[/quote]
,
,
in the Cartesian plane whose coordinates lie inside the unit square defined by![\[ i < x < i+1, \qquad j < y < j+1. \]](http://latex.artofproblemsolving.com/2/3/6/2363248e26b4c9826b4f3e1794c46c358bfd8f42.png)
Find all real numbers 
for which it's possible to choose these points such that for all integers 
and 
,
has perimeter strictly less than 
.
be positive real numbers satisfying![\[ xy + yz + zx = 3xyz. \]](http://latex.artofproblemsolving.com/9/2/0/9200874fd6bf9a32ff0c4e82382a450be68cc444.png)
Prove that![\[
\sqrt{\frac{x}{3y^2z^2 + xyz}} + \sqrt{\frac{y}{3x^2z^2 + xyz}} + \sqrt{\frac{z}{3x^2y^2 + xyz}} \le \frac{3}{2}.
\]](http://latex.artofproblemsolving.com/f/5/f/f5fb85d81e93935601ceb5f30c25aea7aa3f23b2.png)
and 
be real numbers. Let 
be the matrix with entries ![\[a_{ij} = \begin{cases}1,&\text{if }x_i + y_j\geq 0;\\0,&\text{if }x_i + y_j < 0.\end{cases}\]](http://latex.artofproblemsolving.com/c/0/8/c0856fe7b2b113180659952c4d216a01d8849f64.png)
Suppose that 
is an 
matrix with entries 
,
such that the sum of the elements in each row and each column of 
.
.
Prove that
Let 
Prove that![$$ (a+b)(a+1)(b+1) \leq \frac{3}{2}+\sqrt[3]{6}+\sqrt[3]{36}$$](http://latex.artofproblemsolving.com/f/5/e/f5e68204507aa9c3e71c28844bbee3a804546b94.png)
inscribed in 
.
lie on 
such that 
are isogonal in 
.
is the midpoint of 
.
lies on 
such that 
.
intersects 
.
passes through the Euler center of triangle 
such that![\[
f(x + y^2 + z^3) = f(x) + f(y)^2 + f(z)^3
\]](http://latex.artofproblemsolving.com/0/d/7/0d7a6f8586733073f8f543810ca00416770b84cc.png)
for all 
.
are collinear, and 
are arbitrary points
be an acute triangle and 
its orthocenter. Let 
.
and 
as points on the parallels through 
and through 
to 
similarly. If 
are positioned around the sides of 
.
and 
are chosen on segments 
and 
such that 
.
and 
is the midpoint of the arc 
,
.
be a convex quadrilateral inscribed in a circle and satisfying 
.
and 
and 
.
.
Y by epiclucario, TheMaskedMagician, epoch621, spartan168, Adventure10, Mango247
Well everything is in the chart below. Cutoff will be between 100.5-106.5, still deciding. If you notice anything wrong with your score just point it out to me. 34 submissions. Your scores align with the row they are in.
EDIT: You can also use this link to get a better version. https://www.dropbox.com/s/0llz99q941qe8kr/Results%20for%20Mock%20AMC%2010.xlsx?dl=0
EDIT: You can also use this link to get a better version. https://www.dropbox.com/s/0llz99q941qe8kr/Results%20for%20Mock%20AMC%2010.xlsx?dl=0
This post has been edited 1 time. Last edited by joey8189681, Feb 21, 2015, 4:07 AM
,
I tried that for the test.
