,
,
,
satisfy 
and 
.
of 
such that 
.
be a cyclic quadrilateral. A circle centered at 
passes through 
and 
and meets lines 
and 
again at points 
and 
(distinct from 
)
denote the orthocenter of triangle 
Prove that if lines 
are concurrent, then triangle 
and 
are similar.
such that 
, where 
is a power series for which each coefficient 
is 
or 
.
,
must be irrational.
.
s.t. 
.
,![$K=\mathbb{Q}[\cos(\dfrac{2}{3}\arctan(3\sqrt{3}))]$](http://latex.artofproblemsolving.com/a/2/e/a2efd64f27687f9ba5239cc4e0ba49d469d1c8c9.png)
.
.
.
.
;
, ∀x∈[0,1].
be a finite abelian group. There is a magic box 
.
contains only the group identity, 
.
be a positive real number. Divide the positive real axis into intervals 
,
,
,
,
,
satisfying the following differential equations:![\[
u''(x) + 9^2u(x) = 0, \quad \text{for } x \text{ in black intervals},
\]](http://latex.artofproblemsolving.com/2/6/8/268b7bc6150b13eed666d566b540df7e643d8344.png)
![\[
u''(x) + 63^2u(x) = 0, \quad \text{for } x \text{ in white intervals},
\]](http://latex.artofproblemsolving.com/b/4/6/b46d0d00b34cd3fb72ec310625384547b14871c5.png)
with the initial conditions:![\[
u(0) = 1, \quad u'(0) = 1,
\]](http://latex.artofproblemsolving.com/4/1/6/416754efbff23b7efbf02b32149b1e1fd5cb9905.png)
and the continuity conditions:![\[
u(x) \text{ and } u'(x) \text{ are continuous functions}.
\]](http://latex.artofproblemsolving.com/b/3/0/b309f7b8c70cb27683a96af8452ebf9e36f67a4c.png)
Show that![\[
\lim_{\omega \to 0} u\left(\frac{\pi}{45}\right) = 0.
\]](http://latex.artofproblemsolving.com/f/0/b/f0b9105523f0561e0a4d151a2695cbf6917e5cc7.png)
Y by Adventure10, Mango247
Let 
be a natural number. The finite sequence 
of positive integer terms, there are different numbers ( can have repeated terms). Moreover, if from one from its terms any we subtract 1, we obtain a sequence which has, between its terms, at least different positive numbers. What's the minimum value of the sum of all the terms of ?
be a natural number. The finite sequence 
of positive integer terms, there are different numbers ( can have repeated terms). Moreover, if from one from its terms any we subtract 1, we obtain a sequence which has, between its terms, at least different positive numbers. What's the minimum value of the sum of all the terms of ?
,
,
,
.
appears 2 times in the sequence, but 
doesnt appears, then, removing one 
) .
be the number of 
in the sequence.
,
one possible sequence is:
with sum 
.
with sum 
.
,
and 
that alternate. Call them 
and 
and 
be respectively the sum that 
and finishing in 
.
,
integers. The idea is: find expresions for 
,
appear exactly 2 times (but 1 only once). Now, let 
be the biggest number, which is bigger than the half of 
,
for 
for 
