[/quote]
,
Let ![$[n]$](http://latex.artofproblemsolving.com/0/d/e/0deff9086fd7840ef23860f5bc924f1d4d2d2310.png)
denote the set of natural numbers less than or equal to 
![$f(m,n) = \sum_{(x_1,x_2,x_3, \cdots, x_m) \in [n]^{m}} \frac{x_1}{x_1 + x_2 + x_3 + \cdots + x_m} \binom{n}{x_1} \binom{n}{x_2} \binom{n}{x_3} \cdots \binom{n}{x_m} 2^{\left(\sum_{i=1}^{m} x_i\right)}$](http://latex.artofproblemsolving.com/f/d/f/fdf7b2fcfb960e93d59f3f84cef4372262c69499.png)
there exists an 
all of whose digits are odd.
be an acute triangle with 
.
be the centroid of triangle 
and let 
be the foot of the perpendicular from 
.
intersects the circumcircle 
of triangle 
be the point where 
intersects the circle 
,
and 
atoms called usamons. Each usamon either has one electron or zero electrons, and the physicist can't tell the difference. The physicist's only tool is a diode. The physicist may connect the diode from any usamon 
to any other usamon 
.
.
Let 
.![$$ \frac{1}{2}\geq \frac{a}{a^2+3 }+ \frac{b}{b^2+3} \geq \frac{1}{4}(\frac{1}{\sqrt[3]{3}}+\sqrt[3]{3}-1)$$](http://latex.artofproblemsolving.com/f/9/1/f9119e071c12778037616c624c7b9ca75356e8e3.png)
![$$ \frac{1}{2}\geq \frac{a}{a^3+3 }+ \frac{b}{b^3+3} \geq \frac{1}{2\sqrt[3]{9}}$$](http://latex.artofproblemsolving.com/2/b/2/2b2d8a011229fe88a81cb75f9b018ad3a6245faa.png)
![$$ \frac{1}{2}\geq \frac{a}{a^2+ab+2}+ \frac{b}{b^2+ ab+2} \geq \frac{4\sqrt[3]{3}+3\sqrt[3]{9}-6}{17}$$](http://latex.artofproblemsolving.com/5/f/6/5f673cd494b1909ca237db63e96f949c24b6f2ad.png)
![$$ \frac{1}{2}\geq \frac{a}{a^3+ab+2}+ \frac{b}{b^3+ ab+2} \geq \frac{\sqrt[3]{3}}{5}$$](http://latex.artofproblemsolving.com/b/c/3/bc343a07c88dbd98ebffdfb78f311b7b82683d65.png)
-
be a unit square. Points 
are chosen outside 
. Let 
,
.
is at most 
.
and 
lie on segments 
,
and 
intersect at point 
and 
intersect at a second point 
intersect at a second point 
lies on segment 
such that 
.
and 
are similar.
from the non-zero reals to the non-zero reals, such that ![\[f(x+y) \left(f(x) + f(y)\right) = f(xy)\]](http://latex.artofproblemsolving.com/2/f/e/2fe8a09e628f60faedaa07ceb154891375763a83.png)
for all non-zero reals 
such that 
.
.
Let 
.
Let 
.
.
Y by Adventure10, Mango247
Let 
be a natural number. A pyramid 
has base 
and apex 
. The polygon is regular and the point 
is its centre. The line 
is perpendicular to the plane of the base of . A sphere passes through and meets each of the line segments 
internally. For each 
, let 
be the point (other than ) where the sphere meets . Prove that 
be a natural number. A pyramid 
has base 
and apex 
. The polygon is regular and the point 
is its centre. The line 
is perpendicular to the plane of the base of . A sphere passes through and meets each of the line segments 
internally. For each 
, let 
be the point (other than ) where the sphere meets . Prove that 
be the center of sphere and 
it's projection on a plane of poligon.
that
.
its radius, then 
.
are two sets of points with masses associated with them, with the total mass of 
being 
being 
,![\[ \frac{1}{MN}\sum{M_{i}N_{i}|X_{i}Y_{i}|^2 }\]](http://latex.artofproblemsolving.com/7/0/8/7085b87e8edeffe7bf95e472759ec094bd237c78.png)
, with 
being the mean square distance of the points 
,
and 
is![\[ \sigma(X) + |X^{\prime}Y^{\prime}|^2 + \sigma(Y) \]](http://latex.artofproblemsolving.com/4/f/e/4fe5a1a4b362b6f00f98b5d6f9d4ee31db639fc6.png)
.
