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 sequence of independent random variables with distribution 
. Define 
. Prove that 
and 
as 
.
such that there exist four complex numbers 
with 
, and for any complex number 
, if 
, then![\[|az^3+bz^2+cz+d|\le M.\]](http://latex.artofproblemsolving.com/a/9/b/a9b4f124bf07c5a5aaa2d2e46c249ac44b7aaf57.png)
, 
, be 3 complex numbers with 
. Prove that 
be an acute triangle, 
its circumcircle and 
its circumcenter. The altitude from 
intersects in a point 
and the segment 
intersects the circumcircle of 
in a point 
. Finally, let be the midpoint of 
. Show that 
is parallel to 
.
it holds that
be a triangle with circumcentre 
. The points 
and 
are interior points of the sides 
and 
respectively. Let 
and 
be the midpoints of the segments 
and 
. respectively, and let 
be the circle passing through and . Suppose that the line is tangent to the circle . Prove that 
is a positive divisor of other than itself.
consists of positive integers, each of which has at least three proper divisors. For each 
, the integer 
is the sum of the three largest proper divisors of 
.
.
for some odd prime 
. It is known that 
. Prove that must be in the form of 
, where 
is an integer. Give a group 
as an example that satisfies the property.
is defined recursively by ![\[a_0=-1,\qquad\sum_{k=0}^n\dfrac{a_{n-k}}{k+1}=0\quad\text{for}\quad n\geq 1.\]](http://latex.artofproblemsolving.com/f/4/b/f4b8f95e49de7274cccd8ee498e0b496f31ddfea.png)
Show that 
for all 
.
is defined by the formula![\[ a_{i + 1} = \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0;
\]](http://latex.artofproblemsolving.com/5/c/2/5c2ac5789507cdeaabc16284447f28294dd4fbdf.png)
here 
is an arbitrary real number, 
denotes the greatest integer not exceeding 
, and 
. Prove that 
for 
sufficiently large. be an acute triangle with orthocenter 
. Let 
be the point such that the quadrilateral 
is a parallelogram. Let 
be the point on the line 
such that bisects 
. Suppose that the line intersects the circumcircle of the triangle 
at 
and 
. Prove that 
.
, 
and![\[Q_n(x)=\frac{(Q_{n-1}(x))^2-1}{Q_{n-2}(x)}\]](http://latex.artofproblemsolving.com/9/0/b/90bb520063fe59fb095d8cb7a7c23202c6fff329.png)
for all 
Show that, whenever 
is a positive integer, 
is equal to a polynomial with integer coefficients.
+1 w
.
is continuous at 
.
is not continuous at 
.
![$ f(x)= \frac {x - \sqrt{2}}{ [x \sqrt{2} ] } $](http://latex.artofproblemsolving.com/a/9/f/a9f640a2ba0f2c58a1a7085e843e82ae41c72951.png)
?
notation) ......corrections made above. 
be i.i.d with pdf 
.![$$\frac{S_n}{\sqrt{n\log(n)}}\xrightarrow[n\to\infty]{\text{d}}\mathcal{N}(0,1).$$](http://latex.artofproblemsolving.com/9/2/b/92b57d6aabfc2c12f11aaa512d585fd191ec1db2.png)
such that: 
Prove that :
is continous on 
![$ ] \frac {1}{ \sqrt{2} } ; 2 \sqrt{2} [ $](http://latex.artofproblemsolving.com/d/0/4/d04f852d2494e9cd545a1ff20462e7283595c9ce.png)
