AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
. He says to Cecilia: "If you pick any subset of 10 elements from , it will be disjoint with at least one subset from my list." What is the smallest possible number of subsets Miguel's list can have if what he tells Cecilia is true?
be a triangle and 
points in the same plane such that 
is the orthocenter of triangle 
, is the incenter of triangle 
, and is the centroid of triangle 
. If line 
is parallel to 
, prove that 
is the midpoint of segment .
for which the following statement is true: If positive integers 
and 
are such that 
is a multiple of 
and 
is a multiple of , then each of the numbers and is a multiple of ., define
and
meaning 
is the least common multiple of the numbers 
. Determine whether or not there exists a constant 
such that
for every positive integer .
and 
be positive integers with 
. There are tokens placed initially in the leftmost cell of a horizontal row consisting of cells. A move consists of shifting any token 
, 
, 
, 
, 
, or 
positions to the right. Andrés and Beto alternate turns, starting with Andrés. The winner is the player who moves a token into the rightmost cell. Determine who has a winning strategy in terms of and .
be an acute triangle with circumcenter 
. Draw altitude 
, with 
on side 
. The parallel line to 
passing through intersects line 
at point . Prove that point and the midpoints of sides 
and are collinear.
is mundane if there exist positive integers and less than 
such that 
is a positive integer. Find, with proof, all prime numbers that are not mundane.
be a fixed positive integer. Find the greatest real constant 
that has the following property: Any 
real numbers, not necessarily distinct, that lie in the interval ![$[100,101]$](http://latex.artofproblemsolving.com/8/e/3/8e3fe24e3b899bbf2128eebe08756e1ff963aaa0.png)
, can be partitioned into two groups with sums 
and 
such that
.
, 
and 
.
.
-tromino is a figure made up of three squares, obtained by removing one square from a 
board.
board consisting of 
unit segments. A configuration of several -trominoes is optimal if the -trominoes do not overlap, each one covers exactly three squares of the board, and moreover, no unit segment of the board belongs to two -trominoes. Below is an optimal configuration of 5 -trominoes: for which there exists an optimal configuration of L-trominoes on the board.
such that 
is a divisor of 
.
be a point on the plane. Three nonoverlapping equilateral triangles 
, 
, 
are constructed in a clockwise manner. The midpoints of 
, 
, 
are , 
, 
, respectively. Prove that triangle 
is equilateral.
. By smaller letters I'll denote corresponding complex coordinate of a point. Let 
. Then:
. being equilateral is equivalent to:
which is true and easy to check with help of above relations.
be a point on the plane. Three nonoverlapping equilateral triangles , , are constructed in a clockwise manner. The midpoints of , , are , , , respectively. Prove that triangle is equilateral.
What do you think about this ?
,
,
are constructed in a clockwise manner. Given are the middlepoints 
,
,
respectively. Prove that the triangle 
the point 
and I shall choose the origin in the point 
,
a.s.o. Define the complex number 
.
,
,
and 
,
,
,
,
,
and 
,