[/quote]
,
to 
face-down on a table. On each move, the Detective can point to two cards and inquire if the numbers on them are consecutive. The Magician replies truthfully. After a finite number of moves, the Detective points to two cards. She wins if the numbers on these two cards are consecutive, and loses otherwise.
questions.
= 2∠AMP.
is the midpoint of the arc 
,
,
,
be the 
intersects 
.
is perpendicular to 
,
is the incenter of 
with 
and the circle 
of radius 
centered at 
Let 
be the midpoint of 
The line 
intersects 
Let 
be a point on 
Let 
be the intersection of 
and 
cities. For every two cities there exists exactly one direct one-way road between them. We say that a path from 
to 
is a sequence of roads such that one can move from 
and 
be two distinct cities in Anisotropy. Let 
denote the maximal number of paths in a diverse collection of paths from 
denote the maximal number of paths in a diverse collection of paths from 
holds if and only if the number of roads going out from 
and excentres 
and 
,
and 
respectively. The incircle touches 
and 
at 
and 
respectively. Prove that the circles 
and 
have a common point other than 
and 
and 
.
and 
,
.
intersect 
at point 
.
.
such that for all integers 
and 
,![\[ f(2x+f(y))+f(f(2x))=y. \]](http://latex.artofproblemsolving.com/b/0/d/b0d16e87a8b30ebec5997ed12254094cdc3125d5.png)
,
not dividing any of them, and an integer 
and looks at the highest power of 
(which may depend on 
)
.
![$\{x_n\} \subseteq [0,1]$](http://latex.artofproblemsolving.com/8/e/4/8e4206325387485d73765c9d7070ecb1ce18ff60.png)
,
,
and 
could be chosen to satisfy 
and 
.
and there is a morphism 
from the group 
to the monoid 
such that for all 
,
such that 
.
Y by
Consider the real number written in decimal notation:
r = 0.235831...
where, starting from the third digit after the decimal point, each digit is equal to the remainder when the sum of the previous two digits is divided by 10.
Which of the following statements is true?
(a) (10⁶⁰ - 1).r is an integer
(b) (10²⁵ - 1).r is an integer
(c) (10¹⁷ - 1).r is an integer
(d) r is an irrational algebraic number
(e) r is an irrational transcendental number
(Recall that a complex number is called algebraic if it is a root of a non-zero polynomial with integer coefficients.)
r = 0.235831...
where, starting from the third digit after the decimal point, each digit is equal to the remainder when the sum of the previous two digits is divided by 10.
Which of the following statements is true?
(a) (10⁶⁰ - 1).r is an integer
(b) (10²⁵ - 1).r is an integer
(c) (10¹⁷ - 1).r is an integer
(d) r is an irrational algebraic number
(e) r is an irrational transcendental number
(Recall that a complex number is called algebraic if it is a root of a non-zero polynomial with integer coefficients.)
.
is rational). To calculate the period it suffices to consider the Fibonacci numbers mod 
and mod 
.
mod 
,
,
