[/quote]
,
,
,
by the recurrence![\[
a_{n+1} = a_n^2 + 1
\]](http://latex.artofproblemsolving.com/3/1/1/3114595ded8c8961d611afaab9ad19e254973001.png)
Prove that there is no natural number 
such that![\[
\prod_{k=1}^{n} \left( a_k^2 + a_k + 1 \right)
\]](http://latex.artofproblemsolving.com/a/4/f/a4f88ee7e3d4e53e369319f74cd9b12be26c9681.png)
is a perfect square.
written on the board. We apply the following rule: At each step, we will add all the numbers that are the sum of the two numbers on the board so that the sum does not appear on the board. For example, if the two initial numbers are 
;
;
;
,
;
,
be a cyclic quadrilateral with circumcenter 
,
is not a diameter of its circumcircle. The lines 
and 
intersect at point 
,
lies between 
and 
lies between 
and 
is acute and let 
be its orthocenter. The points 
and 
on the lines 
and 
.
,
.
be real numbers such that ![\[ab^2+bc^2+ca^2=6\sqrt{3}+ac^2+cb^2+ba^2.\]](http://latex.artofproblemsolving.com/3/d/0/3d015292c249a69a44d5f923092201d7a271b896.png)
Find the smallest possible value of 
.
distinct positive integers and then writes in blue the 
sums of each two red numbers. At most how many of the blue numbers can be prime?
,
pass through the orthocenter and not through point 
.
.
to 
,
as the line passing through 
and perpendicular to 
as the line passing through 
and perpendicular to 
.
is the reflection of the intersection of 
.
,
be the number of positive integer divisors of 
we have that:![\[ d(a)+d(b) \le d(\gcd(a,b))+d(\text{lcm}(a,b)) \]](http://latex.artofproblemsolving.com/a/3/b/a3bfe462d3888517939eec8a2827ee6bd8d627a1.png)
and determine all pairs of positive integers 
denote the set of positive integers. Find all functions 
such that for positive integers 
and 
![\[f(a^2 + b^2) = f(a)f(b) \text{ and } f(a^2) = f(a)^2.\]](http://latex.artofproblemsolving.com/c/1/f/c1f8ffe04cfcc45b498f3931a4796d5c56dc04d0.png)
.
.
so that any 
.
and 
be concentric circles, with 
to 
)
and 
intersect at a point 
on 
.
intersects line 
and 
.
.
with 
.
and 
and 
.
meets 
meets 
.
.
Y by mjkkra, CoordinateBash13, Adventure10, Mango247, and 1 other user
In a small pond there are eleven lily pads in a row labeled 
through 
. A frog is sitting on pad 
. When the frog is on pad 
, 
, it will jump to pad 
with probability 
and to pad 
with probability 
. Each jump is independent of the previous jumps. If the frog reaches pad it will be eaten by a patiently waiting snake. If the frog reaches pad it will exit the pond, never to return. What is the probability that the frog will escape being eaten by the snake?
through 
. A frog is sitting on pad 
. When the frog is on pad 
, 
, it will jump to pad 
with probability 
and to pad 
with probability 
. Each jump is independent of the previous jumps. If the frog reaches pad it will be eaten by a patiently waiting snake. If the frog reaches pad it will exit the pond, never to return. What is the probability that the frog will escape being eaten by the snake?
This post has been edited 2 times. Last edited by LauraZed, Dec 27, 2018, 4:41 PM
Reason: fixed capitalization
Reason: fixed capitalization
.
and we use our intuition about symmetry to find that 
.
?
be the probability from lily pad 
(so 
,
)
Adding them all, we get![\[ 1 = p_{10} = \left(
\frac{1}{\binom90} + \frac{1}{\binom91}
+ \frac{1}{\binom92} + \dots + \frac{1}{\binom99}
\right) p_1. \]](http://latex.artofproblemsolving.com/2/2/6/2268bed3c36a298b8959e4a42661246ae870cc70.png)
Equivalently, one can also add up to just 
to derive the equivalent![\[ \frac12 = p_{5} = \left(
\frac{1}{\binom90} + \frac{1}{\binom91}
+ \frac{1}{\binom92} + \frac{1}{\binom93} + \frac{1}{\binom94}
\right) p_1. \]](http://latex.artofproblemsolving.com/e/1/c/e1c234216c2dc9c5988d35e86d3b4100eb849016.png)
In any case, we have![\[
p_1 =
\frac{1}{2\sum_{k=0}^4 \binom9k ^{-1} } = \frac{63}{146}.
\]](http://latex.artofproblemsolving.com/e/0/a/e0a6ad11d4ffc7e2e85f89b117aa7fe34ef956e9.png)
and similar equations to what you have above?
.
?
denote the probability that starting from the 
Using our recursion on 
and 
again we obtain 
from which we obtain 
.
