[/quote]
,
and 
.
be a function that satisfies the following equation for all real numbers 
:![\[
f(x^2 + x + 3) + 2f(x^2 - 3x + 5) = 6x^2 - 10x + 17.
\]](http://latex.artofproblemsolving.com/f/6/2/f625b922656061221157fdb5a3b958cfa7d57108.png)
Find the value of 
.
be a given integer. The Mint issues coins of 
different values 
,
is a positive integer (the number of coins of each value is unlimited). A set of values 
is called lucky, if the sum 
can be collected in a unique way (namely, by taking one coin of each value).
with 
(b) Prove that every lucky set of values 
consecutive even numbers, such that the square of their product divides the sum of their squares.
is a given prime number
.
non-empty subsets 
of 
such that 
where 
are non negative integers, such that
for every 
.
Prove that
Where 
Prove that
be real numbers such that 
.
.
with the property that for every real numbers 
and 
it holds that
along the perimeter of a circle, then every word of the length 
consisting of letters 
,
,
,
,
,
,
,
)
-
(i.e. an ordered 
be 
positive integers such that the 
products![\[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\]](http://latex.artofproblemsolving.com/0/2/c/02cbff15c4d0118156a9d24be09360facb264d5a.png)
form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.
be complex numbers such that 
and 
.
be positive real numbers such that 
and 
,
Y by Adventure10, Mango247, Mango247
Let 
and 
be two concentric circles in the plane with radii 
and 
respectively. Show that the orthocenter of any triangle inscribed in circle lies in the interior of circle . Conversely, show that every point in the interior of is the orthocenter of some triangle inscribed in .
and 
be two concentric circles in the plane with radii 
and 
respectively. Show that the orthocenter of any triangle inscribed in circle lies in the interior of circle . Conversely, show that every point in the interior of is the orthocenter of some triangle inscribed in .
,
be any triangle inscribed in 
,
and 
.
.
of 
is given by 
.![\[ |\mathbf{h}| = |\mathbf{a} + \mathbf{b} + \mathbf{c}| \leq |\mathbf{a}| + |\mathbf{b}| + |\mathbf{c}|= 3R, \]](http://latex.artofproblemsolving.com/0/a/0/0a009f1bb4393919b0c369a70134c17a7dc7cdf8.png)
and it follows that 
since the maximumum value of the cos function is 
which exists at angle 
not possible....
assumes every value between 
and 
with equality existing only in the latter.....
is the orthocenter of some triangle inscribed in 
.
and 
with equality existing only in the latter.....
intersects 
,
.
lies in 
.
,
where 
(
it suffices to prove that k can be written as 
.![\[\cos x+\cos y+\cos z=k\]](http://latex.artofproblemsolving.com/4/7/2/47204cc91956209023de48cc60e7afaff018988d.png)
and ![\[\sin x+\sin y+\sin z=0\]](http://latex.artofproblemsolving.com/4/1/2/4124fd59cce423863de14ab9b817fc39c5e3adce.png)
Choosing 
(
)
we get a triangle.
![\[OH < AH + AO = 2R\cos A + R \leq 3R\]](http://latex.artofproblemsolving.com/5/6/2/5620edffd442e97bfe90773fa0704e035c3b6b2e.png)
be any point in 
.
at 
.
,
.
and the reflection of 
lies on 
.
