$ are both in .
$\left\lfloor{\sqrt{m}\right\rfloor}+1\ \in A$. But, for 
, $\left\lfloor{\sqrt{n^{2}+4}\right\rfloor}+1=n+1$. 
and we must have $\left\lfloor{\sqrt{a}\right\rfloor}+1\geq a\geq 2$, which means 
itself.
for some integer 
![$ [\sqrt a] + 1\in A$](http://latex.artofproblemsolving.com/3/5/6/356271de257b46a4898e37e6d66ee8447f406348.png)
![$ [\sqrt a] + 1\ge a$](http://latex.artofproblemsolving.com/c/d/a/cda26b3ec5f5eddca0a8eb6e54b59c9f7d669bfb.png)
![$ \sqrt a\ge[\sqrt a]$](http://latex.artofproblemsolving.com/9/b/f/9bf647b9edd0d03e713d268b36f409ec18276fef.png)
so 
and so 
and so 
in natural numbers m and n.
and 
so 