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| − | 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|]] |
| − | | + | <div style="font-weight: bold; background-color: #B22222; border: 1px solid black; height: 16px; border-radius: 9px; width: 42px; padding-left: 8px; color: white; margin-left: auto; margin-right: auto;">asf</div> |
| − | == January 27, 2011 == | |
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| − | 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?
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| − | 2. Plot 2 points A and B a distance 2 units apart (choose your own unit length).
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| − | (a) Place 6 points in such a way that for every point <math>P</math> of these 6 points, <cmath>AP-BP=0,</cmath>
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| − | i.e. the difference between the distances from P to the two points B is exactly 0.
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| − | (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>
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| − | i.e. the positive difference between the distances from P to the two points A and B is exactly 1.
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| − | == February 3, 2011 ==
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| − | 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?
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| − | 2. Two boys can eat two cookies in two minutes. How many cookies can six boys eat in six minutes?
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| − | 3. (a) Does there exist a triangle with sides of lengths 1, 2, and 3?
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| − | (b) Does there exist a triangle with heights of lengths 1, 2, and 3?
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| − | == February 10, 2011 ==
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| − | 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>.
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| − | 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>
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| − | 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.
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| − | 4. (2009 BAMO-8)
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| − | 5. (2009 BAMO-12)
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| − | == March 3, 2011 ==
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