AMSP Mock AIME/Obligatory Week 2 Post
by djmathman, Jul 14, 2013, 4:08 AM
So yea. We just found out that AMSP is holding a mock AIME on Monday. The questions are going to be written/stolen from obscure contests by 1=2 and dannyhamtx. I'm actually looking forward to it.
In other news, I probably got two out of five questions on each of the tests today. I'm too lazy to type up the solutions, so I'll just post the four problems I got right.
In other news, I probably got two out of five questions on each of the tests today. I'm too lazy to type up the solutions, so I'll just post the four problems I got right.
Combo Problem 1 wrote:
You want to color the numbers from 
with some colors such that no number is divisible by a different number of the same color. What is the smallest possible number of colors you must have?
with some colors such that no number is divisible by a different number of the same color. What is the smallest possible number of colors you must have?Combo Problem 2 wrote:
We are given a polygon 
, a line 
and a point 
on in general position: all lines containing a side of the polygon intersect in distinct points different from . We mark each vertex of the polygon if the sides from it extended will cut the line in two points such that is between them. Show that lies inside the polygon if and only if on each side of there are an odd number of marked vertices.
, a line 
and a point 
on in general position: all lines containing a side of the polygon intersect in distinct points different from . We mark each vertex of the polygon if the sides from it extended will cut the line in two points such that is between them. Show that lies inside the polygon if and only if on each side of there are an odd number of marked vertices.Geometry Problem 1 wrote:
Let 
be the points of tangency of the incircle of a triangle 
with its sides 
respectively. Then, the triangle is equilateral if and only if the centroids of 
and are isogonal conjugates with respect to triangle .
be the points of tangency of the incircle of a triangle 
with its sides 
respectively. Then, the triangle is equilateral if and only if the centroids of 
and are isogonal conjugates with respect to triangle .Geometry Problem 2 wrote:
Let 
be the midpoints of the arcs of triangle which contain the vertices of the triangle. Prove that the Simson lines of with respect to are concurrent. (Hint: these Simson lines are very special cevians in the medial triangle of .)
be the midpoints of the arcs of triangle which contain the vertices of the triangle. Prove that the Simson lines of with respect to are concurrent. (Hint: these Simson lines are very special cevians in the medial triangle of .)This post has been edited 3 times. Last edited by djmathman, Sep 5, 2013, 1:28 AM
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