AoPS Academy
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)
, 
, 
, 
, 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.
, 
and 
are points on sides 
and , respectively, such that 
. If 
and 
, find the area of .
Find 
.
Taking the square root of the second equation, we get
has a right angle at 
, with 
and 
. If 
and 
are points
and 
, respectively, such that 
, find the perimeter of quadrilateral
.
, 
, 
are positive integers such that 
. Find all possible values of 
.
to rearrange the terms and place the higher-degree terms at the front of the expression in order to have a more recognizable form.
![\begin{align*}
a^4+b^4+c^4-2a^2b^2-2a^2c^2-2b^2c^2 &=a^4-2a^2(b^2+c^2)+b^4-2b^2c^2+c^4 \\
&= a^4-2a^2(b^2+c^2)+(b^2+c^2)^2 - 4b^2c^2 \\
&= [a^2 - (b^2+c^2)]^2-(2bc)^2 \\
&= (a^2-b^2-c^2+2bc)(a^2-b^2-c^2-2bc)\\
&=[a^2-(b-c)^2][a^2-(b+c)^2] \\
&= (a+b-c)(a-b+c)(a+b+c)(a-b-c) = -63.
\end{align*}](http://latex.artofproblemsolving.com/0/c/4/0c430b7562ec7571fdf3e492939c354e0c3e7e20.png)
, we get
.
, 
, or 
, we get something of the form![\[
a^2(a+2b)(a-2b)=-63.
\]](http://latex.artofproblemsolving.com/1/0/0/1006c2fabc10f934c25b6e6ae822a59ce2dd2712.png)
and 
are the only square factors of 
, we have 
.
are 
and 
, including all their permutations.
are all distinct positive integers, there are no solutions. are 
.
. We have
is an imaginary number. Thus, N must be equal to +2. We then have
and 
, find the value of 
.
. We have
.
be prime numbers such that their sum isn't divisible by 
. Find the all 
positive integer quadruples satisfy:
is a positive integer, then 
divides 
. Now, see that the 
which is 
or 
; but, as divides 
then divides .
then is clear than 
, 
, 
and is even, or 
, 
, and positive integer.
, then or ; otherwise, and 
are and 
and divides . So, if and 
, sp 
. So 
and 
, 
and 
or 
and 
. If and 
then and .
Prove that
side length of 
area of 
is a right angle, and 
,
.
is a right angle, 
,
.
and 
both contain a right angle and 
through AA postulate. Because they are similar, 
,
,
,
.
.
,
.
and 
Prove that
.
,
is a right angle, using pythagorean theorem, we can get 
.
,
.
from point 
be the intersection of said altitude and 
.
both have a right angle and 
,
through AA Postulate. 
,
,
.
(CD = DE, as stated in the problem), 
,
(they were formed by an altitude, and are therefore both right angles), we can show that 
through HL Theorem. 
corresponds with 
,
.
,
and 
.
.
.
and 
for every 
-
:
