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be a sequence of positive integers with 
, 
and![\[a_n = a_{n-1}^{a_{n-1}a_{n-2}}-1\]](http://latex.artofproblemsolving.com/1/b/f/1bf1a88979e7d2b33530de979d66a4e81032953d.png)
for all 
. Show that if 
is a prime less than 
for some positive integer 
, then there exists 
such that 
.
and 
, there cannot exist infinitely many 
such that![\[
\sqrt{n + a} + \sqrt{n + b} \in \mathbb{Z}.
\]](http://latex.artofproblemsolving.com/5/4/d/54d20546de5736770a9929ac86163e3b06553b7d.png)
![\[
x = \sqrt{n + a} + \sqrt{n + b} \in \mathbb{N}.
\]](http://latex.artofproblemsolving.com/6/6/8/668d53723d307556536736fda0e4a19be848834f.png)
![\[
x^2 = (\sqrt{n + a} + \sqrt{n + b})^2 = (n + a) + (n + b) + 2\sqrt{(n + a)(n + b)}.
\]](http://latex.artofproblemsolving.com/1/9/f/19f2627a432354a757d3c9e8ba42e96f83f46726.png)
So:![\[
x^2 = 2n + a + b + 2\sqrt{(n + a)(n + b)}.
\]](http://latex.artofproblemsolving.com/0/7/3/073602693604bbf3cb1f5c1c2e5051e8994a5e78.png)
![\[
\sqrt{(n + a)(n + b)} \in \mathbb{N}.
\]](http://latex.artofproblemsolving.com/1/1/4/11441fab3ba768fee7a0eef4dbf68124d6637aec.png)
![\[
(n + a)(n + b) = k^2.
\]](http://latex.artofproblemsolving.com/0/e/c/0ec5e7bac2aebe9ec507727ed21f7ca028655222.png)
Assume 
. Then we have:![\[
n + a \mid k \quad \text{and} \quad k \mid n + b,
\]](http://latex.artofproblemsolving.com/7/5/f/75fc52b6662df3a4c0cb2f40f49c4b3351c90a4d.png)
or it could also be that 
.![\[
(n + a)k_1 = k \quad \text{and} \quad kk_2 = n + b.
\]](http://latex.artofproblemsolving.com/1/f/5/1f5936241f7bebcf15c68fed5267bbfad2d40c2b.png)
![\[
k_1 k_2 = \frac{n + b}{n + a}.
\]](http://latex.artofproblemsolving.com/d/4/c/d4c1958e3554b58c3def282070f849842cc10912.png)
, we have:![\[
k_1 k_2 = 1 + \frac{b - a}{n + a}.
\]](http://latex.artofproblemsolving.com/3/0/6/306a6a00c44d4e4db1b36c7e588f684f28fdbf44.png)
, 
must be an integer, which is not possible..
and 
be the incircle and incenter of triangle 
. Let 
, 
, 
be the tangency points of to 
, 
, 
and let 
be the reflection of about . Assume 
intersects the tangents to at and at points 
and 
. Show that 
.![$f:[-1,1] \rightarrow \mathbb{R}$](http://latex.artofproblemsolving.com/d/d/4/dd4186fc296a916a1aedeb1f3b59ce20034e99d3.png)
be a continuous function such that 
. Prove that
![\[I=\int\limits_{-\infty}^{\infty}e^{x-10\cosh (2x)}\mathrm dx.\]](http://latex.artofproblemsolving.com/5/b/6/5b61a79360c164fcd935eb613e12ab056847a93c.png)
be a linear transformation on linear space 
satisfying:
but 
, and 
is the eigensubspace of eigenvalue 
. Prove that can be decomposed to 
-cyclic subspace's direct sum. over the field 
:
and 
functions are given with following properties:
is strict increasing and for each 
there holds 
or 
.
if holds and 
if holds.
and 
such that for all 
we have 
and 
be nonnegative numbers and 
be a nonnegative symmetric matrix such that 
. Given 
, consider the quadratic form
Assume that
there exists 
such that 
-the entry of the 
-th power 
of 
is positive. Show that has a unique minimum and the minimum lies in the open cube 
in 
.
such that for all 
, we have![$$
\sin y f(x)-\sin x f(y) \leq \sqrt[2025]{(x-y)^{20226}}
$$](http://latex.artofproblemsolving.com/8/9/6/896d8a02b5694dd3b18a9dc36a59f9fbcd9737f2.png)
be a ring not assumed to have an identity, with the following properties: that is not nilpotent.
are nonzero elements of , then 
. is a division ring, that is, the nonzero elements of R form a group under multiplication.
, let 
and 
denote the midpoints of and 
. Let be the projection of onto and let be the reflection of over the midpoint of 
. Assume lies in the interior of quadrilateral . Prove that 
.
as the unit circle.
. Also, 
.
is a parallelogram thus, 
and we get, 
. Now we have, 
and 
.
then we want 
. Now,
as desired. Thus 
. (
) 
and 
and 
respectively. Without loss of generality, let 
and 
lie on the unit circle.
Therefore, it follows that the midpoint of 
Then, since we obviously have 
it follows that
Now, in order to show that 
it suffices to show that 
Using all previous results and the fact that 
we see that
Then, taking the conjugate of this, we get 
concluding the proof. 