marbles in 
colors such that there are exactly 
marbles of each color. She wants to arrange them on 
shelves, 
marbles on each shelf such that for any 
colors there is a shelf that has marbles of those colors.
be an integer. The numbers 
are split into 
groups of 
quadruples 
such that they are all in different groups, 
and 
.
,
be positive real numbers so that 
.![\[ \left( a - 1 + \frac 1b \right) \left( b - 1 + \frac 1c \right) \left( c - 1 + \frac 1a \right) \leq 1.
\]](http://latex.artofproblemsolving.com/7/6/e/76e794a62090c8c4b5f3a70ad4d036669418ca69.png)
be an acute triangle with orthocenter 
.
which is parallel to 
and tangent to the incircles of 
and 
.
are three positive real numbers, prove that 
be positive integers. Prove that it is impossible to have all of the three numbers 
to be perfect squares.
be a positive integer. A positive integer 
digits in decimal representation (it cannot have leading zeros).
is a ![\[(20 + 25)^2 = 2025.\]](http://latex.artofproblemsolving.com/5/c/5/5c51bca485cea6c3750c48c09091eb5fd72b5192.png)
Find all 
-
be the set of all positive integers greater than or equal to 
and let 
be a function such that 
for all 
if 
,
and incenter 
.
meets 
at 
and meets 
;
cuts 
.
and 
,
is the midpoint of 
.
and 
intersect at points 
and 
.
or 
.
Y by Adventure10
Let 
and 
be positive real numbers with 
. Let 
be the maximum possible value of 
for which the system of equations
![\[ a^2+y^2=b^2+x^2=(a-x)^2+(b-y)^2\]](//latex.artofproblemsolving.com/1/3/4/13432a7d6e5c1465e08d513ee64c6c7ce99d75f9.png)
has a solution in 
satisfying 
and 
. Then 
can be expressed as a fraction 
, where 
and 
are relatively prime positive integers. Find 
.
and 
be positive real numbers with 
. Let 
be the maximum possible value of 
for which the system of equations![\[ a^2+y^2=b^2+x^2=(a-x)^2+(b-y)^2\]](http://latex.artofproblemsolving.com/1/3/4/13432a7d6e5c1465e08d513ee64c6c7ce99d75f9.png)
has a solution in 
satisfying 
and 
. Then 
can be expressed as a fraction 
, where 
and 
are relatively prime positive integers. Find 
.
and 
.
,
and 
.
and we need to maximize this for 
.
,
.![\[ \rho = \frac {\cos{\frac {\pi}{6}}}{\sin{\frac {\pi}{2}}} = \frac {\sqrt {3}}{2}
\]](http://latex.artofproblemsolving.com/5/e/0/5e049296f93c7894fe4c7c6e64d99319ef7e1e11.png)
and the answer is 
.
,
.
.
.
,
and 
are 
since
is an increasing function over (0, 90), 
is also increasing function over (60, 90).
maximizes at 
=> 
![\[ a^2+y^2=b^2+x^2=2(ax+by)\]](http://latex.artofproblemsolving.com/2/0/e/20ea4850a93a01e6458bd0f2eddc90dde91ef752.png)
,
.
,
.
satisfies the equation. Hence 
,
be maximized at 
?
which matches the other solutions...
,
.
on ![$[0,30]$](http://latex.artofproblemsolving.com/5/c/0/5c0d28e4a7273d9c9719062979757fc6b3272d08.png)
.
for all 
in this range. Then: ![\[\cos x\le k\sin(x+60)=k\cdot\left(\frac{\sqrt{3}}{2}\cos x+\frac{1}{2}\sin x\right)\]](http://latex.artofproblemsolving.com/f/b/8/fb8160a01b7f2675f19e4fe0610994ad2e940ce9.png)
![\[\rightarrow 0\le \left(\frac{\sqrt{3}}{2}-1\right)\cos x+\frac{k}{2}\sin x\rightarrow 0\le (\sqrt{3}k-2)\cos x+k\sin x\]](http://latex.artofproblemsolving.com/c/7/1/c7184c5cb46816cf5e223f36b2fc194db706e665.png)
![\[\rightarrow (2-\sqrt{3}k)\cos x\le k\sin x\rightarrow \frac{2-\sqrt{3}k}{k}\le \tan x,\]](http://latex.artofproblemsolving.com/e/a/8/ea8d072fb7f1f0fa8bfc9566694c32c308a26fe3.png)
for all 
,
must be less than or equal to 
,
.
.
attains a maximum of 
on the interval.
for 
,
and 
in that order (clockwise). The problem statement gives that these three points form an equilateral triangle. Let 
be the line parallel to the x-axis going through 
and let 
be the line parallel to the y-axis going through 
.
.
.
has length 
and width 
,
maximized reduces to finding the triangle such that 
giving us an answer of 
.
