be a triangle with circumcircle 
and let 
be the 
-
and 
be the intersection points of 
and 
be the projections of 
intersects the tangent line at 
at a point 
.
.
and 
: Prove that :
board and made 
moves so that it visited all squares and returned to the starting square. At every moment, the distance from the center of the square the king was on to the center of the board was calculated. A move is called 
if this distance becomes smaller after the move. Find the maximum possible number of pleasant moves. (The chess king moves to a square adjacent either by side or by corner.)
,
.
subsets of 
is to be colored red or blue. (The subset itself is assigned a color and not its individual elements.) For any set 
,
for the number of subsets of 
that are blue.
and 
of ![\[ f(T_1)f(T_2) = f(T_1 \cup T_2)f(T_1 \cap T_2). \]](http://latex.artofproblemsolving.com/1/8/1/18184e9196e6c73601653d85f1cfd6d9798a8e43.png)
is selected, and some positive integers are written on a board. Alice and Bob play the following game. On Alice's turn, she must replace some integer 
on the board with 
,
.
where 
is an odd prime, and let![\[
S_q = \frac{1}{2\cdot 3 \cdot 4} + \frac{1}{5\cdot 6 \cdot 7} + \cdots + \frac{1}{q(q+1)(q+2)}
\]](http://latex.artofproblemsolving.com/c/0/b/c0bf0826fbe8897ebaf0d4b3c861120d502baefd.png)
Prove that if 
for integers 
and 
is divisible by 
(i dont remember exactly if it was 8032038 or some other number, so correct me on this)
(i posted this at 12:29) lol
?
be the sequence defined by 
and 
for each integer 
.
is a prime and 
is a positive integer. Prove that some term of the sequence 
.
Y by Adventure10, Mathlover_1, OronSH, aidan0626, Blue_banana4, PikaPika999, and 7 other users
Given are 
points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least 
scalene triangles with vertices of that color.
points in the plane, no three of them belonging to a same line. Each of these points is colored using one of four given colors. Prove that there is a color and at least 
scalene triangles with vertices of that color.
scalene triangles.
.
to be the number of bases of these triangles (we count three bases for each equilateral and one for each nonequilateral isosceles triangle). Clearly 
.
isosceles triangles, yielding that there are at least 
scalene triangles. 
points of the same color and we apply the lemma.
points on the perpendicular bisector). Hence there are at most 
isosceles triangles, so at lesat 
scalene triangles.
points of that color.
isosceles triangles. At the same time there are 
triangles in total, making 130 non-isosceles AKA scalene triangles. 
triangles that can be formed.
be the number of isosceles triangles in the 
to 
is the maximum number of isosceles triangles. Then the minimum number of scalene triangles is 
,
as a base, since in order for that to happen, 
must lie on the perpendicular bisector of 
isosceles triangles, thus at least 
scalene triangles formed from those points.
isosceles triangles and total triangles equal 
so at least 
triangles.
points in the plane (no three collinear), there are atleast ![\[\binom{x}{3}-2\binom{x}2\]](http://latex.artofproblemsolving.com/e/5/e/e5ebd5bd3558da44fe6d1d25dbe24441be359efa.png)
scalene triangles formed by it.
points and the number of scalene triangles it form are atleast ![\[\binom{13}3-2\binom{13}2=130\]](http://latex.artofproblemsolving.com/3/e/4/3e4c8ec22face221b0219029db3c7ffd7171a46d.png)
as desired.
scalene triangles with same colored vertices.
