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.
and 
such that 
is divisible by 
?
such that there exists a pair 
of positive integers, such that 
is not divisible by the cube of any prime, and 
have the property that both![\[
\left\lfloor \sqrt{2n - 1} \right\rfloor \quad \text{and} \quad \left\lfloor \sqrt{3n + 2} \right\rfloor
\]](http://latex.artofproblemsolving.com/d/e/c/dec527c4673470e7023d6aed00fac90268bd2fcc.png)
are consecutive perfect squares.
, 
, with 
, 
. Prove that there exists 
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For and , the notation 
represents 
. )
be the incentre of acute triangle 
with 
. The incircle 
of is tangent to sides 
, and 
at 
and 
, respectively. The line through 
perpendicular to 
meets at 
. Line 
meets again at 
. The circumcircles of triangle 
and 
meet again at 
.
and 
meet on the line through 
perpendicular to 
.
be a circle, and 
are parallel chords and 
is a in its middle point 
and 
. 
is the middle of 
. Prove that 
with orthic triangle 
, let 
, 
, 
be points such that 
. Let 
be the trilinear polar of the polar conjugate of wrt , and define 
and 
cyclically. Let 
be the triangle bounded by , , . and are bilogic.
, 
, 
, and lie on a line in that order, and points 
and are located outside the line such that 
, 
and 
. Let the circumcircles of triangles 
and 
intersect at points and , and the circumcircles of triangles 
and 
intersect at points 
and 
. Prove that the lines and 
pass through the midpoint of segment .
are collinear with betweeen and . 
is the circle with diameter , and 
is the circle with diameter 
. Another circle touches 
at and meets again at and again at . The line meets again at and again at 
. Show that the lines and 
meet on the perpendicular to at .
be a quadrilateral with 
. Denote by the midpoint of the side , and assume that 
. Let be the foot of the perpendicular from the point to the line . The line 
meets 
at , and the line 
meets at . Show that 
and 
., , , lie on a semicircle in this order with diameter 
, and is not parallel to . Points 
and 
lie on segments and respectively such that 
and 
. A circle 
passes through and is tangent to , and intersects again at 
. Prove that the lines 
, and 
are concurrent.
inscribed in 
with being the midpoint of . The tangents at 
of intersect at . Let be the projection of 
onto . On the perpendicular bisector of , take a point that is not on and different from M. Circle 
intersects at . Lines 
and 
intersect at . Prove that 
is an isosceles triangle.
is fixed on a plane with largest side . Let be an arbitrary diameter of its circumscribed circle, and the point lies on the smaller arc , and the point is on the smaller arc . Points 
are feet of perpendiculars dropped from point to the line , from point to the line and from point to line . Prove that the center of the circumscribed circle of triangle 
lies on a fixed circle.
be points on the sides 
respectively of a triangle . Suppose that 
and 
meet at 
and meet at 
, and 
and meet at 
, such that 
, and 
. Compute the ratio of the area of 
to the area of 
.
be a positive integer. Determine all sequences 
of positive integers such that
for all positive integers 
.
.
such that 
, then by induction it follows that 
, which forms a valid sequence.
. Then, we have:
Hence, we obtain
, where 
, it follows that 
. For sufficiently large , the left-hand side grows linearly while the right-hand side grows quadratically:
which is a contradiction.
.
