Prize Winners

 Grand Prize Winner:
Yang-Yang Chen, 5th Grade,   Bluemont Elementary School

    Grade 5-6
        First Place:  Yang-Yang Chen,  Grade 5,  Bluemont Elementary School
        Second Place:  Joey Dodds,   Grade 6,   Bluemont Elementary School
        Third Place:   Yiyu Zhao,  Grade 6, Lee School

    Grade 9-12
        First Place:  Eric Stewart,  Grade 9, Manhattan High School (East Campus)
        Second Place:  Dennis Goin,  Grade 9,   Manhattan High School (East Campus)
        Third Place:  not awarded

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.    Suppose we want to place the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9, in the circles in the following figure in such a way that the sums of the four numbers on each side of the triangle is the same. If we denote this sum by S, find the biggest and smallest possible value of S, for which such an arrangemnet if possible.

2.    One has 12 matches, each being 1 inch long. Is it possible to arrange them to form a polygon with area equal to 4 in²?

3.     Prove that, when we divide any prime number by 30, we get a remainder which is equal to either 1 or a prime number.

4.    Is it possible to cut an arbitrary triangle into several prieces in such a way that, if we put these pieces together in a different way, we get a rectangle?
 

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.    Find all prime numbers p for which p+10 and p+14 are also prime.

2.    Prove that

3.    Suppose somebody has numbered the squares of chessboard by writing in each square one number from 1 to 64, without repeating any number. Show that there exist at least two neighboring squares such that the difference between their numbers is at least 5.  (Note: Two neighboring squares are two squares which have a common side.)
 
 

4.     Soppuse we have 101 points inside a square with one inch sides, placed in such a way that no three points lie on a straight line. Prove  there exist 3 points such that the triangle they form has area not bigger than 1/100 in².

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.     Prove that if a prime number m  has the property that m²+2  is also prime, then m³+2 must also be prime.

2.     Suppose n is a positive integer. Find a formula to the sum:

3.     John is 3 years old and he knows how to write only the digit 1. Prove that, using only the digit 1, John can write a multiple of 1999. Can you characterize all integer numbers n for which, using only the digit 1, one can write a multiple of n?

4.     Suppose  100,000 straight lines are drwan in the plane, such that any two of them intersect in one point. Suppose also that, whenever P is a common point of two lines, there always exist at least one more line passing through P. Prove that all 100.000 lines have a common point.