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such for any 
:
.
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.
board, a nonnegative real number is written in such a way that the sum of the numbers in each row is equal to 
, and the sum of the numbers in each column is equal to . Define 
to be the largest value in row 
, and let 
. Similarly, define 
to be the largest value in column , and let 
.
?
and in 
steps and has to reach again. But in between one of the s steps and can't go . Find probability. For example 
is valid but 
is not valid.
be a function which is continuous on ![\( [0,1] \)](http://latex.artofproblemsolving.com/0/1/0/01026be294289be8f090c7b2586dde64a0d5b479.png)
and differentiable on 
, with 
. Assume that there is some 
such that 
. Prove that there exists some 
such that 
.
be a polynomial of degree with only real zeros and real coefficients.
we have 
. When does equality occur?
![$f:[0,1]\rightarrow\mathbb{R}$](http://latex.artofproblemsolving.com/9/d/0/9d00bb0f11e69022dded4a8baba1b496f66d272e.png)
,is a convex function and 
![\[\left(\int^{1}_{0}f(x)dx\right)^2\leq \dfrac{3}{4}\int^1_{0}f^2(x)dx\]](http://latex.artofproblemsolving.com/a/a/0/aa0ae3023a71f3452170023452601cc7df785eb9.png)
and 
invertible. Prove that if 
for any positive integer 
then 
given by 
and 
, for all 
. Show that 
Mathematical Gazette
is defined by 
and, for 
![\[a_n=\dfrac{6a_{n-1}^2a_{n-3}-8a_{n-1}a_{n-2}^2}{a_{n-2}a_{n-3}}.\]](http://latex.artofproblemsolving.com/b/c/6/bc6686c63a96c3be5575542d7ac120c2fa7e69b2.png)
Show that, for all , 
is an integer multiple of .
and 
be two points that are diametrically opposite to each other on a unit sphere. right square pyramids are fitted along the line segment 
, such that the apex and altitude of each pyramid , where 
, are 
and 
respectively, and the points 
are collinear. pyramids, with altitudes of equal length, that can be fitted in the sphere, in terms of . pyramids that can be fitted in the sphere, in terms of . tends to infinity.
be a real-valued function that is twice continuously differentiable throughout 
, and define![\[
\rho (x,y)=yh_x -xh_y .
\]](http://latex.artofproblemsolving.com/7/3/6/736993597c4b45cf77b9013185d2a4d931a7d231.png)
Prove or disprove: For any positive constants 
and 
with 
, there is a circle 
of radius whose center is a distance away from the origin such that the integral of 
over the interior of is zero.
and 
of a convex quadrilateral 
intersect at 
. It is known that 
. Let 
and 
be the midpoints of sides and . Prove that the triangle 
is obtuse.
and 
.