[/quote]
,
be the medial triangle of an acute-angle triangle 
.
perpendicular to 
meet 
at 
.
analogously. Let 
.
and 
.
meet the 
,
and 
similarly. Let 
be the circumcenter of 
be the circle with diameter 
,
at 
.
and 
.
are collinear.
lie on a circle centered at 
are coaxial.
be the set of all positive integers. Find all functions 
such that for any positive integers 
and 
,
,
,
,
are equal.
be a triangle with 
.
-
and line 
and 
,
-
.
and 
meet at 
.
.
be three diameters of the circumcircle of an acute triangle 
be the orthogonal projection of 
,
be the point such that 
,
be the point such that 
,
be the point such that 
.
is similar to triangle 
of positive integers such that 
.
players. Every player played every other player once, with no draws. In addition, each player had a numerical rating before the tournament began, with no two players having equal ratings. It turns out there were exactly 
games in which the lower-rated player beat the higher-rated player. Prove that there is some player who won no less than 
and no more than 
games.
,
,
be positive integers. Prove that ![\[ \sum_{i = 1}^n \min\left(\left\lfloor \frac{x}{i} \right\rfloor, m \right) = \sum_{i = 1}^m \min\left(\left\lfloor \frac{x}{i} \right\rfloor, n \right). \]](http://latex.artofproblemsolving.com/b/7/a/b7aebfe11f646d604d2e8cb9d5bd70ec796949e8.png)
of points in the plane is balanced if, for any two different points 
.
.
,
such that
are all powers of 
.
be the complete graph on ![\[ B\le 2A+\frac{n(n-1)}{3}.\]](http://latex.artofproblemsolving.com/6/5/9/659d0199c3d685d43f1ced3704dfa59b02b586b3.png)
(The complete graph on 
edges, with exactly one edge joining every pair of vertices. A triangle consists of the set of 
edges between 
of these 
such that the equation ![\[x^{2}-(a^{2}+b^{2}+c^{2}+d^{2}+1)x+ab+bc+cd+da=0 \]](http://latex.artofproblemsolving.com/9/0/4/904d1b23d5c744118a7d68df1e2e94bddac8c1cd.png)
has an integer solution. Prove that the other solution is integer too and both solutions are perfect squares.
![\[ \left( \cot \dfrac{A}{2} \right)^2 + \left( 2\cot \dfrac{B}{2} \right)^2 + \left( 3\cot \dfrac{C}{2} \right)^2 = \left( \dfrac{6s}{7r} \right)^2, \]](http://latex.artofproblemsolving.com/6/9/b/69b1964be5f92eb44a36b0b8604bf473fe27e210.png)
and 
denote its semiperimeter and its inradius, respectively. Prove that triangle 
whose side lengths are all positive integers with no common divisors and determine these integers.
.
and 
intersect 
and 
be the intersection point of these angle bisectors. Let 
and 
be the respective reflections of 
cards each from this deck. We would like to define a rule that declares one of them a winner. This rule should satisfy the following conditions:
cards: if the cards are laid down in increasing order face down and we are told which card belongs to which player, but not what numbers are written on them, we can still decide the winner.
and 
,
for all 
,
from the deck, 
Y by
There are a) 
; b) 
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??
Proposed by A. Gribalko
; b) 
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??Proposed by A. Gribalko
