All Answers 356736 Answers

Great diagram! Thanks so much, EP!

Thank You!!

Angle YBP  = 180  - 65 - 60  = 55°

Angle BYP  =  180  - 75 - 60 = 45°

So

Angle BYP  =  180  - 55 - 45  = 80°

And Angle CYX is a vertical  angle to BYP  ....so it = 80°

So.... CXY  =  180  - 60  - 80   =   40°

See diagram below

Box 1         Box 2

0                  5

1                  4

2                  3

3 ways ?   If things are indistinguishible....yah?

That is correct! Good job, hectictar! Amazing!

√ [  (xy) / (xy^2z) ] =

√ [   1 / (yz) ]   =

1 / √ [yz]           Multiply  top/bottom  by  √[yz]

√[ yz] / [ yz]

Yes and they are all inside the Radical sign

Can you re-write your question for claification?

Is that:

sqrt  (  (xy)/(xy^2z)  )   ?

Julius got the correct answer....but a little clarification:

Cost per contributor = C = $ 500/n     where n= number of contributors

We want C=20

so    $ 20 = 500/n   now solve for n

      n =$ 500/$20

       n= 25  = number of contributors

Direct sustitution does not work for this example because the denominator would be zero...not allowed.

(x^2-x-2)   the numerator can be factored to  (x-2)(x+1)

  now   (x-2)(x+1) / (x-2)   becomes   (x+1)     the equation of a line  (except x cannot equal 2)

Then  as x approaches 2 the limit becomes  (x+1) = 3

Here is the graph:  (note: I drew a little hole in the graph at x=2)

If you think about it, how many contributors she has already lined up is not necessary. They just put that in to throw you off. 

If each contributor is going to pay an equal amount. Then the cost/# contributors. 

\( C = {500 \over n}\)

So we simply do 500/20 to see how many new contributors Jane must recruit. 

C = 500/20 

C = 25 contributers. 

Umm, so what do we do? 

Triangles ABC and ABD are isosceles with

\(AB=AC=BD\),
and\( \overline{BD}\) intersects\( \overline{AC}\) at \(E\).
If \(\overline{BD}\perp\overline{AC}\), then what is the value of \(\angle C+\angle D\)?

\(\angle D = BDA = DAB \\ \angle C = BCA = CBA \\ \angle \alpha = DBA\\ \angle \beta = BAC \\ \angle \alpha +\angle \beta = 90^{\circ}\)

\(\begin{array}{|rcll|} \hline \angle \alpha &=& 180^{\circ} - 2\times \angle D \\ \angle \beta&=& 180^{\circ} - 2\times \angle C \\ \angle \alpha +\angle \beta &=& 180^{\circ} - 2\times \angle D+180^{\circ} - 2\times \angle C \quad & | \quad \angle \alpha +\angle \beta = 90^{\circ}\\ 90^{\circ} &=& 180^{\circ} - 2\times \angle D+180^{\circ} - 2\times \angle C \\ 90^{\circ} &=& 360^{\circ} - 2\times (\angle C + \angle D) \\ 2\times(\angle C + \angle D) &=& 360^{\circ} - 90^{\circ} \\ 2\times(\angle C + \angle D) &=& 270^{\circ} \\ \mathbf{ \angle C + \angle D } & \mathbf{=}& \mathbf{135^{\circ}} \\ \hline \end{array}\)

The numbers 1,2,3,4,5,6,7,8,9 are arranged in a list so that each number
is either greater than all the numbers that come before it or
is less than all the numbers that come before it.
For example, 4,5,6,3,2,7,1,8,9 is one such list:

notice that (for instance) the 6 is greater than all the numbers that come before it,
and the 2 is less than all the numbers that come before it.
How many are such lists of the numbers 1,2,3,4,5,6,7,8,9 possible?
 

