[/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 Involution, ValidName, Adventure10, Mango247
So I have a very hard geometry problem with lots of points. If the figure is not clear, just ask!
Consider a non-isosceles
. Let 
be the incenter, the 
-excenter, the 
-excenter and the 
-excenter, respectively.
Let
be the intouch triangle.
Let
be the tocuh points of the -excircle and lines 
, respectively.
Let
be the tocuh points of the -excircle and lines 
, respectively.
Let
be the tocuh points of the -excircle and lines 
, respectively.
Lines
form 
. Let 
be the circumcenter of . Furthermore, let 
and 
be the orthocenters of triangles (intouch triangle) and 
.
Prove that if
is the midpoint of 
, then quadrilateral 
is a trapezoid!
Consider a non-isosceles
. Let 
be the incenter, the 
-excenter, the 
-excenter and the 
-excenter, respectively.Let
be the intouch triangle.Let
be the tocuh points of the -excircle and lines 
, respectively.Let
be the tocuh points of the -excircle and lines 
, respectively.Let
be the tocuh points of the -excircle and lines 
, respectively.Lines
form 
. Let 
be the circumcenter of . Furthermore, let 
and 
be the orthocenters of triangles (intouch triangle) and 
.Prove that if
is the midpoint of 
, then quadrilateral 
is a trapezoid!Attachments:
excircle, 
e
excircle. Let 
,
are the 
e
.
be the 
.
is parallel to the line joining 
.
to 
is similar to the excentral triangle of 
seem to be the altitudes of 
be the Bevan point of 
is a parallelogram, so the vector corresponding to 
.
are antiparallel wrt 
,
,
.
parallel to the Euler line of the intouch triangle, which is pretty obvious as this vector is a linear combination of the centroid and the circumcenter of the intouch triangle with sum of coefficients 
and 
.
