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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 . The chords cannot share common endpoints, also, the endpoints of these chords should be among the points.
pairwisely non-intersecting chords can be drawn for each of whom its endpoints differ in at most 
. cannot be improved.
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 
, where![\[
b_k = a^{k^k} \cdot 2^{2^k - k} + 1.
\]](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 . Find the number of positive integers 
less than 
such that 
is divisible by 
but not 
.
be the product of all positive integers less than not divisible by 
. Let 
be the remainder when 
is divided by 
. If 
, find 
.
be a tetrahedron such that 
A sphere of radius 
is circumscribed about tetrahedron 
Given that 
, 
, and 
, prove 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
through are placed on a 
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 an isosceles triangle such that 
. Let 
be a dot on the 
side.
at point intersects the circumcircle of 
at 
. Prove that CD
AB
be an equilateral triangle.M and N are the midpoints of 
and 
respectively. Externally to the triangle an isosceles right triangle APC is constructed, with the angle 
.If point 
is the intersection of 
and 
, show that 
is a bisector of the angle 
implies 
. Then, the converse follows by analogy.
is a parallelogram. Then, 
and 
with directed segments (segments on parallel lines are directed equally). Moreover, 
and 
; in other words, 
and 
.
; on the other hand, and 
. This entails that triangles 
and BCA are homothetic, so that 
, and 
, i. e. 
, and 
. From , we have also 
.
is a parallelogram, and we get 
and 
. We rewrite as 
. On the other hand, , as we have seen before. Furthermore, it is obvious that 
. Thus, the triangles 
and ZYX are homothetic, and we get 
, or, in other words, 
. Rewriting this, we get 
.
, 
, 
be three equal segments on the sides of an equilateral triangle. Prove that in the triangle formed by the lines 
, 
, 
, the segments , , are proportional to the sides in which they are contained.
,
.
be two triangles. Define![\[A_1=BC\cap ZX, A_2=BC\cap XY,\]](http://latex.artofproblemsolving.com/d/d/e/ddef901f1eb9be840b90ea8ae8802f526cb44ea7.png)
![\[B_1=CA\cap XY, B_2=CA\cap YZ,\]](http://latex.artofproblemsolving.com/2/8/7/287b4d9ff91098e3895bccf8203970694e6c806a.png)
![\[C_1=AB\cap YZ, C_2=AB\cap ZX.\]](http://latex.artofproblemsolving.com/d/5/a/d5a12487695a0a4583d24e36f5b4bf5612d3214c.png)
means the point of intersection of two lines 
and 
.
holds if and only if 
.