[/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, Mango247
Let 
be an acute triangle, and let 
, 
, and 
be the feet of the altitudes relative to vertices 
, 
, and 
, respectively. Define 
, 
, and 
as the incenters of triangles 
, 
, and 
, respectively. Let 
, 
, and 
be the intersection of the incircle of triangle with 
, 
, and 
, respectively. Prove that the triangles 
and 
are congruent.
be an acute triangle, and let 
, 
, and 
be the feet of the altitudes relative to vertices 
, 
, and 
, respectively. Define 
, 
, and 
as the incenters of triangles 
, 
, and 
, respectively. Let 
, 
, and 
be the intersection of the incircle of triangle with 
, 
, and 
, respectively. Prove that the triangles 
and 
are congruent.