\(\begin{array}{|rrrr|} \hline \begin{array}{r|r} 1.& 123456789 \\ 2.& 213456789 \\ 3.& 231456789 \\ 4.& 234156789 \\ 5.& 234516789 \\ 6.& 234561789 \\ 7.& 234567189 \\ 8.& 234567819 \\ 9.& 234567891 \\ 10.& 321456789 \\ 11.& 324156789 \\ 12.& 324516789 \\ 13.& 324561789 \\ 14.& 324567189 \\ 15.& 324567819 \\ 16.& 324567891 \\ 17.& 342156789 \\ 18.& 342516789 \\ 19.& 342561789 \\ 20.& 342567189 \\ 21.& 342567819 \\ 22.& 342567891 \\ 23.& 345216789 \\ 24.& 345261789 \\ 25.& 345267189 \\ 26.& 345267819 \\ 27.& 345267891 \\ 28.& 345621789 \\ 29.& 345627189 \\ 30.& 345627819 \\ 31.& 345627891 \\ 32.& 345672189 \\ 33.& 345672819 \\ 34.& 345672891 \\ 35.& 345678219 \\ 36.& 345678291 \\ 37.& 345678921 \\ 38.& 432156789 \\ 39.& 432516789 \\ 40.& 432561789 \\ 41.& 432567189 \\ 42.& 432567819 \\ 43.& 432567891 \\ 44.& 435216789 \\ 45.& 435261789 \\ 46.& 435267189 \\ 47.& 435267819 \\ 48.& 435267891 \\ 49.& 435621789 \\ 50.& 435627189 \\ 51.& 435627819 \\ 52.& 435627891 \\ 53.& 435672189 \\ 54.& 435672819 \\ 55.& 435672891 \\ 56.& 435678219 \\ 57.& 435678291 \\ 58.& 435678921 \\ 59.& 453216789 \\ 60.& 453261789 \\ 61.& 453267189 \\ 62.& 453267819 \\ 63.& 453267891 \\ 64.& 453621789 \\ \end{array} & \begin{array}{r|r} 65.& 453627189 \\ 66.& 453627819 \\ 67.& 453627891 \\ 68.& 453672189 \\ 69.& 453672819 \\ 70.& 453672891 \\ 71.& 453678219 \\ 72.& 453678291 \\ 73.& 453678921 \\ 74.& 456321789 \\ 75.& 456327189 \\ 76.& 456327819 \\ 77.& 456327891 \\ 78.& 456372189 \\ 79.& 456372819 \\ 80.& 456372891 \\ 81.& 456378219 \\ 82.& 456378291 \\ 83.& 456378921 \\ 84.& 456732189 \\ 85.& 456732819 \\ 86.& 456732891 \\ 87.& 456738219 \\ 88.& 456738291 \\ 89.& 456738921 \\ 90.& 456783219 \\ 91.& 456783291 \\ 92.& 456783921 \\ 93.& 456789321 \\ 94.& 543216789 \\ 95.& 543261789 \\ 96.& 543267189 \\ 97.& 543267819 \\ 98.& 543267891 \\ 99.& 543621789 \\ 100.& 543627189 \\ 101.& 543627819 \\ 102.& 543627891 \\ 103.& 543672189 \\ 104.& 543672819 \\ 105.& 543672891 \\ 106.& 543678219 \\ 107.& 543678291 \\ 108.& 543678921 \\ 109.& 546321789 \\ 110.& 546327189 \\ 111.& 546327819 \\ 112.& 546327891 \\ 113.& 546372189 \\ 114.& 546372819 \\ 115.& 546372891 \\ 116.& 546378219 \\ 117.& 546378291 \\ 118.& 546378921 \\ 119.& 546732189 \\ 120.& 546732819 \\ 121.& 546732891 \\ 122.& 546738219 \\ 123.& 546738291 \\ 124.& 546738921 \\ 125.& 546783219 \\ 126.& 546783291 \\ 127.& 546783921 \\ 128.& 546789321 \\ \end{array} & \begin{array}{r|r} 129.& 564321789 \\ 130.