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.
(if we want to extend the 
notation), because 
, we could have 
, and so on.
such that for any positive integer 
, satisfies 
.
such that 
for all real numbers 
and 
be an integer. Let 
, 
, ..., 
be positive real numbers such that ![\[n^2 + 1 > \left( t_1 + t_2 + \cdots + t_n \right) \left( \frac{1}{t_1} + \frac{1}{t_2} + \cdots + \frac{1}{t_n} \right).\]](http://latex.artofproblemsolving.com/1/a/b/1ab410ebc17d9f6c38a3e7ae0f6efa955ffddfd3.png)
Show that 
, 
, 
are side lengths of a triangle for all 
, 
, 
with 
.
and , and 
positive (not neccesarily distinct) real numbers 
, 
. An equation is written on a whiteboard: 
where 
is a fixed positive real number, with exactly asterisks.
, while Rubi fills each asterisk with a number from 
, so that the equation on the whiteboard is correct. Suppose for every positive real number , the number of ways for Ebi and Rubi to do so are equal. and are permutations of each other.
and 
are considered different ways to fill the asterisks, and the chosen terms need not be distinct, for example 
.)
and 
and there exists a positive integer such that 
. Let 
be minimal polynomial 
over 
. Prove that 
, ∀x∈[0,1].
and 
with centres 
and 
are externally tangent at point 
and internally tangent to a circle 
at points 
and 
respectively. Line 
is the common tangent of and at . Let 
be the diameter of perpendicular to , so that 
are on the same side of . Prove that lines 
, 
, 
and are concurrent.
such that for all 
, ![\[xf(y+x)+(y+x)f(y)=f(x^2+y^2)+2f(xy)\]](http://latex.artofproblemsolving.com/5/b/1/5b101ead6195155ccbd99b757016e8f73dac4411.png)
Proposed by Aritra Mondal
be a probability distribution on the set of nonnegative integers. Select a number according to this distribution and repeat the selection independently until either a zero or an already selected number is obtained. Write the selected numbers in a row in order of selection without the last one. Below this line, write the numbers again in increasing order. Let 
denote the event that the number 
has been selected and that it is in the same place in both lines. Prove that the events 
are mutually independent, and 
.
of a variate 
is defined as follows:![\[
F(x) =
\begin{cases}
A, & -\infty < x < -1, \\
B, & -1 \leq x < 0, \\
C, & 0 \leq x < 2, \\
D, & 2 \leq x < \infty.
\end{cases}
\]](http://latex.artofproblemsolving.com/0/9/0/0903946b6a44e9bfc3a1adf8dc0ad5ac013e92c1.png)
are constants. Determine the values of 
and 
.
.
.