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1) Alex wrote ten different digits on one red card and nine blue cards, only one digit per card. If he multiplies the digit on the red card by the sum of all the digits on the blue cards, he gets 296.

What digit is written on the red card?

Alex must have written 8 on the RED card and 1, 2, 3, 4, 5, 6, 7, 9 on the nine BLUE cards, because:

[1+2+3+4+5+6+7+9] =37 and: 8 x 37 = 296.

Well, denominators cannot equal zero so you can make your own start on this.

Come back with what you have done.

Sub in x=1 and get equation (1)

Sub in x=2 and get equation (2)

(2)-(1) will give equation (3) and it will cancel out c so now the only pronumerals will be a and b

Seperate any groups of a+b that are there and put them =1

that will leave you with a little equation with just one variable.

I will let you take it from there.

If you have trouble show what you have done and someone will help you further.

Please no one answer over me.

And do not respond if the asker comes back with a no show of effort response.

To find the product, multiply the magnitudes and add the angles.

As follows:

Of course, it helps to know the summations!  They occur often enough that they are worth memorising.

Thanks Geno,

I think Geno made a small copy error when he told you to plot  (5,-1).  He meant (5,1)

Maybe this will help.

Can you see how vector b has been moved so that it is still the same length and parallel (same direction) but its start point has moved to (3,-1)

This is how vectors are added. It is the pictorial version of what is happening with the numbers.

A complex number in the form: z = a + bi  can be written in polar form as: z = r.e^i.theta, where r = √(a² + b²) and theta = tan^-1(b/a)  (that's arctan(b/a))

You should plug your numbers into these.

I notice the polar form sometimes seems to be written as  r.cis(theta)  which is r.cos(theta) + i.r.sin(theta)  (which I think is more like rectangular form!!).

Let n = 99999 - 10000 + 1 = 90000

n/3 + n/5 - n/15 = 42000    (we subtract n/15 to avoid double counting those that are divisible by both 3 and 5).

hi Silverleaf,

You need to answer some questions for me first.

1) How many possibilities are there if there are no restrictions

2) If the yellow is at one of the ends, what are the restrictions

3) if the yellow is not on the end what are the restrictions.

Can you work out any counts for the restrictions of 2 and 3 ?

Plese let  'silverleaf'  respond properly before anyone comes in with more help.

Erich lists the sequence of non-perfect squares:

\(2,\ 3,\ 5,\ 6,\ 7,\ 8,\ 10,\ 11,\ 12,\ 13,\ 14,\ 15,\ 17,\ \ldots\)
What is the 1000th term in this sequence?

Formula: \(a_n = n + \lfloor 0.5 + \sqrt{n} \rfloor \)

\(\begin{array}{|rcll|} \hline \mathbf{a_n} &=& \mathbf{n + \lfloor 0.5 + \sqrt{n} \rfloor } \quad | \quad n=1000 \\\\ a_{1000} &=& 1000 + \lfloor 0.5 + \sqrt{1000} \rfloor \\ a_{1000} &=& 1000 + \lfloor 0.5 + 31.6\ldots \rfloor \\ a_{1000} &=& 1000 + \lfloor 32.1\ldots \rfloor \\ a_{1000} &=& 1000 + 32 \\ \mathbf{a_{1000}} &=& \mathbf{1032} \\ \hline \end{array}\)

As follows:

a=1;b=0;c=0;d=0;e=0;p=0; cycle:n= a*10000+b*1000+c*100+d*10+e;if(n%3==0 or n%5==0 or n%15==0, goto loop, goto next); loop:printn,", ",;p=p+1; next:e++;if(e<10, goto cycle, 0);e=0;d++;if(d<10, goto cycle, 0);e=0;d=0;c++;if(c<10, goto cycle,0);c=0;d=0;e=0;b++;if(b<10, goto cycle,0);e=0;d=0;c=0;b=0;a++;if(a<10, goto cycle,0);print"Total = ",p

OUTPUT = 42,000 such numbers.

1 - Start with the first digit from the left. You have 6 choices:2, 3, 4, 5, 6, 7

2 - Start with the 2nd digit from the left. You have 9 choices: 0, 1, 3, 4, 5, 6, 7, 8, 9 (when it begins with 2).

3 - Start with the 3rd digit from the left. You have 8 choices: they could be any of remaining 8 digits as long as they are different from the 1st and 2nd digits. if the 2nd is 3, then you have: 0, 1, 4, 5, 6, 7, 8, 9.

4 - So, the total number will be: 6 x 9 x 8 = 432 such numbers.

Between 10,000 and 99,999 there are 42,000 numbers that are either multiples of 3, or multiples of 5, or multiples of 15.

     x^2 - y^2 + z^2 - 2xz + x + y - z = (x - y - z + 1)(x + y - z)

"...Graph x(θ)=2cosθ,y(θ)=5sinθ , where [0,π] ...."

Here's the graph. You can use it to answer the questions.

Solve 5*sqrt(1 + x) + 5*sqrt(1 - x) = 7*sqrt(2).

Area =  9 * 9 * 3.5    

CP = 0.562

BP = 0.632

The formula is n(n-3)/2 where n is the number of the sides, so plug and chug!

20(17) / 2 = 10 * 17 = 170 diagonals

quite a lot huh?

There are 432 such  numbers.

I'm going to change each of these into rectangular coordinates.

x  =  42 + (2/3)t   --->   solve for t   --->   x - 42  =  (2/3)t    --->    3x - 126  =  2t   --->   t  =  (3x - 126)/2

y  =  8 + (1/6)t   --->   substitute   --->   y  =  8  +  (1/6)(3x - 126)/2   --->   y  =  (1/4)x - 15/6

x  =  12 + t   --->   solve for t   --->   t  =  x - 12

y  =  5 + (1/4)t   --->   substitute   --->   y  =  5 + (1/4)(x - 12)   --->   y  =  (1/4)x + 2

Since they have the same slope, they don't intersect  --  they do not collide!

Please graph this ....

You'll find that you have a rectangle.

One point of intersection of the two graphs is the point (-2.5, 2.5).

Use the distance formula to find the distance from the origin to this point. This will be one of the sides of the rectangle.

Another point of intersection is the point (1.5, 1.5).

Use the distance formula to find the distance from the origin to this point. This will be another side of the rectangle.

Knowing the two sides of the rectangle, you can find the area ...

From 7/8 to 4/3 is     32/24 -21/24 = 11/24

     from 1 to 4/3 is    1/3   

1/3   out of   11/24     =   1/3 /  11/24 = 24/33 chance=    8/11

x component of trajectory =   73 cos 34   f/s   

    Now how long will it take to travel 49 yards (= 147 feet) ?

          147 / (73 cos 34) = 2.43 seconds

INITIAL y component of trajectory = 73 sin 34 f/s = 40.82 f/s  

     but THIS velocity is acted upon by gravity

        y = y0 + v_ot - 1/2 a t^2       

                          y0 = 0        Now we need to know the y value at 2.43 seconds to see if it will clear the uprights

        y = 40.82 (2.43)  - 1/2 (32.174)(2.43)²  = 4.2 Feet      Tooo low !   Missed it by  5.8 feet