[/quote]
,
,
is a prime.
board, a nonnegative real number is written in such a way that the sum of the numbers in each row is equal to 
,
to be the largest value in row 
,
.
to be the largest value in column 
.
?
we consider a 
rectangular frame, consisting of the 
boundary squares of a 
for which Renate has a winning strategy.
cats consisting of five different breeds. He records the number of cats of each breed and after multiplying these five numbers he obtains the number 
.
,
is a perfect square.
,
be all positive integers smaller than 
such that 
for all 
is the largest positive integer that divides both 
and 
.
.
be an integer. In a configuration of an 
board, each of the 
cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate 
counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of 
and 
.![\[P=\frac{1}{5-2a}+\frac{1}{5-2b}+\frac{1}{5-2c}.\]](http://latex.artofproblemsolving.com/c/3/7/c3708768a84c2bbb1e4fae502a9bb1e92cb13d02.png)
student. We know that in this school every two students have attended exactly in one common class. Also due to smallness of school each class has less than 
is not a perfect square, prove that there exist a student that has attended in at least 
classes.
,
,
,
'![\[n=a_1+a_2+\dots+a_m,\]](http://latex.artofproblemsolving.com/6/c/2/6c2a979a5ff6c40bf2d7152c8c32088fc36f848b.png)
where 
,
,
,
and 
.
and 
Prove that
where 
be a function defined on 
taking values in 
such that
for all 
for all 
;
identical-looking coins in a circle. It is known that exactly two of them are fake. Real coins weigh the same, fake ones too, but they are lighter than the real ones. How can you determine in three weighings on a cup scale without weights whether there are fake coins lying nearby or not??
Y by Adventure10 and 1 other user
Two circles with different radii intersect in two points 
and 
. Let the common tangents of the two circles be 
and 
such that 
lie on the first circle, and 
on the second. Prove that the orthocenters of the triangles 
, 
, 
and 
are the four vertices of a rectangle.
and 
. Let the common tangents of the two circles be 
and 
such that 
lie on the first circle, and 
on the second. Prove that the orthocenters of the triangles 
, 
, 
and 
are the four vertices of a rectangle.
on their center line 
Let the line XA meet the circles 
The triangles 
are centrally similar with the similarity center X, hence, 
and 
The radical axis AB of the circles 
Translate the triangle 
by the distance 
and reflect it in the radical axis AB to obtain an oppositely congruent triangle 
such that 
Since 
we have 
and the quadrilateral ASA'T is cyclic with a circumcircle (O) and one diagonal 
be the orthocenters of the triangles 
respectively. The common circumcircle (O') of the triangles 
is the reflection of the circumcircle (O) in the line ST. Since the circles 
are congruent and 
(all perprendicular to ST), we have 
Thus 
is a parallelogram, 
and 
Transforming the triangle 
back to the oppositely triangle 
the orthocenter 
of the triangle 
of the triangle 
in the radical axis AB. Since the triangles 
are reflections of the triangles 
in the center line 
respectively, the orthocenters 
of the triangles 
of the triangles 
is a rectangle with the midlines 
are in one circle, and 
and 
be 
.
and 
are symmetric about this line, so our figure is an iscoceles trapezoid. I claim now that the midpoint of 
is the midpoint of 
,
is parellel to, and equal in length, to 
.
using Alternate Segment Theorem a few times, so using definition of orthocentre the quadrilaterals 
and 
are concyclic, so consider the circumcircles of 
and 
.
are reflection of each other in 
where 
are resp orthocentres of 
.
is a parallelogram. Also 
would be a reflection of 
wrt perp bisector of 
be the center of the first circle and 
be the center of the second one.
over line 
.
= 
.
,
.
is a parallelogram and it suffices to show that 
.
,
is a parallelogram. Do the same for 
.
.
