AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
be a sequence of positive real numbers such that for every 
, we have:![\[
a_n = \max_{1 \leq i \leq 2025} a_{n-i} - \min_{1 \leq i \leq 2025} a_{n-i}
\]](http://latex.artofproblemsolving.com/e/3/0/e30e9e533eb9a41cf847484e676ac4522edda665.png)
Prove that there exists a natural number 
such that for all 
, the following holds:![\[
a_n < \frac{1}{1404}
\]](http://latex.artofproblemsolving.com/0/1/f/01f1a81eed2230332d51a15501ef2a6ad3a7ec82.png)
into rectangular form.
, 
, 
, 
, where and 
are odd primes and![\[2^ap^b=(p+2)^c-1.\]](http://latex.artofproblemsolving.com/5/8/5/58518a8ebed14836b72af743ff100eae2efba5e2.png)
, such that for any positive integer 
and non-negative reals 
, the following inequality always holds:
Here 
and 
denote the median and arithmetic mean of 
, respectively.
be an isosceles triangle with 
, and 
an arbitrary point on side 
. The internal angle bisector of 
meets the circumcircle of 
again at 
, and the internal angle bisector of 
meets it again at 
. Show that lines 
, 
and 
are concurrent.
be non-negative real numbers. Prove that
be a triangle such that 
. Points 
lie on segments 
, respectively, such that lines 
and 
are the angle bisectors of triangle 
. Find the measure of angle 
.
satisfy 
prove in equality.
has circumcircle 
and circumcenter 
. A circle 
with center 
intersects the segment at points 
and 
, such that 
, , , and 
are all different and lie on line in this order. Let 
and 
be the points of intersection of and , such that , , , , and lie on in this order. Let 
be the second point of intersection of the circumcircle of triangle 
and the segment 
. Let 
be the second point of intersection of the circumcircle of triangle 
and the segment 
.
and 
are different and intersect at the point 
. Prove that lies on the line 
.
be a point inside a triangle 
such that![\[ \angle APB - \angle ACB = \angle APC - \angle ABC.
\]](http://latex.artofproblemsolving.com/a/6/e/a6e3dc27d0457682fa22af4f918f58d9cd8403bc.png)
, 
be the incenters of triangles 
, 
, respectively. Show that the lines 
, 
, 
meet at a point.
on ![$[1,n]$](http://latex.artofproblemsolving.com/2/2/4/2240bb122bdadbe086c0a6291dcd5a3eb82ad5cf.png)
as a word with 
in some order, hence 
is a permutation on 5 symbols. We write 
for the wellness of namely the amount of pairs 
that appear in the same order in the word. Here 
.
for the number of 
with wellness 
. Hence we want to determine 
. hence 
. The symbol k adds 
to the wellness hence now we count the permutations on ![$[1,7]\setminus \{k\}$](http://latex.artofproblemsolving.com/2/a/3/2a3fb0361b0937dbfba193cae8ae422eb3328c31.png)
with wellness . Since we do not consider the symbol anymore, we may as well count the number of permutations on ![$[1,6]$](http://latex.artofproblemsolving.com/a/b/0/ab036e7da17bee704b3b7ee0ca5ea79f424465e5.png)
with wellness . This equals 
.
.. If the first symbol is 
then it contributes 
to the wellness hence 
. We find 
. The lower bound may be 
, in that case we start counting at 
. This reduces the problem to finding the numbers 
. Etc.
and then work your way upwards.
of 
by a unique 
-tuple of integers 
such that 
for all valid , where each represents the number of pairs of integers 
satisfying 
. Thus, we seek the number of -tuples such that for all valid and 
.
in the expansion of 
. But notice that 
is a root of 
. Hence, the sum of the coefficients of the terms of with degree divisible by is 
. Observe that the coefficients of 
and 
are trivially each 
, and the coefficients of 
and 
are equal due to symmetry. Thus, the coefficient of is simply 
.
Prove that
satisfying
Prove
satisfying 
and 
.