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.
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.
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 .
, 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 .
such that 
for any relatively prime natural numbers 
.
is described on a circle 
, and is not a trapezius inscribed in a circle. Let the tangency points of and 
be 
respectively. A circle with a center 
, different from is tangent to the segement and lines 
. A circle with center 
, different from is tangent to segment and lines 
. A circle with center 
, different from is tangent to segment 
and lines . A circle with center 
, different from is tangent to segment 
and lines . Prove that the lines 
are concurrent.
such that 
is a divisor of 
.
, such that 
and 
.
such that 
is prime, 
with 
and ?
. 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?
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 . 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 .
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 
.
.
sides, we choose a point different from the vertices and we obtain a new polygon of sides. For which values of can we obtain a polygon such that the internal angles are all equal but the polygon isn't regular?
sides, we choose a point different from the vertices and we obtain a new polygon of sides. For which values of can we obtain a polygon such that the internal angles are all equal but the polygon isn't regular?
.
be the vertices and 
the point on 
(all indices mod n). If , if we take the side to be 1 just take 
and 
. Therefore since 
are all isosceles triangles, we have 
so all angles are equal, however if 
sides are of alternating lenghts., let 
(
) be the point on 
such that 
for 
and also 
. We notice that if 
moves towards , 
moves towards 
(this is because by induction 
moves towards 
and 
towards 
. Therefore ther can be at most one point such that 
. However if 
with fixed 
we have a regular polygon (by simmetry) which also yields a solution for each 
in 
, which are all the possible values of . Therefore, there can be no nonregular such polygon.
