[/quote]
,
,
,
,
,
,
initially contains one coin. The following operations are allowed
,
,
;
,
,
and 
.
coins.
and 
be positive integers with 
.
,
is a subset of 
such that every element of 
![\[ f(x)=x^n+a_{1}x^{n-1}+a_{2}x^{n-2}+\cdots+a_{n-1}x+a_n \]](http://latex.artofproblemsolving.com/0/9/d/09d80fe5c25c5c6ea57c545b68c7e11b319cbada.png)
with integer coefficients 
,
,
,
and 
such that ![\[ f(a)=f(b)=f(c)=f(d)=5, \]](http://latex.artofproblemsolving.com/8/2/d/82d0c5d0c7a6dd48802ad8fbe04b211278c1fa58.png)
show that there is no integer 
such that 
.
be positive real numbers so that 
.![\[ \left( a - 1 + \frac 1b \right) \left( b - 1 + \frac 1c \right) \left( c - 1 + \frac 1a \right) \leq 1.
\]](http://latex.artofproblemsolving.com/7/6/e/76e794a62090c8c4b5f3a70ad4d036669418ca69.png)
such that 
divides by 
s
n
r
t
hats, numbered from 
hats. After that, each wise man (in turn: first the first, then the second, etc.) loudly calls a number, trying to guess the number of his hat. The numbers mentioned should not be repeated. When all the wise men have spoken, they take off their hats and check which one of them has guessed. Can the sages to act in such a way that at least three of them knowingly guessed?
.
be the midpoint of 
be the midpoint of 
and let 
be the centroid of the triangle 
be a circle through 
.
and 
are collinear.
be the incentre of acute-angled triangle 
,
,
respectively. Let line 
intersect the circumcircle of the triangle at 
,
lies between 
.
is internally tangent to 
at 
and 
.
and 
meet at 
and intersect 
respectively. The ray 
meets the circumcircle of triangle 
again at 
.
.
such that points 
lie on the line 
is tangent to the segments 
and 
and to the circle 
is tangent to the segments 
and 
and to the circle 
and 
intersect at point 
.
be the set of all positive integers. Find all functions 
such that 
divides 
for all positive integers 
board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each 
or 
.
are given. Losyash likes some of the cells of the 
checkerboard. In addition, he is interested in any checkered rectangle with a perimeter of 
,
Y by Adventure10, Mango247
Let the Nagel point of triangle 
be 
. We draw lines from 
and 
to so that these lines intersect sides 
and 
in 
and 
respectively. 
and 
are midpoints of segments 
and 
respectively. 
is the second intersection point of circumcircles of triangles 
and 
. 
and 
are perpendicular lines to 
and 
in points and respectively. Prove that lines and intersect on the circumcircle of triangle .
Proposed by Nima Hamidi
be 
. We draw lines from 
and 
to so that these lines intersect sides 
and 
in 
and 
respectively. 
and 
are midpoints of segments 
and 
respectively. 
is the second intersection point of circumcircles of triangles 
and 
. 
and 
are perpendicular lines to 
and 
in points and respectively. Prove that lines and intersect on the circumcircle of triangle .Proposed by Nima Hamidi
This post has been edited 1 time. Last edited by goodar2006, Sep 28, 2012, 1:25 PM
at 
then 
.
,
.
,
are concyclic together with 
intersect at 
,
and 
.
,![\[\angle ABG=\angle AMG+\angle BGM=\angle ATG+\angle CGT=\angle ACG\]](http://latex.artofproblemsolving.com/e/c/9/ec90911fa84290260df01aab5a5540451916c5eb.png)
, so 
are indeed concyclic.
so 
and 
are cyclic. Thus 
and 
.
so 
is congruent to 
.
are corresponding medians, so 
so 
is cyclic. Thus 
is cyclic so 
.
is congruent to 
so 
.
and since 
is cyclic, 
so by SAS 
is congruent to 
.
so 
so 
intersect on the circumcircle of 
.
