[/quote]
,
,
,
,
be distinct positive integers, 
.
and 
such that 
does not divide any of the numbers 
,
,
.
board, 
pieces are placed (vertically or horizontally) so that they occupy entirely 
boxes on the board and do not have a common point.
,
be the sum of the digits of 
such that there are infinite positive integers 
.
,
be fixed positive integers. There are 
ducks sitting in a
be the circumcenter of triangle 
.
intersects the altitude from 
at point 
.
be the midpoints of 
,
respectively. If 
intersects 
at 
,
meets 
at 
,
is cyclic.
![\[
\text{Find all functions } f : \mathbb{R} \to \mathbb{R} \text{ such that:} \\
f(a^4 + a^2b^2 + b^4) = f\left((a^2 - f(ab) + b^2)(a^2 + f(ab) + b^2)\right)
\]](http://latex.artofproblemsolving.com/2/a/7/2a73a5e58eab4a994fbd83160fb79a0b00152951.png)
and circumcentre 
be the touchpoints of the incircle with 
,
is the Euler line of 
.
with its incircle 
.
and 
be the points of tangency of 
and 
,
and 
such that 
is tangent to 
.
is the intersection point of the lines 
and 
.
and 
are parallel or coincide.
and let 
be the midpoint of 
Suppose that 
different from 
such that 
Prove that the orthocenter of 
lies on the circumcircle of 
.
that satisfy
for all integers 
and rational numbers 
.
are one of such 
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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.
"
is the average degree, then we want to delete vertices until the minimum degree is at least 
.
...)" 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 
for most transfers, but decreases by more if one transfers from 
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 
