[/quote]
,
towns. For every pair of towns, there is a narrow road going from one town to the other. One day, all the roads are declared to be “one way” only. Alice has no information on the direction of the roads, but the King of Hearts has offered to help her. She is allowed to ask him a number of questions. For each question in turn, Alice chooses a pair of towns and the King of Hearts tells her the direction of the road connecting those two towns.
questions.
be any point on side 
of 
.Let 
and 
be points on 
and 
such that 
and 
respectively. Prove that the locus of circumcenter of 
is a line.
and 
pebbles, respectively. In the first round Barry chooses a positive integer 
and Andrew takes away 
pebbles from one of the two heaps (if 
be a triangle with 
.
lies inside the triangle in such a way that 
.
be the midpoint of 
.
,
at 
.
similarly.Prove that 
are collinear.
grid with 
rows and 
columns, and each square of the grid is filled with a non-negative integer. Let 
the there exist two identical rows (meaning same numbers in the same order).
![$$\left(a^{3}+b^{3}+c^{3}-3abc\right)^{2}-\left[a^{3}+b^{3}+c^{3}+3abc-ab(a+b)-bc(b+c)-ca(c+a)\right]^{2}\ge 2k\cdot(a-b)^{2}(b-c)^{2}(c-a)^{2}$$](http://latex.artofproblemsolving.com/d/2/4/d24b46e65a587fbd977467b6e4e31838a0eb57aa.png)
forall 
such that p is a prime, 
and 
is a divisor of 
.
such that![\[ \frac {2^n + 1}{n^2}
\]](http://latex.artofproblemsolving.com/9/2/2/922444e7669c326efcad5fde3ea00f8c2d9fa5d8.png)
is an integer.
and 
Find the ratio 
,
is maximized.
and 
be the altitudes. Let the points 
and 
fulfil the following requirements:
is the circumcentre of triangle 
and 
are equal to the circumradius of triangle 
has the same direction as the oriented segment 
.
has the same direction as 
,
has the same direction as 
.
,
be the intersections of the perpendicular bisectors of 
and 
.
and 
is a divisor 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.
"
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 
