AoPS Academy
points are given on a circle. On a player's turn, that player must draw a triangle using three unused points as vertices, without creating any crossing edges. The first player who cannot move loses. If Elmo's clone goes first and players alternate turns, who wins? (Your answer may be in terms of 
.) greater than 10. Then, she multiplies all the digits of and obtains an odd number. Find all possible values of the units digit of .
and 
be prime numbers. Prove that if 
is a perfect square, then 
is also a perfect square.
cups labeled 
, where the 
-th cup has capacity liters. In total, there are liters of water distributed among these cups such that each cup contains an integer amount of water. In each step, we may transfer water from one cup to another. The process continues until either the source cup becomes empty or the destination cup becomes full.
Prove that from any configuration where each cup contains an integer amount of water, it is possible to reach a configuration in which each cup contains exactly 1 liter of water in at most 
steps.
Prove that in at most 
steps, one can go from any configuration with integer water amounts to any other configuration with the same property.
be prime numbers such that![\[
\frac{1}{pq}+\frac{1}{qr}+\frac{1}{pr}=\frac{1}{839}
\]](http://latex.artofproblemsolving.com/b/6/f/b6f882f3c93b0877419db625c2a427fc9561605b.png)
Hence find the numbers of ordered triples 
,there are 
equal to 
, and 
equal to 
. Find the sum of all the products 
, where 
.
for all real numbers 
satisfying the equation 
is:
be positive integers such that
Find the maximum possible value of 
.
be real numbers, such that![\[
a^2+b^2+c^2+abc=5
\]](http://latex.artofproblemsolving.com/7/c/8/7c8044347b7e25cb21dbef42c65ec351cefd4699.png)
![\[
a+b+c=3
\]](http://latex.artofproblemsolving.com/6/4/a/64a3034ae02dd28aa6220ee5c37a22d608074173.png)
Prove that atleast one of the numbers is equal to 
.
(base-10) into base-
, where is the smallest integer that has 8 factors. The answer will be the code to the safe.[/quote]
, and let the spheres be 
. We can assume that are each tangent to three faces of the sphere, and are in corners 
respectively.
scale factor 
that takes 
and 
, so that 
is inscribed in 
. Translate by 
to 
, which circumscribes 
. Note that 
, so 
from which the result follows.
,
.
.
has all edges of length less than 
,
.
.
- locus of centers of all spheres with diameters 
,
.
.
where 
-
,
so 
,