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.
, inscribed in the circle 
, satisfies 
and 
, and 
is the intersection point of the diagonals 
and 
. The circle with center 
and radius 
intersects in two points 
and 
. Prove that the line 
is tangent to the circles with diameters 
and 
.
such that 
and 
for any real numbers 
and 
.
, let 
denote the sum of squares of digits of . Prove that the sequence 
is eventually periodic.
grid, where is a composite integer.
unit squares are divided up into 
disjoint sets of 
unit squares arbitrarily such that 
. Denote this family of sets as 
. unit squares are again divided up into 
disjoint sets of 
unit squares arbitrarily such that 
. Denote this family of sets as 
.
unit squares such that no two unit squares are in the same set of or the same set of ?
be a polynomial with integer coefficients. Define a sequence 
of integers such that 
and 
for all 
. Prove that if there exists a positive integer 
for which 
then either 
or 
.
be a triangle, and let 
be an arbitrary point on line 
, where 
is the circumcenter of . Define 
and 
as the orthocenters of triangles 
and 
. Prove that 
passes through a fixed point which is independent of the choice of .
be collinear points (in order) and 
a point in plane. Consider the disc 
of center and radius 
, for some 
. Prove that ![$\mathcal{D}\cap [AC]$](http://latex.artofproblemsolving.com/0/5/6/0569d4130943000380a87b72be211965b5ff02ad.png)
is either the empty set or a segment of length at most 
. be a positive integer and ![$P(X)\in\mathbb{C}[X]$](http://latex.artofproblemsolving.com/9/9/e/99e8cde28b0b9bc9414425ebc30fa2dc0386fb37.png)
be a polynomial of degree . Prove that ![\[\sup_{x\in[0,1]}|P(x)|\le(2n+1)^{n+1}\int\limits_{0}^{1}|P(x)|\mathrm{d}x.\]](http://latex.artofproblemsolving.com/9/1/6/91647cd75888c5e9d291280c8ae3f199cc6d1bfd.png)
be a right triangle in with 
. Let 
be the midpoint of 
and a point on such that 
. Let 
.
and the ratio 
.
be positive real numbers satisfying 
. Prove that![\[
3(x^{\frac{2}{3}} + y^{\frac{2}{3}}) \geq 4 + 2x^{\frac{1}{3}}y^{\frac{1}{3}}.
\]](http://latex.artofproblemsolving.com/f/3/c/f3cd114f909f388eacee142d3f90d07f6b149131.png)
be the set of the differentiable functions 
satisfying 
for any 
then the set of solutions of the equation 
is infinite.
![$f\colon[0,1]\rightarrow \mathbb{R}$](http://latex.artofproblemsolving.com/f/d/b/fdb1fb0216ab9d17007b1b7b03d20d1060b88531.png)
be a continuous function. Suppose that for each 
, the function ![\[f_t\colon[0,1-t]\rightarrow\mathbb{R}, f_t(x)=f(x+t)-f(x)\]](http://latex.artofproblemsolving.com/4/f/c/4fc6a66d0600a999c000136a26ea23dfcc269cc3.png)
has an unique global minimum point, which we will denote by 
. Prove that if 
, then 
is constant zero.
and 
Prove that
and 
Prove that
