[/quote]
,
and let 
be positive integers such that 
.
and determine when equality holds.
prove that the triangle must have a right angle.
candidates and 
judges, where 
is an odd integer. Each candidate is evaluated by each judge as either pass or fail. Suppose that each pair of judges agrees on at most 
candidates. Prove that ![\[{\frac{k}{m}} \geq {\frac{n-1}{2n}}. \]](http://latex.artofproblemsolving.com/3/8/7/387c8824b28f6422306e3c25b34fccb0a841f4cd.png)
be real numbers such that![\[ a^2+x^2=b^2+y^2=c^2+z^2=(a+b)^2+(x+y)^2=(b+c)^2+(y+z)^2=(c+a)^2+(z+x)^2 \]](http://latex.artofproblemsolving.com/7/e/b/7ebfb2f663d390714310d3c43b9c68e1b101d632.png)
.
,
,
and 
,
is the intersection point of the diagonals 
and 
.
and radius 
intersects 
and 
.
is tangent to the circles with diameters 
and 
.
of non-negative real numbers such that the following three conditions are all satisfied:
;
;
of 
[/list]
is a permutation of 
,
.
for which there exist real numbers 
satisfying 
,
and
for 
.
and 
are such that, for every positive integer ![\[ a_n > 0,\qquad\ b_n>0,\qquad\ a_{n+1}=a_n+\dfrac{1}{b_n},\qquad\ b_{n+1} = b_n+\dfrac{1}{a_n}. \]](http://latex.artofproblemsolving.com/4/f/c/4fc9fd1cb31a25378268f20e019e639667e52910.png)
Prove that 
.
of positive integers![\[a_{n+2}=a_{n+1}+\sqrt{a_{n+1}+a_{n}} \]](http://latex.artofproblemsolving.com/4/a/9/4a9cc873fd72a001413f0bc1558025e0673ea9cd.png)
for every positive integer 
be an infinite sequence of positive integers. Prove that there exists a unique integer 
such that![\[a_n < \frac{a_0+a_1+a_2+\cdots+a_n}{n} \leq a_{n+1}.\]](http://latex.artofproblemsolving.com/9/4/f/94fc5a51b7e68588123c5b527fe75183bc4c4937.png)
be two positive sequences defined by 
and![\[ \begin{cases} {{x}_{n+1}}{{y}_{n+1}}-{{x}_{n}}=0 \\ x_{n+1}^{2}+{{y}_{n}}=2 \end{cases} \]](http://latex.artofproblemsolving.com/2/0/1/2012dc6ff3a41478547cea4aa9b6bccbe17a3623.png)
for all 
.
of real numbers such that 
and
members, each of whom is fluent in the same 
be the positive number satisfied 
Find the minimum of 
Y by MENELAUSS
Triangle 
has circumcircle 
and incircle 
, where is tangent to 
at 
, respectively. 
is an arbitrary point on . 
meets 
at 
, 
meets again at 
, 
meets again at 
. 
is a point on such that 
are colinear. Let 
be the intersection of 
and , prove that 
is tangent to .
Proposed by chengbilly
has circumcircle 
and incircle 
, where is tangent to 
at 
, respectively. 
is an arbitrary point on . 
meets 
at 
, 
meets again at 
, 
meets again at 
. 
is a point on such that 
are colinear. Let 
be the intersection of 
and , prove that 
is tangent to .Proposed by chengbilly
and 
(clearly 
)
Other way, 
These two cross-ratio implies that 
