such that![\[
f(f(x)+y) = x+f(f(y))
\]](http://latex.artofproblemsolving.com/9/d/7/9d780faa5901646f7035e88400a9e3d8023f04ff.png)
for all 
.
points in a circle and number them arbitrarily with numbers from 
to 
pairwisely non-intersecting chords can be drawn for each of whom its endpoints differ in at most 
.
satisfies that for any positive integer 
and 
Suppose there is 
non-negative reals 
such that for 
Determine the least possible value of 
is chosen. Prove that there exists a prime number that divides infinitely many terms of the sequence 
,![\[
b_k = a^{k^k} \cdot 2^{2^k - k} + 1.
\]](http://latex.artofproblemsolving.com/1/7/5/1751a60482d729a36c71b77ac9c978e724f40da0.png)
be the polynomial with leading coefficent 1 and rational coefficents, such that ![\[p\left(\sqrt{3 + \sqrt{3 + \sqrt{3 + \ldots}}}\right) = 0,\]](http://latex.artofproblemsolving.com/0/f/4/0f472be9f2cede0221e1b4d970d65de94dba01b9.png)
and with the least degree among all such polynomials. Find 
.
such that 
is a positive integral power of a prime positive integer."
for positive integral values of 
be the largest perfect square that divides 
less than 
such that 
is divisible by 
but not 
.
be the product of all positive integers less than 
.
be the remainder when 
is divided by 
.
,
.
be a tetrahedron such that 
A sphere of radius 
is circumscribed about tetrahedron 
Given that 
,
,
,![$$r^2+\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \ge \frac{9\sqrt[3]{4}}{4}$$](http://latex.artofproblemsolving.com/6/c/6/6c665b4611f5fc8ebeba03ac649bfee4e2634950.png)
with equality at ![$a=b=c=\sqrt[3]{2}.$](http://latex.artofproblemsolving.com/b/b/9/bb9c9671b23cb45e5f85ae01bf25d566c2e43e91.png)
th and 
th toss, is
square such that all rows and columns sum to the same value. Please note that diagonals do not need to sum to the same value. How many ways can this be done?
,
.
.
be a dot on the 
at point 
at 
A
.
is the intersection of 
and 
,
is a bisector of the angle 
Y by Adventure10, Mango247
In an acute triangle 
point 
is the point of intersection of altitude 
and side 
, and points 
are orthogonal projections of point on sides 
and 
. Lines 
and 
cross the circumcircle of triangle at points 
and 
. Prove that point is the center of the incircle of 
.
point 
is the point of intersection of altitude 
and side 
, and points 
are orthogonal projections of point on sides 
and 
. Lines 
and 
cross the circumcircle of triangle at points 
and 
. Prove that point is the center of the incircle of 
.This post has been edited 1 time. Last edited by j___d, Jul 27, 2016, 10:25 PM
thus 
so 
bisects 
and 
is midpoint of arc 
.
which is easy since we know 
.
and 
onto 
be 
respectively and 
be the circumcenter of 
.
is cyclic, isogonality and Reim's yields 
so 
.
so 
The Incenter-Excenter Lemma finishes. 
denote the reflections of 
and 
are collinear.
and 
.
and center 
may lead to another solution, as 
and 
are orthogonal. In particular, 
should be related to the 
-
