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 circle inscribed inside a rhombus 
. Let 
and 
be variable points on 
and 
respectively, such as 
is always the tangent line to . and the area of triangle 
is the same.
, find the smallest integer 
such that any graph with 
vertices and at least edges must contain at least two disjoint cycles (cycles with no common vertices).
passing through point 
is an oblique strophoid.
be a cyclic quadrilateral for which 
. Construct two squares 
and 
externally on the sides 
and 
respectively.
meets 
at 
, 
meets 
at 
and 
denotes circumcenter of . Prove that 
and 
meets again on 
.
be a positive integer. Show that any polynomial ![$Q\in\mathbb{Z}[X]$](http://latex.artofproblemsolving.com/e/8/2/e8286b6f2b9790f39c88849a1415d86f85c0b92c.png)
with all positive coefficients such that
is a perfect square for any 
must be a constant polynomial.
+1 w
be an acute triangle and let 
be the foot of the -bisector. Moreover, let 
be the midpoint of . The circle 
with diameter 
meets 
at 
, while the circle 
with diameter meets 
at 
. Assume that and lie inside . Prove that 
, , , 
are concyclic.
of degree 
having 
distinct real roots (
), such that for every root 
: 
.
defined on all real numbers and taking real values such that ![\[f(f(y)) + f(x - y) = f(xf(y) - x),\]](http://latex.artofproblemsolving.com/f/5/2/f528fc4047e21247663d4993913eadf346315be7.png)
for all real numbers 
. Prove that there does not exist a pair of rational numbers 
such that 
.
![$P\in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/7/4/c/74cb2247069abb8a22666e87fcbd69a55a015004.png)
for which there exists 
such that for all 
we have: ![\[P\left(x+\frac1n\right)+P\left(x-\frac1n\right)=2P(x).\]](http://latex.artofproblemsolving.com/9/0/9/90954cdeade9d13149e1aaa03df2c87dff157041.png)
, the points and 
are taken such that the point lies between the points and , and 
. On segments 
and 
, points and are taken so that 
and 
. Prove that 
.
, define a sequence of triples 
by
.Show that each of the sequences 
converges to a limit and find these limits.
, define a sequence of triples by
,.Show that each of these sequences converges to a limit and find these limits.
are the three positive sidelengths or a triangle, we can rather easily show that :
are the sides of a new triangle with same area
, 
and 
are decreasing sequences
have same limits, sides of an equilateral triangle of same area
![$l=\sqrt[4]{\frac{(x_0+y_0+z_0)(x_0+y_0-z_0)(x_0-y_0+z_0)(-x_0+y_0+z_0)}3}$](http://latex.artofproblemsolving.com/7/4/9/749d0a7955dd294a78fcf172aa31ed1225df1dc7.png)
, this formula gives 
. Also, the thing under the radical suspiciously looks like the product of length of tangent from a vertex and some other thing. This motivates the introduction of a triangle with sides 
. Usually we need some kind of invariant in multivariable limits. So, we try the squared area of the triangle since it is a polynomial in , more precisely, 
. This surprisingly comes out to be invariant. Also to show the convergence, it seems better to choose the difference between squares.
![\[
\text{Find all prime numbers } (p, q) \text{ such that } p^2 + 3pq + q^2 \text{ is a fifth power of an integer.}
\]](http://latex.artofproblemsolving.com/c/f/8/cf855cb087102024a50d41f49c67e7fd1ef22767.png)
