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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.
, satisfying
are consecutive integers. is an integer.
, satisfying are consecutive integers. is an integer.
, with 
. By Heron's formula, the area of the triangle is 
. In order for this to be an integer, we must have 
for some integer . Using 
, we get that both 
and are even (If is odd, then 
, which cannot result in a quadratic residue of 
). Thus, 
is divisible by , so if exists, then is an integer.
since 
. Thus, let 
and 
. Substituting these values in gives us the Pell equation 
. This has fundamental solution 
. Thus, using the recursive formula 
, we get the following solutions: 
. These give the side lengths triplets of 
.
