Can Anyone Solve These Challenging Math Problems?

[marmot83457]

I need some help with some problems urgently...

Hi my teacher assigned me some math problem due very soon but i could not figure out how to solve them so can anyone please help me with these? thank you very much

Problem 1
An increasing arithmetic sequence with infinitely many terms is determined as follows.
A single die is thrown and the number that appears is taken as the first term. The die is thrown again and the second number that appears is taken as the common difference between each pair of consecutive terms. Determine with proof how many of the 36 possible sequences formed in this way contain at least one perfect square.

Problem 2
George has six ropes. He chooses two of the twelve loose ends at random (possibly from the same rope), and ties them together, leaving ten loose ends. He again chooses two loose ends at random and joins them, and so on, until there are no loose ends. Find, with proof, the expected value of the number of loops George ends up with.

Problem 3
Let r be a nonzero real number. The values of z which satisfy the equation
R^4z^4 + (10r^6 - 2r^2)z^2 - 16r^5z + (9r^8 + 10r^4 + 1) = 0 are plotted on the complex plane (i.e. using the real part of each root as the x-coordinate and the imaginary part as the y-coordinate). Show that the area of the convex quadrilateral with these points as vertices is independent of r, and find this area.

Problem 4
Homer gives mathematicians Patty and Selma each a di_erent integer, not known to the other or to you. Homer tells them, within each other’s hearing, that the number given to Patty is the product ab of the positive integers a and b, and that the number given to Selma is the sum a + b of the same numbers a and b, where b > a > 1. He doesn’t, however, tell Patty or Selma the numbers a and b. The following (honest) conversation then takes place:
Patty: “I can’t tell what numbers a and b are.”
Selma: “I knew before that you couldn’t tell.”
Patty: “In that case, I now know what a and b are.”
Selma: “Now I also know what a and b are.”
Supposing that Homer tells you (but neither Patty nor Selma) that neither a nor b is greater
than 20, find a and b, and prove your answer can result in the conversation above.

Problem 5
Given triangle ABC, let M be the midpoint of side AB and N be the midpoint of
side AC. A circle is inscribed inside quadrilateral NMBC, tangent to all four sides, and that circle touches MN at point X. The circle inscribed in triangle AMN touches MN at point Y , with Y between X and N. If XY = 1 and BC = 12, find, with proof, the lengths of the sides AB and AC.

 

[Hurkyl]

Well, you have to start by telling us your thoughts on the problems -- what you've tried already, what ideas you think you might be able to use, that sort of thing.

 

[marmot83457]

Well me and my friends mostly tried a trial and error aproach. For example with #1 we know that if 1 or 4 is the first roll u then 2nd roll doesn't matter and you have 6+6=12 out of the 36. if the second roll is 1 then there would be (6+6+6)-2 =14 out of the 36 possible.But then we have to write a proof and we are not sure. We were thinking like a+bx where a = first roll b= 2nd roll and x = term of the sequence. and then find a way to prove it's a perfect square but none of use know much about these properties. We also tried taking that approach of listing the different possible combinations for the other problems but it doesn't really work well. for 5 we figure out a few random facts like MN =6 but don't really know how to move on from there. So if u can help me with some of the concept it would be great.