AoPS Academy
be a sequence of positive integers and let 
be the sequence of real numbers given by
Show that, if there exists at least one term among every million consecutive terms of the sequence that is an integer, then there exists some 
such that 
.
satisfying 
the following inequality holds:
of the triangle 
consider a mobile point 
. Let 
the orthogonal projection of 
on 
. If the mobile points 
and 
are such that 
is a paralellogram, find the locus of point 
when goes through 
.
be an equilateral triangle, and is a point on its incircle. Let 
be the circle tangent to 
passing through and 
. Similarly, let 
be the circle tangent to passing through and , and 
be the circle tangent to 
passing through and 
., , has a common tangent line.
and 
Prove that 
be real numbers such that 
. Find the minimum of 
and 
Prove that 
people around in a circle, who at 
each have one penny. After each second every person gives deals penny they have to the left or right, or keeps that penny with equal probability. What is the expected value of seconds such that one person has all pennies? (Express answer in terms of ).
but I’m not sure. Purely guessing.
the answer is 
because from any arrangement there is exactly 
way to get to any of the other arrangements (consider the problem from the coin’s perspective as they switch humans. Label the humans 
. Therefore the initial arrangement is . Every time we add or subtract or leave alone each of the numbers. We can only obtain, say, 
(where every coin is in the second person’s hand), in one way: that is 
. This is true for every possible arrangement of coins from any possible arrangement of coins. Hence the probability of landing on 
or or 
is 
and the answer is .
are positive real numbers such that 
then 
.
![\[
\textbf{Problem:}
\]](http://latex.artofproblemsolving.com/5/c/3/5c311b6a956843df04ab5bdf4117e1089aecb9b9.png)
be the roots of the polynomial![\[
P(x) = x^{20} - 7x^3 + 1.
\]](http://latex.artofproblemsolving.com/1/8/c/18c475088b173b3cb3bf72d1d83ef8f2d40b8e36.png)
![\[
S = \frac{1}{r_1^2 + 1} + \frac{1}{r_2^2 + 1} + \dots + \frac{1}{r_{20}^2 + 1}
\]](http://latex.artofproblemsolving.com/b/d/6/bd63906510edd807858482b9c7ca3ca202da52d6.png)
, where 
and 
are coprime positive integers. Determine the value of 
.
and 
be elements of the set 
such that 
, 
is a divisor of 
, and 
is a divisor of 
. Determine the largest value that 
can take.
and 
, it follows that 
and 
divide 
. It is clear that cannot be less than or equal to 
and so 
, so
from which we obtain that 
. Then,
deducing that 
. To ensure that 
is the maximum possible value for , it is sufficient to show the following example:
![\[
19^{17^{15^{13^{11^{9^{7^{5^3}}}}}}} \mod 100
\]](http://latex.artofproblemsolving.com/d/3/d/d3d8657e2ab69cdc0886e2498ab03f7fd959056f.png)
then prove 