& 564327189 \\ 131.& 564327819 \\ 132.& 564327891 \\ 133.& 564372189 \\ 134.& 564372819 \\ 135.& 564372891 \\ 136.& 564378219 \\ 137.& 564378291 \\ 138.& 564378921 \\ 139.& 564732189 \\ 140.& 564732819 \\ 141.& 564732891 \\ 142.& 564738219 \\ 143.& 564738291 \\ 144.& 564738921 \\ 145.& 564783219 \\ 146.& 564783291 \\ 147.& 564783921 \\ 148.& 564789321 \\ 149.& 567432189 \\ 150.& 567432819 \\ 151.& 567432891 \\ 152.& 567438219 \\ 153.& 567438291 \\ 154.& 567438921 \\ 155.& 567483219 \\ 156.& 567483291 \\ 157.& 567483921 \\ 158.& 567489321 \\ 159.& 567843219 \\ 160.& 567843291 \\ 161.& 567843921 \\ 162.& 567849321 \\ 163.& 567894321 \\ 164.& 654321789 \\ 165.& 654327189 \\ 166.& 654327819 \\ 167.& 654327891 \\ 168.& 654372189 \\ 169.& 654372819 \\ 170.& 654372891 \\ 171.& 654378219 \\ 172.& 654378291 \\ 173.& 654378921 \\ 174.& 654732189 \\ 175.& 654732819 \\ 176.& 654732891 \\ 177.& 654738219 \\ 178.& 654738291 \\ 179.& 654738921 \\ 180.& 654783219 \\ 181.& 654783291 \\ 182.& 654783921 \\ 183.& 654789321 \\ 184.& 657432189 \\ 185.& 657432819 \\ 186.& 657432891 \\ 187.& 657438219 \\ 188.& 657438291 \\ 189.& 657438921 \\ 190.& 657483219 \\ 191.& 657483291 \\ 192.& 657483921 \\ \end{array} & \begin{array}{r|r} 193.& 657489321 \\ 194.& 657843219 \\ 195.& 657843291 \\ 196.& 657843921 \\ 197.& 657849321 \\ 198.& 657894321 \\ 199.& 675432189 \\ 200.& 675432819 \\ 201.& 675432891 \\ 202.& 675438219 \\ 203.& 675438291 \\ 204.& 675438921 \\ 205.& 675483219 \\ 206.& 675483291 \\ 207.& 675483921 \\ 208.& 675489321 \\ 209.& 675843219 \\ 210.& 675843291 \\ 211.& 675843921 \\ 212.& 675849321 \\ 213.& 675894321 \\ 214.& 678543219 \\ 215.& 678543291 \\ 216.& 678543921 \\ 217.& 678549321 \\ 218.& 678594321 \\ 219.& 678954321 \\ 220.& 765432189 \\ 221.& 765432819 \\ 222.& 765432891 \\ 223.& 765438219 \\ 224.& 765438291 \\ 225.& 765438921 \\ 226.& 765483219 \\ 227.& 765483291 \\ 228.& 765483921 \\ 229.& 765489321 \\ 230.& 765843219 \\ 231.& 765843291 \\ 232.& 765843921 \\ 233.& 765849321 \\ 234.& 765894321 \\ 235.& 768543219 \\ 236.& 768543291 \\ 237.& 768543921 \\ 238.& 768549321 \\ 239.& 768594321 \\ 240.& 768954321 \\ 241.& 786543219 \\ 242.& 786543291 \\ 243.& 786543921 \\ 244.& 786549321 \\ 245.& 786594321 \\ 246.& 786954321 \\ 247.& 789654321 \\ 248.& 876543219 \\ 249.& 876543291 \\ 250.& 876543921 \\ 251.& 876549321 \\ 252.& 876594321 \\ 253.& 876954321 \\ 254.& 879654321 \\ 255.& 897654321 \\ 256.& 987654321 \\ \end{array}\\ \hline \end{array}\)

