be an acute scalene triangle with circumcenter 
.
be the feet of the altitudes from 
to 
respectively. Let 
be the midpoint of 
.
be the circle with diameter 
.
be the intersection of 
and 
.
be the orthocenter of 
be the intersection of 
and 
be the lines through 
tangent to 
respectively. Let 
be the intersection of 
and 
.
be the intersection of 
and 
.
be the line through 
and let 
be the reflection of 
is tangent to 
.
,
,
terms is divisible by 
be a positive integer divisible by 3, and let 
be non-negative real numbers satisfying 
.![\[
M_n = \min_{1 \le i \le n} \{x_i x_{3i}\},
\]](http://latex.artofproblemsolving.com/d/b/b/dbbf0f6bf20f506d6cb63357b06ec6c52bebed10.png)
where the indices are taken modulo 
.
be pairwise distinct numbers. Prove that someone can divide the difference of these number into two sets with equal sum.
)
be two 
complex matrices of the same rank, and let 
if and only if 
.
to find the third side 
He uses logarithms as follows. He finds 
and doubles it; adds to that the double of 
subtracts the sum of the logarithms of 
and 
divides the result by 
and takes the anti-logarithm. Although his method may be open to suspicion his computation is accurate. What are the necessary and sufficient conditions on the triangle that this method should yield the correct result?
be a sequence of positive numbers and suppose that 
is divergent. Let further 
.
of positive numbers such that 
is convergent and
.
be given real numbers and let 
for all 
.
converges and find its limit.
.
be an 
.
is zero.
of positive integers fast-growing if 
for all 
.
of real numbers and a sequence 
of positive integers, let us denote![\[
S(a, b) = \sum_{n=1}^{\infty} \left| a(b_n) + a(b_n + 1) + \cdots + a(b_{n+1} - 1) \right|.
\]](http://latex.artofproblemsolving.com/2/f/c/2fc080e56fa9d8d6b1cf8a0bb2cb404a65d9ec73.png)
such that for every sequence ![\[
\sum_{n=1}^{\infty} a(n), \quad S(a, b) \quad \text{and} \quad S(a, c)
\]](http://latex.artofproblemsolving.com/0/8/a/08aa112d16a9c772145ff03a74fb512b4d02340d.png)
are convergent, then the series 
is also convergent?
such that for every sequence ![\[
S(a, b), \quad S(a, c) \quad \text{and} \quad S(a, d)
\]](http://latex.artofproblemsolving.com/5/8/9/589be33c766fde9e227f56d280d02322585e2ba3.png)
are convergent, then the series 
be a ring with 
its centre 
)
the set of invertible elements and 
.
has the property 
if there exists a subgroup 
of group 
such that 
,
and 
for any 
.
has the property 
,
.
and 
are the identity elements for 
and 
does the series 
converge (and absolutely converge)?
Y by Funcshun840
A computer science teacher has asked his students to write a program that, given a list of 
numbers 
, calculates the list 
where 
is the number of times the number 
appears in the list. So, for example, for the list 
, the program returns the list 
.
Next, the teacher asked Alexandre to run the program for a list of
numbers. Then he asked him to apply the program to the resulting list, and so on, until a number greater than or equal to 
appears in the list. Find the largest value of for which, whatever the initial list of positive integers 
, it is possible for Alexander to do what the teacher asked him to do.
numbers 
, calculates the list 
where 
is the number of times the number 
appears in the list. So, for example, for the list 
, the program returns the list 
.Next, the teacher asked Alexandre to run the program for a list of
numbers. Then he asked him to apply the program to the resulting list, and so on, until a number greater than or equal to 
appears in the list. Find the largest value of for which, whatever the initial list of positive integers 
, it is possible for Alexander to do what the teacher asked him to do.
.
and we are done.
