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.
, 
is biased so that, when tossed, it has probability 
of falling heads. If the 
coins are tossed, what is the probability that the number of heads is odd? Express the answer as a rational function .
be an equilateral triangle, 
be a point inside it, and 
be the intersections of 
with the sides of . If 
are the midpoints of 
, 
, 
, show that there is a triangle with sides 
, 
and 
.
be two integers and 
be a positive integer such that 
. Show that ![\[(n-t)(d-t)>\dbinom{n-t+r}{r+1},\]](http://latex.artofproblemsolving.com/e/5/f/e5f9486f980870ff72858b57f7b50cd2aa9d142a.png)
for 
be a continuous function such that 
for all 
. Show that 
for 
.
with at least four elements such that for every 
with 
we have![\[\sum_{x\in S}x=\prod_{x\in A^*\setminus S}x.\]](http://latex.artofproblemsolving.com/f/f/d/ffde3b8dbaa280887b741069db7722a56a5fcbee.png)
, 
, 
we have ![\[ 3(a+b+c) \ge 8\sqrt[3]{abc} + \sqrt[3]{\frac{a^3+b^3+c^3}{3}}. \]](http://latex.artofproblemsolving.com/a/9/8/a98b6c29802355c491016bcce58228a020423526.png)
Prove that 
When does equality hold?
satisfy that: 
and 
. Find the maximum: 
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 positive real numbers such that 
.![\[
\left( \frac{a + b}{c} \right) \left( \frac{b + c}{a} \right) \left( \frac{c + a}{b} \right) \geq 9.
\]](http://latex.artofproblemsolving.com/7/7/f/77f8bffb0d9b617655c7a8f4a8eb7245d60d8d9f.png)
of matrices by the following recurrence: 
where 
is the 
identity matrix.
has 
distinct integer eigenvalues 
with multiplicities 
, respectively.
. Proof of lemma comes easily.
we obtain 
for all 
. Furthermore from 
, the result follows!
are the same as those of 
has eigenvalues 
with multiplicities 
, respectively. To make the base of induction easier, we define 
.
has eigenvalues 
with multiplicities 
and eigenvectors 
respectively.
Since 
are linearly independent, so are 
. Hence that is the complete eigenspace for .
with mulitplicities 
respectively.
and 
Prove that
, such that 
Prove that :
equality cases : ?
