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6. '''The Bridges of Konigsberg.''' The river Pregel flows through the town of Konigsberg in Prussia, as shown below. Is it possible to walk through the town in such a way to cross eery bridge exactly once? | 6. '''The Bridges of Konigsberg.''' The river Pregel flows through the town of Konigsberg in Prussia, as shown below. Is it possible to walk through the town in such a way to cross eery bridge exactly once? | ||
| + | |||
| + | [diagram n/a yet] | ||
7. Let <math>x</math> be an integer such that <cmath>x=7^a+7^b,</cmath> where <math>a</math> and <math>b</math> are chosen independently from the integers 1 through 100 inclusive. Assuming that each integer has an equal likelihood of being chosen, what is the probability that <math>x</math> is an integral multiple of 5? | 7. Let <math>x</math> be an integer such that <cmath>x=7^a+7^b,</cmath> where <math>a</math> and <math>b</math> are chosen independently from the integers 1 through 100 inclusive. Assuming that each integer has an equal likelihood of being chosen, what is the probability that <math>x</math> is an integral multiple of 5? | ||
Revision as of 19:14, 7 July 2011
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)
March 3, 2011
1. Three Utilities Problem. There are three houses and three utilities (gas, water, and electricity). You must draw a line from each house to each utility, without the lines crossing. Can you connect the houses to the utilities?
2. Three Utilities Problem on a Torus. Continuing the problem above, what if the houses and the utilities and the lines connecting them lie on a torus? (Recall that a torus is the surface of a doughnut.)
3. Find a way to position four points on a sheet of paper so that when every pair of points is joined by a curved or straight line segment, none of the segments cross.
4. Can you position fie points on a sheet of paper and connect each pair of points with a curved or straight line segment in such a way that none of the segments cross? What if the five points and the segments connecting them all lie on a torus?
5. Which two surfaces are obtained by gluing the edges of each of the following triangles as shown? Note that you'll get two different surfaces, one for each triangle. Side 
is not glued to anything.
[diagram n/a yet]
6. The Bridges of Konigsberg. The river Pregel flows through the town of Konigsberg in Prussia, as shown below. Is it possible to walk through the town in such a way to cross eery bridge exactly once?
[diagram n/a yet]
7. Let be an integer such that ![\[x=7^a+7^b,\]](http://latex.artofproblemsolving.com/b/5/d/b5d1d7711782155966f6b755b005866a3be7e8ea.png)
where 
and are chosen independently from the integers 1 through 100 inclusive. Assuming that each integer has an equal likelihood of being chosen, what is the probability that is an integral multiple of 5?
