Prize Winners

 Grand Prize
Yang-Yang Chen,  6th Grade,   Bluemont Elementary School, Manhattan
                                                                                    Teacher: Melisa J. Hancock

    Grade 5-6
        First Place:  Yang-Yang Chen, Grade 6,  Bluemont Elementary School, Manhattan
                                                                     Teacher: Melisa J. Hancock
        Second Place: Jordan Bishop,  Grade 6,  Bluemont Elementary School, Manhattan
                                                                    Teacher: Melisa J. Hancock
        Third Place:   Cochise Fant,  Grade 5, Bluemont Elementary School, Manhattan
                                                                    Teacher: Melisa J. Hancock
       Honorable Mention:  Najee Lewis, Grade 5, Westwood Elementary School. Junction City
        Honorable Mention:  Maia Williams, Grade 5, Marlatt Elementary School, Manhattan

        First Place: Nicholas Ma,  Grade 8, Curtis Middle School, Wichita
        Second Place:   Frank Jiang,  Grade 8, Susan B. Anthony Middle School,  Manhattan
        Third Place:   Adam Gelroth,  Grade 8, Eisenhower  Middle School,  Manhattan
        Honorable Mention:   Helen Cox,  Grade 8, Northfield Middle School, Wichita
        Honorable Mention:   Eddie Fonner,   Grade 8, Topeka Collegiate Middle School, Topeka

    Grade 9-12
        First Place: Greg Hayrapetyan, Grade 10, Manhattan High School, Manhattan
        Second Place: Geoff Tims,  Grade 12 ,  Shawnee Mission West High School, Lenexa
        Third Place:  Adam Brown,  Grade 9, Topeka High School, Topeka
        Third Place:   Neset Tamer,  Grade 12, Manhattan High School, Manhattan
        Third Place:  Jonathan West,  Grade 9, Topeka High School, Topeka

Put your name on all papers  you use and turn them all in.

Try to solve as many problems as you can, in any order you choose. For any problem you try, give as complete an answer as you can. Include a clearly written explanation of how you found yous answer and why it is  true. You may use drawings or calculations as part of your justification.
 

1.  You take a spoon of milk from a bucket of milk and you put it into your cup of tea. Then you return a spoon of the (nonuniform) mixture of tea and milk from your cup to the bucket.  Now you have some milk in the cup, and some tea in  the bucket. Which is larger: the quantity of milk in the tea cup, or the quantity of tea in the milk bucket?

2. Find all triples (a,b,c) of integers a>b>c>0  for which   ¹/_a+ ¹/_b+ ¹/_c is an integer.

3.   The numbers 1, 2, ... , 1999  are written on the blackboard in the classroom. Every time a leprichaun enters the classroom, he chooses two numbers on the blackboard, say  a   and b, with  a >= b, then erases them and writes  the difference  a-b  somewhere on the blackboard.  After this procedure is carried on  1998  times, there will be only one number left on the blackboard. Prove that the last remaining number must be even.

4.  Cut the figure

Put your name on all papers  you use and turn them all in.

Try to solve as many problems as you can, in any order you choose. For any problem you try, give as complete an answer as you can. Include a clearly written explanation of how you found yous answer and why it is  true. You may use drawings or calculations as part of your justification.
 

1.  In the triangle ABC  we take M  to be the midpoint of  AB, and we take N  to be the midpoint of AC. (The midpoint of a segment is the point which divides it into two equal segments). We are given that the segments CM  and BN  are equal. Prove that the segments AB  and AC are  equal.

2.  Alice and Bob started at sunrise and each walked at a constant velocity. Alice went from the town A to town  B and Bob went from B to A (on the same road). They met at noon and, continuing without stop, arrived respectively at B at 4 p.m.  and at A at 9 p.m. At what time was the sunrise on this day?
 

3.  There are six points inside of a circle of radius 1 cm.  Prove that there are two points that are no more than 1 cm apart.
 

4.  Steve is trying to fill in a Magic Square starting as follows:

Put your name on all papers  you use and turn them all in.

Try to solve as many problems as you can, in any order you choose. For any problem you try, give as complete an answer as you can. Include a clearly written explanation of how you found yous answer and why it is  true. You may use drawings or calculations as part of your justification.

1.  The plane is divided into regions by n different straight lines. Prove that one can color these regions, using the colors red and blue,  in such a way that no two regions with a common edge have the same color.

2.  You are given a regular hexagon inscribed in the circle of radius 1  (``Regular'' means that all edges are equal and all angles are equal. ``Inscribed'' means that all vertices are on the circle).  For any point on the circle, one computes the sum of the squares of the distances from this point to all six vertices of the hexagon. Prove that this quantity does not depend on the point.

3.  Let us consider the product 2000x2001x 2002x...x3998. What is the maximal power of 2  which divides the product?

4.  In the sequence

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