[/quote]
,
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??
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 
;
be triangle, inscribed in parabola. Tangents in points 
forms 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)
and 
doubloons, where 
.
coins. Find the number of lucky numbers.
with no parallel sides is inscribed in a circle 
.
are inscribed in triangles 
,
and 
,
,
,
.
be a quadrilateral inscribed in a circle 
such that 
.
and 
be the midpoints of 
and 
respectively. The line 
meets 
.
are concurrent.
Y by
Let 
be the circumcircle of the acute-angled triangle 
. The bisector of angle 
intersects for the second time at the point 
is the diameter . Let 
be a point symmetric to the orthocenter 
of the triangle wrt the line 
. Prove that the center of the circumcircle of the triangle 
belongs to the line 
.
(D. Hilko)
be the circumcircle of the acute-angled triangle 
. The bisector of angle 
intersects for the second time at the point 
is the diameter . Let 
be a point symmetric to the orthocenter 
of the triangle wrt the line 
. Prove that the center of the circumcircle of the triangle 
belongs to the line 
.(D. Hilko)
This post has been edited 1 time. Last edited by parmenides51, Jun 27, 2021, 7:23 AM
