[/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 
,
.
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
Let 
be a triangle and the points 
and 
on 
, 
and 
on 
, and 
and 
on 
be such that 
and 
. The intersections of 
with 
and 
are 
and 
respectively, and the intersections of 
with 
and 
are 
and 
, respectively. Prove that the lines 
and 
are concurrent.
be a triangle and the points 
and 
on 
, 
and 
on 
, and 
and 
on 
be such that 
and 
. The intersections of 
with 
and 
are 
and 
respectively, and the intersections of 
with 
and 
are 
and 
, respectively. Prove that the lines 
and 
are concurrent.
,
,
,
.
,
,
multiplying everything gives 
So from Menelaus 
are collinear. Since 
,
,
are collinear 
are concurrent(From Desaurges Theorem). So 
are collinear. Similar for 
and 
.
, 
,
are collinear 
are concurrent(From Desaurges Theorem). We are done.
and let 
.
passing through the point 
,
,
.![[asy]
import geometry;
size(10cm);
pair A=dir(120);
pair B=dir(210);
pair C=dir(330);
pair L= .2A+.8B;
pair K=.8A+.2B;
pair N=.3B+.7C;
pair M=.7B+.3C;
pair P=.25A+.75C;
pair Q=.25C+.75A;
pair R=intersectionpoint(line(K,N),line(M,Q));
pair T=intersectionpoint(line(K,N),line(L,P));
pair D=intersectionpoint(line(P,N),line(M,L));
pair E=intersectionpoint(line(P,N),line(K,Q));
pair X=intersectionpoint(line(E,B),line(D,R));
pair F=intersectionpoint(line(K,Q),line(L,M));
pair S=intersectionpoint(line(M,Q),line(L,P));
draw(conic(K,L,M,P,Q),deepgreen);
draw(B--E,grey); draw(A--D,grey); draw(C--F,grey); draw(N--K); draw(Q--M); draw(L--P);
draw(A--B--C--cycle, red);
draw(D--E--F--cycle);
dot("A",A,dir(A));
dot("B",B,dir(B));
dot("C",C,dir(C));
dot("K",K,dir(120));
dot("L",L,dir(185));
dot("M",M,dir(250));
dot("N",N,dir(-50));
dot("P",P,dir(0));
dot("Q",Q,dir(30));
dot("R",R,dir(10));
dot("T",T,dir(60));
dot("D",D,dir(280));
dot("E",E,dir(20));
dot(X);
dot("F",F,dir(165));
dot("S",S,dir(120));
[/asy]](http://latex.artofproblemsolving.com/b/1/4/b148db34973f8fc27cf21c4755da3ff5bd2cf2dd.png)
