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 
be prime numbers. Prove that if 
is a perfect square, then 
is also a perfect square.
and 
, where 
. She observes that these two graphs and 
axis divides the positive side of the plane (
) into two triangles and a quadrilateral. Find the ratio of sums of two triangles' areas to the area of quadrilateral.
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 .![\[\frac{x^3+1}{(y-1)(z+1)}+\frac{y^3+1}{(z-1)(x+1)}+\frac{z^3+1}{(x-1)(y+1)}\]](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 .
for some real number and ![\[ a_n=na_{n-1}+(n-1)(n!(n-1)!-1) \]](http://latex.artofproblemsolving.com/4/1/f/41f0cc7b041a8bba056882b5c794503195c374fd.png)
for integers 
. Given that 
, and given that 
, where and are positive integers whose greatest common divisor is 
, compute 
and 
are in the coordinate plane such that 
. Let 
denote the locus of all points 
in the coordinate plane satisfying 
, and let 
be the midpoint of . Points 
and 
are on such that 
and 
. The value of 
can be expressed in the form 
, where 
and 
are relatively prime positive integers. Find 
.
Prove that
Let 
Prove that
be the foot of the altitudes from 
.
,
.
from 
and subtracting 
,
in terms of 
,
.