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.
, such that every positive integer appears exactly once as a member of the sequence and 
divides 
if and only if 
?
be a sequence of real numbers, and let 
be a fixed positive integer less than 
. We say an index 
with 
is good if there exists some 
with 
such that 
, where the indices are taken modulo . Let 
be the set of all good indices. Prove that 
.
such that for any positive real numbers 
and 
,
, with 
.Initially, he folds the sheet along a straight line 
, where 
is a point on the side 
, so that vertex 
is located on side 
, as shown in the figure. Then folds the sheet again along a straight line 
, where 
is a point on side , so that vertex 
lies on the line ; and finally folds the sheet along the line 
. Matías observed that the vertices and 
were located on the same point of segment after making the folds. Calculate the measure of the angle 
.
, 
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.
be a triangle, and let 
be its altitudes. Let 
be its orthocenter, and let 
and 
be the circumcenters of triangles 
and 
. Let 
be the second intersection of the circumcircles of triangles 
and 
. Prove that the lines 
, , and 
-median of 
are concurrent.
be the feet of the altitudes of the triangle from 
, respectively. 
is the point on the circumcircle of the triangle , is the orthocenter of the triangle , and the incircle of triangle 
has radius 
. Let 
be the point such that and are on the opposite side of 
, line 
is perpendicular to the line , and the distance from to line is . Define 
and 
similarly. Prove that 
are collinear.
be the midpoint of on triangle . Point 
lies on segment 
such that 
and 
on line 
such that 
. Prove that 
.
be a convex quadrilateral. The diagonals 
and 
intersect at 
, with 
. Let 
and 
be the midpoints of sides 
, and 
, respectively. Find the greatest positive real number 
such that![\[
EG + 3FH \geq k \cdot d + (1 - k) \cdot s,
\]](http://latex.artofproblemsolving.com/b/e/f/bef24b0d482f95d2e6ef22b3797a432c69d98af1.png)
where 
is the semiperimeter of and 
is the sum of the lengths of its diagonals (i.e., 
). Determine when equality holds.
and 
passes through the center of the other one. Triangle is inscribed into , and lines 
touch . Prove that 
.
is the circumcircle of 
wwith diameter 
. The tangents of at points and intersect at . The line perpendicular to 
at point intersects the extension of at point . Extend 
at length 
. If 
, find the measure of 
.
is 
. Line 
meets line at point , and there is a point on 
such that 
. Line 
intersects at point . The perpendicular bisector of line segment intersects line at point 
, and line intersects 
at point 
.
is tangent to 
.
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.
.
