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Problems from 2009


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The length of the hypotenuse of a right triangle is the square root of 2009. The other two sides have integer (whole number) lengths. What are they?
Six 7's for 2009Use six 7's and any combination of +, , × ,÷, and parentheses make 2009. 
The figure to the left shows the outline of a ten by ten square using 36 points. How many lines can be drawn that pass through at least two of the points?
The figure to the right shows 100 points in a ten by ten square. A point at one of the corners is red. How many lines can be drawn that pass through the red point and at least one other point in the square?
Find a rule that describes the numbers in the following sequence and fill in the missing terms..
1, 2, 3, 6, 11, 20, 37, ___ , 125, 230, ___ , 778, . . .
Find a rule that describes the numbers in the following sequence and fill in the missing terms..
1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, ___ , 10, 10, 10, 10, 10, 11, 11, 11, 11, 11, ___ , 12, . . .
In chess the king attacks any pieces one square away from the square it occupies, either horizontally, vertically, or diagonally. For example, on the 4x4 chessboard to the right, the white king at b3 attacks all the black kings except the one at d1. It also attacks the white king at c2. Can you place 16 black and white kings on the 4x4 chessboard so that each black king attacks an even number of other black kings and each white king attacks an even number of other white kings? (Zero is an even number.)
Can you place 16 black and white kings on the 4x4 chessboard so that each black king attacks an even number of other black kings and each white king attacks an odd number of other white kings?
Replace each letter below with a different digit to create a valid summation. (The first digit in each number must not be zero.)
M  O  O  
+  M  O  O 


C  O  W  S 
Replace each letter below with a different digit to create a valid summation. (The first digit in each number must not be zero.)
E  E  K  
+  E  E  K 


M  I  C  E 
There are five checkers in a row numbered from one to five. The goal is to reverse the order of the checkers by using a threecycle moves. A three cycle move rotates three consecutive checkers by moving the first checker to the end and the other two back one, or vice versa. What is the fewest number of threecycles needed to reverse the checkers?
Slide ReversalThere are five checkers in a row numbered from one to five on a checker board with six squares. The goal is to reverse the order of the checkers by moving checkers. A move can be a movement to an open adjacent square or by jumping over one checker. At the end, the five checkers must be in the five rightmost squares (the five squares in which they started). What is the fewest number of moves needed to reverse the checkers? 
The figure to the right shows the results of applying a mystery operation to the numbers 1 to 12. Find a possible definition for the operation and the value of the operation when applied to 20.
The figure to the left shows the results of applying a mystery operation to the numbers 1 to 12. Find a possible definition for the operation and the value of the operation when applied to 20.
The figure to the right shows three treasure boxes. One contains treasure while the other two are empty. Each box is labeled with a statement. The three statements are consistent with the location of the treasure. Which box has the treasure?
The figure to the left shows three treasure boxes. One contains treasure while the other two are empty. Each box is labeled with a statement. The three statements are consistent with the location of the treasure. Which box has the treasure?
Jamie invites some of his friends to a party. At the party, only those who were invited attend. Jamie spends time talking with everyone at the party. When he comes to you and he states,
"Everyone here knows a different number of people at the party!"
"Including you?" you ask.
"Yes, including me," Jamie says
"Jamie, you must have miscounted," you say.
Why do you believe Jamie is mistaken? (You may assume that Jamie didn't include in his count that a person knows him/herself. You may also assume that if person A knows person B, then B also knows A.)
"Let's see. there are twelve people at the party. Oh, yes, you're right!" Jamie exclaims, "I forgot about Jessie. Just a second while I check with her."
"You don't need to ask Jessie. I can tell you how many people at the party Jessie knows." you claim.
Jamie looks at you quizzically and asks, "how do you know?"
How many people does Jessie know at the party?
NOLIES is an operation that takes an integer and returns a positive integer. The figure to the left shows the results of NOLIES being applied to some numbers. When the NOLIES operation is applied to a number in the figure, an arrow is drawn from the circle around the number to the circle around the result. What number goes in the circle without a number? (Hint: NOLIES is an acronym for the operation.)
TTPOTD is an operation that takes an integer and returns an integer ≥ 0. The figure to the right shows the results of TTPOTD being applied to some numbers. When the TTPOTD operation is applied to a number in the figure, an arrow is drawn from the circle around the number to the circle around the result. What number goes in the circle without a number? (Hint: TTPOTD is an acronym for the operation.)
 
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