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.
of positive integers, for which![$$(a, b) + 3[a, b] = a^3 - b^3$$](http://latex.artofproblemsolving.com/b/8/3/b83f52a0152dc39b6d966880a1d6ca824a0f3786.png)
denotes the greatest common divisor of , and ![$[a, b]$](http://latex.artofproblemsolving.com/d/a/2/da2e551d2ca2155b8d8f4935d2e9757722c9bab6.png)
denotes the least common multiple of .
is an acute angle triangle such that 
and 
. Let's denote by 
the center of the circumscribed circle of the triangle and 
the intersection of altitudes of this triangle. Line 
intersects 
in point 
and 
in point 
. Find the value of the ration 
.
. Let 
be a point on side 
, and 
be a point on the extension of 
such that 
. Let be the circumcenter of 
, and be a point on the side extension of 
satisfying 
. Line BP intersects AC at point Q. Prove that 
of positive integers with 
prime and ![\[ a^p=b!+p. \]](http://latex.artofproblemsolving.com/9/f/6/9f674dabcb364872cef4153198e2e362ae40a2fd.png)
satsfying 
and:![\[ f(2n)f(2n+1)=9f(n)^2+3f(n)\]](http://latex.artofproblemsolving.com/b/c/1/bc18d6ad72589f012dcab7836228b3ed7ee061c2.png)
be positive reals such that 
. Prove that ![\[18\sum_{\text{cyc}}\frac{1}{(3-c)(4-c)}+2(ab+bc+ca)\ge 15. \]](http://latex.artofproblemsolving.com/5/d/5/5d5c47a52d2d6ea7d13ce1c2fa14a08425bbfa5d.png)
Proposed by David Stoner
such that 
. Find the value of 
consisting of a prime number and two positive integers 
and 
such that 
and 
are both powers of .
board and made 
moves so that it visited all squares and returned to the starting square. At every moment, the distance from the center of the square the king was on to the center of the board was calculated. A move is called 
if this distance becomes smaller after the move. Find the maximum possible number of pleasant moves. (The chess king moves to a square adjacent either by side or by corner.)
be a tetrahedron such that 
, 
be the radius of its inscribed sphere and be the orthocenter of triangle . Prove that 
be a tetrahedron such that 
, 
, 
and be an orthocenter of the triangle .![$$OH\leq\frac{\sqrt[3]{HA\cdot HB\cdot HC}}{\sqrt2}.$$](http://latex.artofproblemsolving.com/c/9/f/c9fa62742e000e3e52308c61006371b6a4aea8af.png)
be a tetrahedron such that , , and be an orthocenter of the triangle .
be a tetrahedron such that , , and be an orthocenter of the triangle . Then foot of on plane is orthocenter of triangle .
where 
.
be a tetrahedron such that , , and be an orthocenter of the triangle .
and let 
be reals such that 
satisfying
Prove
plane 
is perpendicular to line 
.
.
hence plane 
is perpendicular to line 
.
.
is perpendicular to plane