# Need some math help very difficult problems

Hi my teacher assigned me some math problem due very soon but i could not figure out how to solve them :grumpy: 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.

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What are your thoughts on each one, what have you tried? What have you thought about? What other problems do these look like? Etc.... We can't just flat out do your homework.....(only help..)

BTW, I'm willing to bet if you look into your textbook, you will find examples pretty close to those, with the steps listed.

Well actually these aren't just standard textbook problem but from former contests. I'm only taking pre-calculus math so this is way beyond that in terms of difficulty. Me and my friends have tried this for 2 days but i've never written this sort of proofs before so i dont really know where to start. These are just extra credit problems and aren't worth that much but I really wanna do them but seems like it's a little beyond my abilities. If youre not willing to do these then could you help me with some hints, concept or just tell me where to start? I would greatly appreciate it.

EnumaElish
Homework Helper
In 1 and 2 I don't see a way other than the brute force approach: writing down all possible outcomes and then calculating the answer (essentially by counting how many outcomes satisfy the desired condition, out of how many total outcomes).

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EnumaElish
Homework Helper
In 4, since you know 1 < a < b < 20, again you can try out all possible integer combinations for {a,b} and check each combination against the conversation.

a = 2, b = 3 through 20 ---> 20 - 3 + 1 = 18 combinations to check
a = 3, b = 4 through 20 ---> 20 - 4 + 1 = 17 combinations to check
...
a = 19, b = 20 through 20 ---> 20 - 20 + 1 = 1 combination to check

In all, you have 171 combinations (sum of 1 through 18) to check.

If you can use Excel it shouldn't take more than 2 minutes to generate all these combinations.

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Zurtex
Homework Helper
Problem 4 is wrong. There is no solution for it.

I've done the problem before, even though the integers themselves lie within the range (1, 20], the problem does not have a solution unless you assign them something of a range something around (1, 80]. If you would like me explain why, I could go and look up a forum I once posted this on but if I remember correctly the proof is somewhat tedious and long and is just a matter of taking in to account of all the things you can do to reduce the number pair and showing that it doesn’t work for numbers which have number ranges where the upper bound is less than about 80.

Problem 5 is beyond my knowledge of geometric proofs and although I could probably work it out algebraically that would be very long and messy. However a little advice when you are drawing it:

- Draw a circle
- Draw a long line tangent to this circle at the bottom of the circle; this will be an extension to line BC
- Draw a long line parallel to line you just drew and also tangent to the circle (but at the top of the circle), this will be an extension to the line MN.
- Draw a line tangent and to the left of the circle, perpendicular through the lines you just drew, this will be an extension to your line AB
- Mark the point B that crosses the bottom horizontal line with your new vertical line
- Mark the point M that crosses the top horizontal line with your new vertical line
- Find and mark the point on the line you just drew, that is above the circle and makes twice the distance from B, as BM, this will be your point A
- Draw a line from point A that make a tangent with your circle and crosses all 3 lines you have drawn so far (at point A, N and C)
- Mark the point N as the point that crosses with the line you just drew and the top horizontal line
- Mark the point C as the point that crosses with the line you just drew and the bottom horizontal line
- Mark the point X as the point where the top of the circle touches the horizontal line (the line MN)
- Get out a pair of compass and draw a circle that is tangent with all three sides inside the triangle AMN
- Mark the point Y where the new circle and the line MN touch
- Write down that XY = 1
- Write down that BC = 12

Unfortunately my geometric knowledge is not enough to go any further than this and I don’t really want to prove it all for you algebraically (probably be here for hours on end trying to prove it algebraically anyway.

Couple of little comments I would also like to make, I don’t know why a right angled triangle seems to work such that once you have drawn line AB then line AC just fits in to place and the mid point ends up just on the right place, I imagine there is some geometric proof for such thing but to me it just seemed right when I started drawing triangles and circles. However because I have no proof you can not assume the triangle is right angled in any proof you attempt unless you yourself can find out why.

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First of all, I would like to say that I am saddened by this thread simply because I do not condone cheating. The USAMTS is supposed to be work done on your own. I will NOT say anything about the answers except that problem 4 can be done with numbers 1 through 20 assuming you have the propper logic behind it. As far as you being in pre-calc, all I have taken is algebra 1 and basic geometry and I figured out problems 1,2, and 4 right off the bat. DO YOUR OWN WORK! Zurtex
Homework Helper
frog_139 said:
First of all, I would like to say that I am saddened by this thread simply because I do not condone cheating. The USAMTS is supposed to be work done on your own. I will NOT say anything about the answers except that problem 4 can be done with numbers 1 through 20 assuming you have the propper logic behind it. As far as you being in pre-calc, all I have taken is algebra 1 and basic geometry and I figured out problems 1,2, and 4 right off the bat. DO YOUR OWN WORK! Erm, as I understand this, this is an old paper they've been asked to see if they can do and if bother to read the thread no one has given them any answers at all, the most I've done is explain how to draw the diagram but if they couldn't figure that out it's not likely that they'll get any further but I thought it might be a nice push to see if they can work it out.

As for problem 4, I am quite sure that this is wrong, I thought I was going to have to write a very long proof on why I thought it was wrong but with such a low bound as 20 it doesn't seem I need to. I feel it quite pointless not writting it here as it can be goggled in moments or even searched on this very forum. But I will PM it you for now.

Yea I think this is last year's questions

These look like olympiad problems, I would hate to have a teacher that assinged such questions for marks.