User:Asf
This page is a collection of problems (without solutions from me yet) from a math circle because I don't know where else to put them.
January 27, 2011
1. Place 4 points on the plane in such a way that every triangle with vertices at these 4 points is isosceles. Could you do the same with 5 points? More than 5 points?
2. Plot 2 points A and B a distance 2 units apart (choose your own unit length).
(a) Place 6 points in such a way that for every point 
of these 6 points, ![\[AP-BP=0,\]](http://latex.artofproblemsolving.com/8/a/1/8a141786a4633d635b9724b0126648f549658d93.png)
i.e. the difference between the distances from P to the two points B is exactly 0.
(b) Place 6 points in such a way that for every point of these 6 points either ![\[AP-BP=1\text{ or }BP-AP=1,\]](http://latex.artofproblemsolving.com/d/7/5/d75a171f634d99c679f4bb97841ba76a477f7576.png)
i.e. the positive difference between the distances from P to the two points A and B is exactly 1.
February 3, 2011
1. A hungry caterpillar climbs up a tree that is 14 meters tall. During the day, she goes up 6 meters, and during the night, she drops 4 meters. In how many days will she reach the top of the tree?
2. Two boys can eat two cookies in two minutes. How many cookies can six boys eat in six minutes?
3. (a) Does there exist a triangle with sides of lengths 1, 2, and 3?
(b) Does there exist a triangle with heights of lengths 1, 2, and 3?
February 10, 2011
1. In the interior of triangle 
with area 1, points 
, 
, and 
are chosen such that is the midpoint of 
, is the midpoint of 
, and is the midpoint of 
. Find the area of the triangle 
.
2. Find all ordered pairs 
such that both of the following equations are satisfied. ![\[xy+9=y^2 \\ xy+7=x^2\]](http://latex.artofproblemsolving.com/0/9/c/09cd2abf59da6fb1731f87e5eca4541b9fc4b0fe.png)
3. Let 
be a function whose domain is 
, and whose range is contained in 
. Compute the number of different functions which have the following property: no range value 
comes from more than three arguments 
in the domain. For example, ![\[f=\{(1,1),(2,1),(3,1),(4,4),(5,4),(6,6)\}\]](http://latex.artofproblemsolving.com/c/a/b/caba8db93ef9f63c0ec0a46b8b6c1bfc89e7dcbd.png)
has the property, but ![\[g=\{(1,1),(2,1),(3,1),(4,1),(5,3),(6,6)\}\]](http://latex.artofproblemsolving.com/4/a/4/4a4fa8c2fb435e88c8abd67374c13bf3ed5078fa.png)
does not.
4. (2009 BAMO-8)
5. (2009 BAMO-12)
