AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
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 
. How many cats of each breed does he have?
, 
is a perfect square.
, let 
be all positive integers smaller than that are coprime to . Find all 
such that 
for all 
is the largest positive integer that divides both 
and 
. Integers and are coprime if 
.
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 , the maximum number of good cells over all possible starting configurations.
classes and 
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 students. If 
is not a perfect square, prove that there exist a student that has attended in at least 
classes.
, prove that there exists a positive integer 
, such that for any integer 
, can be expressed by a sum of positive integers 
's as![\[n=a_1+a_2+\dots+a_m,\]](http://latex.artofproblemsolving.com/6/c/2/6c2a979a5ff6c40bf2d7152c8c32088fc36f848b.png)
where 
, 
, 
, 
and 
.
where is an odd integer. Let 
be a function defined on 
taking values in 
such that
for all 
for all 
; b) 
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??
points of general position marked on the coordinate plane (i.e., points among which there are no three lying on the same straight line). Is there a polynomial of two variables 
a) of degree 
; b) of degree such that it equals to zero exactly at these marked points?
be triangle, inscribed in parabola. Tangents in points 
forms triangle 
. Prove that 
. (
is area of triangle 
).
such that there exist integers 
with ![\[x^3_1+x^3_2+\,\ldots\,+x^3_t=2002^{2002}\,?\]](http://latex.artofproblemsolving.com/e/f/a/efaa98e77ac0273284ee3446e4335d8cc4f681cb.png)
be a triangle with 
and 
.
inside triangle satisfies the following conditions:![\[
XA^2 + BC^2 \leq XC^2 + AB^2 \leq XB^2 + AC^2.
\]](http://latex.artofproblemsolving.com/0/3/1/031aa58d7d3dabe636dfb4245ef2456a2a3d5df6.png)
Find the maximum value of 
such that![\[
\frac{XA}{AC} \geq m.
\]](http://latex.artofproblemsolving.com/c/5/e/c5ed2892b224aff3b2f3e20b4c9424707ff88690.png)
and 
.![\[P=\frac{1}{5-2a}+\frac{1}{5-2b}+\frac{1}{5-2c}.\]](http://latex.artofproblemsolving.com/c/3/7/c3708768a84c2bbb1e4fae502a9bb1e92cb13d02.png)
and 
Prove that
