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 positive integer. Paul has a 
rectangular strip consisting of 
unit squares, where the 
square is labelled with 
for all 
. He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then translate (without rotating or flipping) the pieces to obtain an 
square satisfying the following property: if the unit square in the row and 
column is labelled with 
, then 
is divisible by 
.
cities in a country, where 
is an integer. Some pairs of cities are connected by direct (two-way) flights. For two cities 
and 
we define:
A 
between and as a sequence of distinct cities 
, 
, such that there are direct flights between 
and 
for every 
;
A 
between and as a path between and such that no other path between and has more cities;
A 
between and as a path between and such that no other path between and has fewer cities. and in the country, there exist a long path and a short path between them that have no cities in common (except and ). Let 
be the total number of pairs of cities in the country that are connected by direct flights. In terms of , find all possible values 
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 a positive integer. Find all real solutions 
to the system:![\[a_1^2 + a_1 - 1 = a_2\]](http://latex.artofproblemsolving.com/a/4/e/a4e26b83c929b82800576bac11e12de0c3aa1cf6.png)
![\[ a_2^2 + a_2 - 1 = a_3\]](http://latex.artofproblemsolving.com/9/9/5/995e8eec93828b82901b0639463f907f5ff95d08.png)
![\[\hspace*{3.3em} \vdots \]](http://latex.artofproblemsolving.com/e/e/1/ee15e1fefe2ad550d3c9b1c51207eaf66fbfa41f.png)
![\[a_{n}^2 + a_n - 1 = a_1\]](http://latex.artofproblemsolving.com/7/c/7/7c727d1f2044c1bf7c80981e1074bb3cde807b8f.png)
, 
, 
, ... be an infinite sequence of real numbers satisfying the equation 
for all 
, where and are two different positive reals., , , ... be bounded? and 
in a non-isosceles triangle 
with incenter 
intersect its circumcircle 
at 
and 
, respectively. The line through , parallel to 
, intersects 
at 
. Prove that 
is tangent to .
be the circumcircle of right-angled triangle (
). The tangent to at point intersects the line 
at point . Suppose that 
is the midpoint of the minor arc 
, and 
intersects for the second time in 
. The tangent to at point intersects at 
. Prove that 
.
such that 
and 
, evaluate 
.
, 
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 fixed points on this order on a line. Let be a variable circle through and 
and suppose it meets the perpendicular bisector of 
at the points 
and 
. Let 
and 
be the other points of intersection of 
and 
with . Prove that 
passes through a fixed point independent of .
is divided by the diagonal into two similar, but not equal, triangles. What can be the length of the diagonal if the lengths of the sides and 
are 
and 
, respectively?
is divided by the diagonal into two similar, but not equal, triangles. What can be the length of the diagonal if the lengths of the sides and are and , respectively?
show that![\[\frac{a_1}{a_1+a_2}+\frac{a_2}{a_2+a_3}+\cdots +\frac{a_n}{a_n+a_1}\ge\frac{a_1+a_2+\cdots +a_n}{a_1+a_2+\cdots +a_n-(n-2)\sqrt[n]{a_1a_2\cdots a_n}}>1.\]](http://latex.artofproblemsolving.com/c/e/2/ce286b2083f9fcbb7a976b511eef0a223a464a77.png)
such that:
and 
.
