[/quote]
,
,
be the point which minimizes the sum of squares of distances from the sides of the triangle. Let 
the projections of 
is the barycenter of 
.
be the set of all positive real numbers. Find all strictly monotone (increasing or decreasing) functions 
such that there exists a two-variable polynomial 
with real coefficients satisfying
for all 
.
.
.
be the midpoint of the arc 
not containing 
and define 
similarly. Show that the orthocenter of 
is the incenter 
of 
sides 
.
consecutive vertices is constant and its value is 
.
is marked with 
and 
is marked with 
,
.
intersects circle 
at 
,
intersects circle 
at 
.
and 
are isogonal conjugates in triangle 
,
.
.
are such that 
. Prove that 
.
,
.
denote the set of positive real numbers. Find all functions 
such that![\[x+f(yf(x)+1)=xf(x+y)+yf(yf(x))\]](http://latex.artofproblemsolving.com/1/6/c/16c10ac3b45b6b3f864f9314c3aeae654db514db.png)
for all 
is the midpoint of the hypotenuse 
.
is chosen on the leg 
so that the line segment 
meets 
again at 
)
be the reflection of 
meet again at 
)
.
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 
,
,
