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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 
.
+1 w
be an acute triangle with incenter 
and circumcenter 
. The incircle touches sides 
and 
at 
and 
respectively, and 
is the reflection of 
over . The circumcircles of 
and 
meet at 
, and the circumcircles of 
and meet at a point 
, where 
is the midpoint of 
. Prove that if 
and meet at 
, then 
.
be a function, two times derivable on 
for which there exist 
such that![\[ \frac { f(b)-f(a) }{b-a} \neq f'(c) ,\]](http://latex.artofproblemsolving.com/a/3/1/a31861ef909a6e17ed8b495b9b9b16d887ba573f.png)
for all 
.
.
be positive real numbers .
, whether or not the polynomial 
irreducible over 
?
such that the equation in the unknown 
:
admits 
roots: 
s.t. 
.
does there exist an 
matrix whose entries are all in 
, such that 
is the matrix of all ones?
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 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 .
to be on the three medians of the triangle.
. By Ceva's theorem, if 
and 
, then 
(
, 
, 
). Let the side length be 
, then![\[A'A''=\left|BA'-BA''\right|=\left|\frac b{b+1}-\frac12\right|=\frac{|b-1|}{2(b+1)}.\]](http://latex.artofproblemsolving.com/f/6/3/f639ba63d9b8b789471e9073da82530d4382d2bc.png)
Similarly we obtain 
and 
. Multiplying by 
, we need to show that![\[\frac{|a-1|}{a+1}+\frac{|b-1|}{b+1}=\frac{(a+1)|b-1|+(b+1)|a-1|}{(a+1)(b+1)}>\frac{|a-b|}{a+b}.\tag{*}\]](http://latex.artofproblemsolving.com/b/5/8/b580ae87663620b5fc0f193a4773ebb952c1a37e.png)
On the one hand,
where
On the other hand,
where![\[(2)\Leftrightarrow2(a+b)>(a+1)(b+1)\Leftrightarrow(a-1)(b-1)<0.\]](http://latex.artofproblemsolving.com/2/5/7/25716af9dc6ff71bcb238991b68fe4e78108c64b.png)
At least one of 
and 
holds, so we are done.
?
and 
.![\[P=\frac{1}{5-2a}+\frac{1}{5-2b}+\frac{1}{5-2c}.\]](http://latex.artofproblemsolving.com/c/3/7/c3708768a84c2bbb1e4fae502a9bb1e92cb13d02.png)
![$\underset{n\to \infty }{\mathop{\lim }}\,\sqrt[n]{lcm(1,2,3,...,n)}=e$](http://latex.artofproblemsolving.com/a/1/3/a139e5cb990f3386de109c701f8edc71b4054712.png)
