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- Problem #221 - Posted Saturday, February 12, 2005
- Square, Not a Square. Sorry about the long delay - Dan (back to top)
- a) Find the first three squares that are in arithmetic progression.
- b) Find four squares in arithmetic progression, if that's possible.
- c) Find 100 square-free integers in arithmetic progression. def below
- Here, a "square" is the square of a positive integer. A "square-free" integer is not
- divisible by any square > 1. Show reasoning clearly.
- Problem #221 - Posted Saturday, February 12, 2005
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- Problem #222 - Posted Monday, March 7, 2005
- Spare Sum Squares? (back to top)
- Find three distinct positive integers, whose sum is a perfect square, and the
- sum of any pair is a square. a) Find all such solutions with total sum less than 1000.
b) Can a solution exist so that the original three numbers are also squares? (Bonus pt this part)- Here, a "square" is the square of a positive integer. Show reasoning clearly.
- Problem #222 - Posted Monday, March 7, 2005
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- Problem #223 - Posted Friday, April 1, 2005
- Divisor Take-Away (back to top)
- Here's the game: the first player is given an integer n > 1 , then the second player subtracts
- a proper divisor d < n of that number, telling the first player the difference n - d , then the
- first player subtracts a proper divisor of that new number, and so on. The player who
- announces a difference of 1 is the winner. Does either player have a winning strategy?
- If not explain why not; if so, which player is it, does it depend on n, and what's the strategy?
- Show reasoning clearly.
- Problem #223 - Posted Friday, April 1, 2005
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- Problem #224 - Posted Wednesday, May 11, 2005
- Fractionacci Numbers? (back to top)
- Let's define a Fibonacci-esque sequence of fractions: ao = 1 ; an+1 = (an) / (1 + n an)
- i) List the first ten 'Fractionacci Numbers' ii) Give an explicit formula (and proof) for
- the exact value of an (the nth Fractionacci). Show reasoning clearly
- Problem #224 - Posted Wednesday, May 11, 2005
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- Problem #225 - Posted Wednesday, June 8, 2005
- 3 Sums of 2 Nums (back to top) Answer these questions three; send the sums to me.
- a) The product, the quotient, and the difference of two real numbers are all the same.
- Find the sum of the two numbers.
- b) If [ x + \/(x^2 + 1) ] [ y + \/(y^2 + 1) ] = 1, find the value of the sum of x and y.
- c) The equation log3x(3) + log27(3x) = - 4/3 has two positive solutions. Find their sum.
- Show reasoning clearly. Notation: \/(a) is sqrt[a].
- Problem #225 - Posted Wednesday, June 8, 2005
- Problem #226 - Posted Wednesday, June 22, 2005
- Con-hex-utive Numbers (back to top)
- These three tasks require you to fill in the integers from 1 to 9 in the hexagons at right, using each one exactly once for each part, so that, in turn:
- a) No two adjacent hexagons contain either consecutive numbers nor numbers whose names have the same number of letters.
- b) No two adjacent hexagons contain digits whose sum is a multiple of four or five.
- c) In any hexagon, the total of the numbers in the surrounding hexagons is a multiple of the number in the original hexagon.
- Each part has a unique solution up to rotation or reflection. Give each answer in three rows of 3, starting with left-to-top. Show your reasoning please.
- Photo Mosaics! (back to top)
- I have lots of photo prints, in two sizes: (a) 4 x 6 inches, and (b) 5 x 7 inches.
- I put photos up on the wall, each one can be vertical or horizontal, so that they tile into a big rectangle
- (with no overlapping or cutting, of course). i) Prove I can't make a 19 x 19-inch "photo-square."
- ii) Prove theoretically that a 29" x 29" square is possible. iii) Show me how to tile the 29 x 29 photo-square.
- Bonus point: Prove a 31 x 29 rectangle is impossible (hard prob!)
- No pictures necessary, just give the upper-left coordinates of each photo, starting with (1, 1).
- Show your reasoning. Part ii) can (and should) be done without using part iii). No attachments please, except simple gifs.
- I have lots of photo prints, in two sizes: (a) 4 x 6 inches, and (b) 5 x 7 inches.
