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#1 Re: Help Me ! » Complex numbers » 2017-07-13 10:54:56

bob bundy wrote:

hi mathster

What you have discovered about the number and its complex conjugate is generally true for all complex numbers.  But this one has what as its magnitude?

Then you can apply de Moivre's theorem. https://en.wikipedia.org/wiki/De_Moivre%27s_formula

LATER EDIT:

Use a calculator to evaluate (in degrees) ACOS((root3 +1)/(2root2))

Now can you prove this result analytically?

Bob

Hi, bob. Thanks, but I don't really understand trigonometry this well. We haven't studied it in class yet, so I know there's an easier solution. Could you please explain it without using the theorem?

#2 Help Me ! » Complex numbers » 2017-07-13 02:48:36

mathster
Replies: 3

The value


is a positive real number. What real number is it?



The problem states that there is a shortcut and big calculations are unnecessary and probably part of a wrong solution.
I have found that the complex number without the power of 72 multiplied by its conjugate is the its magnitude squared. So, the magnitude must be a real number. However, I don't know what to do with this information.

#3 Re: Help Me ! » Problem. » 2016-07-06 22:52:49

thickhead wrote:

I get it now. But one thing you didn't include: When 4 girls sit together it should be N3=4!(ways to arrange the girls)*6!(5 boys and the 4-girl block).
So N1+N2+N3=103680.
So bobbym was right. neutral
Thanks anyway, though.

#4 Re: Help Me ! » Problem. » 2016-07-05 00:02:50

I'm torn between two different answers, if possible, could you please provide a solution on how you got them?

#5 Help Me ! » Problem. » 2016-07-03 01:12:42

mathster
Replies: 8

Hi, I'm struggling with the following problem:

There are 5 boys and 4 girls in my class. All of them are distinguishable.
In how many ways can they be seated in a row of 9 chairs such that at least 3 girls are all next to each other?

Please help. Thanks in advance.

#7 Help Me ! » Series. » 2016-01-29 08:58:50

mathster
Replies: 3

Compute the sum


Help will be very much appreciated!

#8 Help Me ! » Geometry problems. » 2015-12-21 06:16:40

mathster
Replies: 3

In the diagram below, we have $\angle ABC = \angle ACB = \angle DEC=\angle CDE$, $BC = 8$, and $DB = 2$. Find $AB$.

Angle bisectors $\overline{TX}$ and $\overline{UY}$ of $\triangle TUV$ meet at point $I$. Find all possible values of $\angle V$ if $\angle  TIU = 109^\circ$. As your answer, enter the number of degrees in $\angle V$. If you find more than one possibility, list the possible values in increasing order, separated by commas.

The interior angle measures of a pentagon form an arithmetic progression. The difference between the largest and smallest angle measures is $44^\circ$. Find the measure of the smallest angle, in degrees.

Let $ABCD$ be a convex quadrilateral, and let $P$, $Q$, $R$, $S$, $T$, $U$, $V$, and $W$ be the trisection points of the sides of $ABCD$, as shown.
http://latex.artofproblemsolving.com/8/3/d/83df21d419d8743fb5b781faac136a6c8676d4ef.png
If the area of quadrilateral $ABCD$ is 180, then find the area of hexagon $AQRCUV$.

In the triangle shown, $n$ is a positive integer, and $\angle A > \angle B > \angle C$. How many possible values of $n$ are there?

d4c0ef5eaa79738f8864060f1490f3a9f85b0939.png

Help very greatly appreciated!

#9 Re: Help Me ! » Algebra problems. » 2015-10-19 04:13:31

bob bundy wrote:

hi mathster,

"Simon's Favorite Factoring Trick".  I've not met this before.  Looked up on AoPS but still none the wizer.

Please would you post how you used it.

Bob

Multiplying

by
gives
.

Using the trick, it's factorable to

.

We would probably want to set

and
to
, but we remember that
.

The next closest factors are

and
.

This gives

and
. Therefore
and
. Finally we come to
.

#10 Re: Help Me ! » Algebra problems. » 2015-10-19 03:33:08

Thanks so much guys! Question 3 was quite confusing, although I don't know calculus, apparently you can solve this with SFFT (Simon's Favorite Factoring Trick). Quite clever.

#11 Help Me ! » Algebra problems. » 2015-10-18 07:05:21

mathster
Replies: 9

Two trains travel directly toward each other. One of the trains travels at a rate of 12 km/h while the other travels at a rate of 20 km/h. When the trains are 72 km apart a conductor at the front of one of the trains releases the insane pigeon Hyde. Hyde flies first from the slower of the two trains to the faster train at which point Hyde doubles back toward the slower train. Hyde continues to fly back and forth between the trains as they approach, always at a constant speed of 48 km/h. Assuming the trains never change speed until they meet and magically stop, how many kilometers has Hyde flown when the trains meet?

The terms of a particular sequence are determined according to the following rules:
* If the value of a given term is an odd positive integer s, then the value of the following term is 3s - 9
* If the value of a given term is an even positive integer t, then the value of the following term is 2t - 7.
Suppose that the terms of the sequence alternate between two positive integers (a,b,a,b,...). What is the sum of the two positive integers?

Given positive integers

and
such that
and
, what is the smallest possible positive value for
?

Help would be very appreciated. Thanks!

#12 Help Me ! » Quick problem. » 2015-10-15 23:06:15

mathster
Replies: 5

Alana is making a ball out of rubber bands. When the ball has 54 rubber bands, it has a diameter of 3 cm. How many rubber bands should Alana add to the ball to increase its diameter by 1 cm? Assume that all of Alana's rubber bands have the same volume.

I am getting 74. Could anyone please tell me if that's correct? Thanks.

(with 54 rubber bands).

(with new amount of rubber bands).


Is my solution correct?

#13 Help Me ! » 3D Geometry. » 2015-10-09 06:10:59

mathster
Replies: 1

Let

be a cube of side length 5, as shown. Let
and
be points on
and
, respectively, such that
and
. The plane through
,
, and
intersects
at
. Find
.
6c649fb67c97a8ea3699eb865305964a51fd4f45.png
Could someone please give me the answer with a solution? I don't understand vectors. sad
Thanks.

#14 Re: Help Me ! » Hard area problem. » 2015-10-09 04:45:52

Wow, drawing BF never occurred to me. Thanks a lot everyone!

#15 Help Me ! » Hard area problem. » 2015-10-09 02:27:52

mathster
Replies: 6

What is the area in square inches of the pentagon shown?
d5aa8865969ffc2e6a05763fe8ca2a0d771cabcb.png
Could someone please tell me the answer? Solution preferred, but optional.
Thanks in advance.

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