|
|
| Line 1: |
Line 1: |
| − | 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.
| + | [[Image:Asf.png||center|50px|My current avatar]] |
| | | | |
| − | == January 27, 2011 == | + | == Introduction == |
| | | | |
| − | 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?
| + | I spend my time doing math, going on AoPS, and reading. |
| | | | |
| − | 2. Plot 2 points A and B a distance 2 units apart (choose your own unit length).
| + | This page is a collection of statistics from various things. |
| | | | |
| − | (a) Place 6 points in such a way that for every point <math>P</math> of these 6 points, <cmath>AP-BP=0,</cmath>
| + | == Games == |
| − | 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 <math>P</math> of these 6 points either <cmath>AP-BP=1\text{ or }BP-AP=1,</cmath>
| + | Played/playing: |
| − | i.e. the positive difference between the distances from P to the two points A and B is exactly 1.
| |
| | | | |
| − | == February 3, 2011 == | + | * <url>viewtopic.php?f=535&t=409544 The Letter Market II</url> by bluecarneal |
| | + | * <url>viewtopic.php?f=535&t=416484 Payoff</url> by smallpeoples343 |
| | | | |
| − | 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?
| + | <h4>Settlements</h4> |
| | | | |
| − | 2. Two boys can eat two cookies in two minutes. How many cookies can six boys eat in six minutes?
| + | Settlements is a game I created where four players seek to eliminate all the other players. It is currently being tested in my blog. |
| | | | |
| − | 3. (a) Does there exist a triangle with sides of lengths 1, 2, and 3?
| + | <h5>Rules</h5> |
| | | | |
| − | (b) Does there exist a triangle with heights of lengths 1, 2, and 3?
| + | <url>blog.php?page=download&mode=download&id=1509 Starting Positions</url> |
| | | | |
| − | == February 10, 2011 ==
| + | The game |
| | | | |
| − | 1. In the interior of triangle <math>ABC</math> with area 1, points <math>D</math>, <math>E</math>, and <math>F</math> are chosen such that <math>D</math> is the midpoint of <math>AE</math>, <math>E</math> is the midpoint of <math>BF</math>, and <math>F</math> is the midpoint of <math>CD</math>. Find the area of the triangle <math>DEF</math>.
| + | == Blog == |
| | | | |
| − | 2. Find all ordered pairs <math>(x,y)</math> such that both of the following equations are satisfied. <cmath>xy+9=y^2 \\ xy+7=x^2</cmath>
| + | <url>blog.php?u=80033 as(d)f's blog</url> |
| | | | |
| − | 3. Let <math>f</math> be a function whose domain is <math>S=\{1,2,3,4,5,6\}</math>, and whose range is contained in <math>S</math>. Compute the number of different functions <math>f</math> which have the following property: no range value <math>y</math> comes from more than three arguments <math>x</math> in the domain. For example, <cmath>f=\{(1,1),(2,1),(3,1),(4,4),(5,4),(6,6)\}</cmath> has the property, but <cmath>g=\{(1,1),(2,1),(3,1),(4,1),(5,3),(6,6)\}</cmath> does not.
| + | I try to keep a views/entries ratio higher than 10. |
| | | | |
| − | 4. (2009 BAMO-8)
| + | <h4>CSS Skins</h4> |
| − | | |
| − | 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 <math>b</math> 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 <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?
| |
| − | | |
| − | 8. In a plane, we have two sets of parallel lines such that the lines in the first set are not parallel to those in the second. There are eight parallel lines in the first set and an unknown amount in the second. However, we know that their intersection forms a total of 420 parallelograms, many of which overlap one another. How many parallel lines are there in the second set?
| |
