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.
that satisfies,
, let 
be the second smallest positive integer such that 
is a perfect square. Prove that 
is a perfect square.
be a triangle. Points 
, 
, and 
are placed on sides 
, 
, and 
respectively such that 
. The line 
meets the circumcircle of 
again at 
. Similarly, the line 
meets the circumcircle of 
again at 
. If 
is the reflection of across the midpoint of , prove that the four points , , 
, and 
lie on a circle.
be non-negative real numbers. Prove that 
satisfying 
, one has![\[1 \le \frac{x}{1-yz}+\frac{y}{1-zx}+\frac{z}{1-xy} \le \frac{9}{8}.\]](http://latex.artofproblemsolving.com/d/9/c/d9cb40fd1427d3ce31af7525728b9d9a5031e2c6.png)
+1 w![$A\in [0,1]^n$](http://latex.artofproblemsolving.com/0/7/e/07eed58d80cf78c7741d61ff142ffffea77a5edf.png)
be an invertible 
matrix with entries between 
and 
. The Frobenius norm of a matrix is defined as![\[\Vert X \Vert_F = \sqrt{\mathrm{tr}(XX^T)} = \left(\sum_{i,j=1}^n X_{ij}^2\right)^{1/2}.\]](http://latex.artofproblemsolving.com/a/2/a/a2ac3606933c81aebd6e9b26af765e7202a23aec.png)
Prove that![\[\Vert A^{-1} \Vert_F \ge \frac{2n}{n+1}\]](http://latex.artofproblemsolving.com/9/6/b/96b79f60d6d54816ac3c7e7cd7b6b95f823d4b75.png)
for (a) 
and (b) sufficiently large odd 
.
such that 
is a prime number. be inscribed in a circle 
, and let 
be a point on the perpendicular bisector of 
. Denote by 
the incenters of triangles 
and 
, respectively. Let 
be the midpoint of arc 
of . Prove that the points 
, 
, 
, and lie on the same circle.
and 
not is a perfect square for all 
.
and so on to create any number!
be a scalene triangle, and 
a point inside 
and not on the sidelines of .
be the circumcevian triangle of wrt .
be the radius of the circle internally tangent to , and segments 
, 
. Define 
, 
cyclically.
be the radius of the circle internally tangent to , and segments 
, . Define 
, 
cyclically.
, then, the pedal circle of wrt is tangent to the incircle of . is outside of )
,
