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.
be a triangle, and let 
and 
be points on segment 
symmetric with respect to the midpoint of . Let 
denote the circle passing through and tangent to line 
at 
. Similarly, let 
denote the circle passing through and tangent to line 
at 
. Let the circles and intersect again at point 
(
). Prove that 
.
and 
, let us consider the equation
[list=a]
and 
, find the least positive integer 
satisfying this equation. and , there exist infinitely many positive integers satisfying this equation.
denotes the greatest common divisor of positive integers 
and 
.)
events 
that are each continuous and last on a certain time interval. Each event has a weight 
However, one can only choose to attend activities that do not overlap with each other. The goal is to maximize the sum of weights of all activities attended. Prove or disprove that the following algorithm allows for an optimal selection:
consider 
the sum of 
over all 
such that 
and are not compatible.
are 
spheres centered at 
, with radius of 
and , which is remarked as 
and 
, respectively. Given 
points 
outside of (not including the surface of ).
as the number of sets 
such that the midpoint of 
lies entirely inside of .
. There after each term is derived from the previous term by substituting 
for and for 
Thus the first few terms are , , 
,
?
is called frameable if it is possible to draw a regular polygon with sides all whose vertices lie on these lines, and no line contains more than one vertex of the polygon.
are frameable.
is not frameable.
is frameable.
, determine all polynomials 
with integer coefficients such that 
divides 
for every positive integer .
with the smaller base are constructed the squares 
and 
. Prove that the perpendicular bisector of passes through the midpoint of 
.
such that
for all real numbers 
satisfying the equation
has circumcenter 
. The lines 
and 
intersect at 
, while 
and 
intersect at 
and 
and 
intersect at 
. Show that if the triangles and 
are similar(with vertices in that order), than is equilateral.
candidates and judges, where 
is an odd integer. Each candidate is evaluated by each judge as either pass or fail. Suppose that each pair of judges agrees on at most candidates. Prove that ![\[{\frac{k}{m}} \geq {\frac{n-1}{2n}}. \]](http://latex.artofproblemsolving.com/3/8/7/387c8824b28f6422306e3c25b34fccb0a841f4cd.png)
be a triangle. Let 
be the feet of the altitudes from 
respectively. Let 
be the projections of onto line 
. Show that 
.
have the property that 
is a perfect square for infinitely many positive integers 
?
such that 
. There is a point 
lying on the segment 
, and 
. The point 
is symmetrical to 
with respect to 
. Prove that:![\[\frac {AC}{CD} = \frac {BE}{BD - AD}\]](http://latex.artofproblemsolving.com/1/e/8/1e85d6296526b3a32cfa06c1b0520fd47c445598.png)
on such that 
and let 
be the intersection between the perpendicular from to line and .
, and so 
and 
.
be the midpoint of 
. We have 
and 
is cyclic.
, but 
and so 
, from which 
is cyclic too.
and triangles 
and 
are similar and the thesis follows.
be a 
- isosceles triangle. For a point 
, denote of the point w.r.t. the line . Prove that 
.
, i.e. is cyclically. Apply the Ptolemaios' theorem in the quadrilateral :
.
, then the quadrilateral 
is cyclically and 
. See here.
be a - isosceles triangle. For a point , denote of the point w.r.t. the line . Prove that . , i.e. is cyclically. Apply the Ptolemaios' theorem in the quad.rilateral : . , then the quadrilateral is cyclically and .
, 
is cyclic. Then by Ptolemy's, 
The desired condition is then equivalent to 
but this is clear as each pair of lengths are respectively equal.
?
be the mipoint of 
,
the point such that 
.
and since 
,
is cyclic. Also, since 
and 
and also since 
and 
.
and 
,
,
on 
,
,
.
,
.
,
.