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 
,
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 
.
?
and in 
steps and has to reach 
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 
,
.
such that 
.
such that 
.
be a polynomial of degree 
we have 
.
![$f:[0,1]\rightarrow\mathbb{R}$](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)
invertible. Prove that if 
for any positive integer 
then 
given by 
and 
,
.
Mathematical Gazette
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. 
,
,
and 
respectively, and the points 
are collinear.
be a real-valued function that is twice continuously differentiable throughout 
,![\[
\rho (x,y)=yh_x -xh_y .
\]](http://latex.artofproblemsolving.com/7/3/6/736993597c4b45cf77b9013185d2a4d931a7d231.png)
Prove or disprove: For any positive constants 
and 
with 
,
of radius 
over the interior of 
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Hi everyone, I am currently writing a book on combinatorics for people preparing for national and international math competitions, especially the IMO and selection tests leading up to it. The book is intended to expose readers to a variety of ideas, techniques and problem solving strategies, ranging from the intuitive “greedy algorithms” in the first chapter to the powerful Probabilistic Method in chapter nine.
I am uploading chapter one here, and would appreciate your feedback and any suggestions. Over the coming weeks, I will be uploading the remaining chapters one at a time.
The only prerequisites are familiarity with basic graph theoretic concepts and terminology, algebraic inequalities, induction and the pigeonhole principle. Experience with invariants and the extremal principle is also helpful.
EDIT: CHAPTER 9 IS OUT! Since only 3 attachments are allowed per post, I have uploaded chapters 4, 5 and 6 in my comment below (10th on this page), and chapters 7, 8, and 9 further below (comment number 49 on this page).
NOTE: The solution to example 8 in chapter 1 is incorrect, and will be corrected in the final version of the book. For now, ignore this example.
Full book (uploaded by green_dog_7983):Dead Link
[Amir: new link]
I am uploading chapter one here, and would appreciate your feedback and any suggestions. Over the coming weeks, I will be uploading the remaining chapters one at a time.
The only prerequisites are familiarity with basic graph theoretic concepts and terminology, algebraic inequalities, induction and the pigeonhole principle. Experience with invariants and the extremal principle is also helpful.
EDIT: CHAPTER 9 IS OUT! Since only 3 attachments are allowed per post, I have uploaded chapters 4, 5 and 6 in my comment below (10th on this page), and chapters 7, 8, and 9 further below (comment number 49 on this page).
NOTE: The solution to example 8 in chapter 1 is incorrect, and will be corrected in the final version of the book. For now, ignore this example.
Full book (uploaded by green_dog_7983):
[Amir: new link]
This post has been edited 15 times. Last edited by Amir Hossein, Jan 23, 2020, 7:55 PM
.
,
,
,
,
is usually not tight. I think it's useful to point out in this way that the greedy algorithm usually does not always perform optimally.
must be nonempty.
"
and 
.
...)" should be stuffed into a parenthetical.
seems more conventional than 
.
does not change the sum of the weights.
and 
is actually tricky, initially I thought that 
could just pass in either direction and it would still work. You might want to show explicitly that this is not the case -- that is, explain why 
which decreases by 
so that you don't have to keep repeating yourself now or later.
vectors 
are independent over 
.
dimensional vector space over 
.
elements 
with addition 
.
.
such that the subgraphs induced on 
and 
have all vertices of even degrees. It's possible the above problem to be proved as a corollary of Gallai's theorem. In 
