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.
, where 
, and 
are positive integers that satisfy
and 
.
+1 w
be real positive numbers such that: 
. Prove that: 
: 
such that:
for all x,y E 
, 
, 
, 
be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, 
. Prove that 
for all 
.
with rational coordinates lie on a plane. It turned out that the distance between every pair of points is an integer. Prove that there exist points 
with integer coordinates such that 
for every pair 
be a trapezoid inscribed in circle 
, 
. Let 
is the symmetric point of 
across 
, 
intersects at 
. Let 
is midpoint of and 
is orthocenter of triangle 
. On 
take a point 
so that 
. Prove that: 
coincide
be integer numbers 
satisfy that:
. Prove that: are even
is a perfect square number
can’t be any power 
of a positive integer number
, 
be the orthocenter of it and 
be any point on the side 
. The points 
are on the segments 
, respectively, such that the points 
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
. Prove that the points 
and lie on the same line.
is given where are positive reals. Prove that:
with the following property:
number is a divisor of the number if and only if the number is a divisor of the corresponding three-digit number 
.
and need not be different.
satisfying![\[f(x+f(y))=x+f(f(y))\]](http://latex.artofproblemsolving.com/8/d/a/8daa6bcb3155768ee1ba4bd22beb53a85a6e3923.png)
and 
, with the additional constraint 
.
be positive odd integers. Prove that 
doesn't divide 
.
is given, construct an equilateral triangle 
in space such that lines 
pass through a given point.
such that
and 
Find the minimum 
