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.
children, no two of the same height, lined up. Let us call a pair of different children 
good if between them there is no child whose height is greater than the height of one of 
and 
, but less than the height of the other. What is the greatest number of good pairs that could be formed? (Here, and 
are considered the same pair.)
satisfy 
prove in equality.
![\[\sum_{cyc}(x+y)\sqrt{(z+x)(z+y)} \geq 4(xy+yz+zx),\]](http://latex.artofproblemsolving.com/0/4/7/047adb616a6be9f9ed52f2385fa592fb5464e420.png)
and 
.
and 
be elements of the set 
such that 
, 
is a divisor of 
, and 
is a divisor of 
. Determine the largest value that can take.
such that the following equation holds:![\[
7^a + 1 = 2b^2
\]](http://latex.artofproblemsolving.com/e/0/8/e08d3fc7feefd7b4acc4a0f9c7e8bcfa328b5791.png)
have the property that 
is a perfect square for infinitely many positive integers 
?
and 
respectively be the incentre, circumcentre and circumcircle of triangle 
. Points 
are chosen on , such that 
, and point 
is the foot of the altitude from 
to 
. If 
are similarly defined, prove that lines 
and 
concurr on 
.
that is not isosceles at 
, let 
be its incircle, which is tangent to 
at 
, respectively. The lines 
and 
intersect the line passing through and parallel to 
at 
and 
, respectively. The lines passing through 
and perpendicular to 
intersect 
and 
at 
and 
, respectively.
are collinear and that 
are concurrent.
positive integer numbers such that the least common multiple (LCM) of all these numbers is not a perfect square. Mykhailo consecutively hides one of these numbers and writes down the LCM of the remaining 
numbers that are not hidden. What is the maximum number of the written numbers that can be perfect squares?
is a prime number and 
are positive integers. Prove that there exist infinitely such that 
are divisible by 
, let 
if is not squarefree and 
if is a product of 
primes, and let 
be the sum of the divisors of . Prove that for all we have![\[\left|\sum_{d|n}\frac{\mu(d)\sigma(d)}{d}\right| \geq \frac{1}{n}, \]](http://latex.artofproblemsolving.com/8/e/b/8eb88de7cd7ea8e2853dbc80cae24bdc6b77f9be.png)
be a triangle such that 
. The incircle with centre touches 
at 
. Line 
intersects the circumcircle of at 
, and 
again meets at 
. Find 
at 
. Since is the midpoint of the arc 
not containing 
we have 
and 
so is the center of the spiral similarity sending 
to and to 
. Moreover 
so since triangle 
and 
are both isosceles triangles, this spiral similarity sends to aswell. This implies that 
so 
is cyclic, hence 
.
we will denote the power of point 
WRT circle .We have 
It's well known that is the circumcenter of 
,so 
From the above two relations we deduce that 
.This is equivalent to 
,implying that 
.
.
.
But 
,so 
.
.
at let the circumcircle of 
touches the circumcircle of at we want to show that 
are colinear let 
cuts at 
by the kown lemma 
is perpendicular to 
we khow angles 
we know that 
so triangles 
are similar so angles 
so we are done
and let 
and 
be tangency points of the incircle of 
and 
and 
.
,
and 
,
.
and 
be feet of perpendiculars from 
and 
respectively. Prove that 
,
and 
are concurrent.
satisfying: 
.
It is well-known that the midpoint 
is the circumcenter of 
Thus, the inversion 
with center 
fixes 
Hence, 
and 
Therefore, 
and 
Hence, ![\[\angle API = \angle MPA - \angle MPI = \angle MED - \angle MID = \angle IDE = 90^{\circ}.\]](http://latex.artofproblemsolving.com/e/0/9/e09be5ecaa2074fc6e8c087995a6ec272e192c55.png)
sends 
to 
, orthogonal to line 
. Hence 
is internal bisector of 
= 
= 
= 
then: 
= 
or 
= 
)
= 