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 
, where 
. She observes that these two graphs and 
axis divides the plane into two triangles and a quadrilateral. Find the ratio of sums of two triangles' areas to the area of quadrilateral.
, the following inequality holds 
(
denotes the semiperimeter of a triangle)
is given, such that could be expressed as difference of two perfect squares (
). Is it possible for 
to be expressed as difference of two perfect squares?
be a cyclic quadrilateral and let the intersection point of lines 
and 
be 
. Let the points 
and 
be arbitrary points on sides and respectively, which satisfy the conditions
Prove that 
.
square is given, and a quadratic trinomial with a positive leading coefficient is placed in each of its cells. There are 
coefficents in total, and these coefficents are chosen from the set ![$[-66;47]$](http://latex.artofproblemsolving.com/d/d/9/dd92777f4ba30299d3326b285f1f73656aba544b.png)
, and each coefficient is different from each other. Prove that there exists at least one column such that the sum of the six trinomials in that column has a real root.
satisfying the following two conditions:
such that, for all indices 
, 
. and for any index , the number![\[
a_n a_{n+1} \cdots a_{n+p-1} - a_{n+p}
\]](http://latex.artofproblemsolving.com/5/4/a/54ac17063104439114f3c7b07de8fd27450ea1eb.png)
is a multiple of . and 
be prime numbers. Prove that if 
is a perfect square, then 
is also a perfect square.
![\[\frac{x^3+1}{(y-1)(z+1)}+\frac{y^3+1}{(z-1)(x+1)}+\frac{z^3+1}{(x-1)(y+1)}\]](http://latex.artofproblemsolving.com/5/c/6/5c6894b617b990ec69c62ed1cf7d062d4660af53.png)
where 
are reals.
be a triangle and let 
denote the circumcircle of . The foot of altitude from 
to 
is 
. The foot of altitudes from to and 
are 
, respectively. Let 
intersect at 
, and let 
intersect at 
. Prove that is the incenter of triangle 
to the following system:
, which lie on a circle, are marked by distinct reals 
. Let 
be 
good for a 
on the circle different than , if and only if 
is the greatest number on at least one of the two arcs 
. Let the score of be the number of good points on the circle. Determine the greatest 
such that regardless of the values of , there exists a point with score at least .
called ‘nice’ if and only if 
and 
are both perfect squares. How many positive integers 
such that 
are composites for all nice numbers 
?
and 
are natural numbers such that
is a prime number. Find all possible values of ,,.
be a positive integer. Aslı and Zehra are playing a game on an 
grid. Initially, 
stones are placed on some of the unit squares of this grid., regardless of the initial placement of the stones, can Aslı guarantee that every square ends up with the same number of stones?
such that there exist positive integers 
for which the following holds 
for all 
.
