Some Problems

Asakura · 2008-09-22 06:56:51

1. Show that if a rectangle, which is twice as long as it is broad, can fit diagonally into a square, then it can also fit into the square with its sides parallel to the sides of the square. Also, prove this is not true if the rectangle is three times as long as it is wide.

2. A queue of slow moving traffic is 5 miles long. It takes 15 minutes to pass a particular road sign. A police car takes a total of 20 minutes driving at constant speed to travel from the back of the queue to the front and return to the back. How fast does the police car travel?

3. In 1946, an American numerologist, Prof. W, predicted the downfall of the USA in the year 2141 based on what he called a profound mathematical discovery depending on the following expression:
1492^n - 1770^n - 1863^n + 2141^n
He spent many months calculating the value of this for n = 1, 2, 3 and so on up to 1945 and found the remarkable fact that the result was always divisible by 1946. Since the years 1492, 1770 and 1863 are all important in American history, he claimed that 2141 would also be significant - hence his prediction.
Show how he could have saved himself months of work.

4. A circle of radius 15cm intersects another circle of radius 20cm at right angles. What is the difference of the areas of the non-overlapping portions? Hence, what is the sum of the areas of the non-overlapping portions?

5. Show that the square of any interger leaves a remainder of 0, 1, 4, or 7 when divided by 9.
Use this to establish the following condition that a number which is a perfect square must satisfy the following:
for a number that is a perfect square, add up its digits to form a second number. If that number has more than one digit, add up its digits to form a third number. Continue until you obtain a single digit number. That final number must be 1, 4, 7 or 9.

Quick responses will be appreciated.

gnitsuk · 2008-09-22 21:10:47

[Answer removed]

Last edited by gnitsuk (2008-09-22 21:12:38)

gnitsuk · 2008-09-22 21:52:39

[Removed]

Last edited by gnitsuk (2008-09-22 21:53:36)

Asakura · 2008-09-24 03:14:23

thank you very much, but does anyone have any more insight on 3, 4, and 5?

mathsyperson · 2008-09-24 06:53:19

[Removed]

Asakura · 2008-09-25 03:37:39

mathsyperson: thank you very much, i think i get it now

but can anyone help me out with 3 and 4? i got a hint from my friend for 3, it's: ((x^n)-(y^n)) = (x-y)(x^(n-1)+....+y^(n-1))

i've got no idea what that formula means unfortunately

as for 4, i've got no idea as to even how to approach that question....

Last edited by Asakura (2008-09-25 07:11:55)

stoobzy · 2008-09-25 06:33:58

This is first round of the scottish maths challange. I might have helped if you had only asked for help o none question, but considering you are asking for ALL the answers this seems a bit like cheating...

stoobzy · 2008-09-25 06:38:49

I've just realised how many answers are being posted here. I think a moderator should remove them because it seems a bit unfair that someone who didn't know how to do any of the questions is going to get them all right, whereas someone who has taken the effort to do them themselves, may not. This is an official competition from the Scotish Mathematical Council, so please honour it by removing these answers!

mathsyperson · 2008-09-25 07:00:55

True point, stoobzy. I've got rid of all the answers that were here (sorry gnitsuk).

The deadline for entries is the 3rd of October, so discussion should be safe after that.

Asakura · 2008-09-25 07:17:45

i am extremly sorry for this, i did not realize this was from a competition. (I live in Peru) my friend from the UK sent me these problems and asked me to see if i can do them, when i realized i had no idea how to do them, i thought it was a bit embarrasing and asked for help here. i have not sent any of the answers back so he won't be getting an unfair advantage.

again, very sorry for this

stoobzy · 2008-09-25 08:13:39

I have to give you credit for thinking up that excuse, Asakura! Considering one of your posts once read " i got a hint from a teacher for 3" and was changed to " i got a hint from my friend for 3" seems to suggest that what ur saying isn't exactly true. The hint you refer to is actually on the question paper...

Don't blame you though. It's really hard stuff. To be honest I was hoping this forum would give me a clue to Q2. Its the only one I have left to do but watch out because the answer to it that someone posted here was wrong. He said that by going 30mph (10mph faster than the traffic) the car could catch up with the front of the traffic and return to the back. Common sense tells you that if he is catching up at 10mph he wont even catch up the 5 miles in 20mins, never mind return to the back.

And for 3, try rearranging the term into a term involving (x^n - y^n), and then use the hint (you don't need to find out the missing terms in the hint)

But thats all the help ur getting from me

mathsyperson · 2008-09-25 09:17:16

I've just done that one by writing 1/3 = [time to get to the front] + [time to get back].

Both the parts of the right side are in terms of the police car's speed.
Once you've got that, you can rearrange it into a quadratic equation and then solve.

gnitsuk · 2008-09-25 22:18:34

Quite right stoobzy, my answer was wrong.

It may be instructive to see why it is wrong.

I assumed that the distance travelled by the queue during the time it took for the police car to travel from the back of the queue to the front was the same as the distance travelled by the queue during the time it took the police car to travel from the front of the queue to the back. It is not.

The police car's velocity relative to the queue is different on each pass. My answer was wrong because I indirectly set this relative velocity the same on each pass.

Indeed the answer is greater than my original.

(If only the queue had not been moving at all, then my original answer would have been right!)

Last edited by gnitsuk (2008-09-26 01:25:23)

Math Is Fun Forum

#1 2008-09-22 06:56:51

Some Problems

#2 2008-09-22 21:10:47

Re: Some Problems

#3 2008-09-22 21:52:39

Re: Some Problems

#4 2008-09-24 03:14:23

Re: Some Problems

#5 2008-09-24 06:53:19

Re: Some Problems

#6 2008-09-25 03:37:39

Re: Some Problems

#7 2008-09-25 06:33:58

Re: Some Problems

#8 2008-09-25 06:38:49

Re: Some Problems

#9 2008-09-25 07:00:55

Re: Some Problems

#10 2008-09-25 07:17:45

Re: Some Problems

#11 2008-09-25 08:13:39

Re: Some Problems

#12 2008-09-25 09:17:16

Re: Some Problems

#13 2008-09-25 22:18:34

Re: Some Problems

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