[/quote]
,
board, show that there is a 
block on the board in which at least 3 of the squares are colored red.
be a quadrilateral such that 
and 
are acute and 
.
,
.
and let 
be a point on the shorter arc 
and 
and 
have equal areas.
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 
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 
.
,
.
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 
.
Y by Adventure10
In the tetrahedron 
, points 
are the midpoints of the sides 
respectively, . The lengths of the segments 
and 
are equal to 
and 
respectively, and the length of the segment 
equals to 
. Find the length of the side 
of the tetrahedron.
, points 
are the midpoints of the sides 
respectively, . The lengths of the segments 
and 
are equal to 
and 
respectively, and the length of the segment 
equals to 
. Find the length of the side 
of the tetrahedron.This post has been edited 2 times. Last edited by parmenides51, Nov 7, 2022, 9:32 PM