The answer is 256

The last digit has to be either a one (1) or nine (9), then are two (2) options (>,<) for the remaining eight (8) digits, so there are 2^8 =256 arraignments where each digit is either greater than or less than the preceding digits.

GA

Computing...

1.

\(\text{Numbers}:\\ \begin{array}{rrrrrrr} 31 & 30 & 29 & 28 & 27 & 26 & \\ 32 & 13 & 12 & 11 & 10 & 25 & \\ \ldots & 14 & 3 & 2 & 9 & 24 & \\ \ldots & 15 & 4 & 1 & 8 & 23 & \\ \ldots & 16 & 5 & 6 & 7 & 22 & \\ \ldots & 17 & 18 & 19 & 20 & 21 & \\ \end{array} \)

2.

\(\text{Coordinates of the lattice points}:\\ \begin{array}{rrrrrrr} \text{Start at $(0, 0) = 1$ }\\\\ (-3,3) & (-2,3) & (-1,3) & (0,3) & (1,3) & (2,3) & \\ (-3,2) & (-2,2) & (-1,2) & (0,2) & (1,2) & (2,2) & \\ \ldots & (-2,1) & (-1,1) & (0,1) & (1,1) & (2,1) & \\ \ldots & (-2,0) & (-1,0) & (0,0) & (1,0) & (2,0) & \\ \ldots & (-2,-1) & (-1,-1) & (0,-1) & (1,-1) & (2,-1) & \\ \ldots & (-2,-2) & (-1,-2) & (0,-2) & (1,-2) & (2,-2) & \\ \end{array}\)

3.

\(\text{Algorithem to get the lattice point coordinates:} \\ \begin{array}{|r|r|} \hline \text{lattice point number} & \text{coordinate }(x,y) \\ \hline 1.& (0,0) \\ 2.& (0,1) \\ 3.& (-1,1) \\ 4.& (-1,0) \\ 5.& (-1,-1) \\ 6.& (0,-1) \\ 7.& (1,-1) \\ 8.& (1,0) \\ 9.& (1,1) \\ 10.& (1,2) \\ 11.& (0,2) \\ 12.& (-1,2) \\ 13.& (-2,2) \\ 14.& (-2,1) \\ 15.& (-2,0) \\ 16.& (-2,-1) \\ 17.& (-2,-2) \\ 18.& (-1,-2) \\ 19.& (0,-2) \\ 20.& (1,-2) \\ 21.& (2,-2) \\ 22.& (2,-1) \\ 23.& (2,0) \\ 24.& (2,1) \\ 25.& (2,2) \\ 26.& (2,3) \\ 27.& (1,3) \\ 28.& (0,3) \\ 29.& (-1,3) \\ 30.& (-2,3) \\ 31.& (-3,3) \\ 32.& (-3,2) \\ \ldots \\ 1843.& (21,15) \\ \ldots \\ 2017.& (22,14) \\ \ldots \\ 2018.& (22,15) \\ \ldots \\ 2019.& (22,16) \\ \ldots \\ 2201.& (23,15) \\ \ldots \\ \hline \end{array}\)

4.

short algorithm to get (x,y) the index in a grid-net of the spiral in c++:

    int n_max = 2818;
    int y_coordinate = 0;
    int x_coordinate = 0;
    for(int i = 0; i < n_max; ++i) {
        // output Number i+1
        // output x_coordinate
        // output y_coordinate
        if(abs(y_coordinate) <= abs(x_coordinate) && (y_coordinate != x_coordinate || y_coordinate >= 0))
            y_coordinate += ((x_coordinate >= 0) ? 1 : -1);
        else
            x_coordinate += ((y_coordinate >= 0) ? -1 : 1);
    }

Find the lattice point coordinate of 2018.

We find (22,15).

The four numbers directly adjacent to the number “2018” in
this spiral are (21,15), (22,14), (22,16), (23,15).

The Numbers are 1843, 2017, 2019, 2201.

6)

Going through a similar process

(5, -3)  =  [  ( -5, - 3)  - (-3, -3) ] *SF  +  (-3, -3)

(5, - 3) - (-3, -3)  =  [ ( -2, 0)] *SF

(8,0)   =   (-2, 0 ) * SF

8 / - 2   =   - 4   =  SF

Impressive, hectictar  !!!

After the last round, A has 20 counters, B has 10, and C has 9 . So...

(p + q + r) * the num of rounds  =  20 + 10 + 9   =   39

p + q + r   =   39 / the num of rounds

Since  p + q + r  must be an integer, the num of rounds must be  3,  13,  or  39 .

If the num of rounds = 13 ,  p + q + r  =  3  , but no 3 different positive integers add to 3 .

If the num of rounds = 39 ,  p + q + r  =  1  , but no 3 different positive integers add to 1 .

So the num of rounds must be 3 , and  p + q + r = 13

By trial and error, I think I found 3 different integers that work:  1, 4, and 8

sum of counters earned in 3 rounds for player A   =   8 + 8 + 4  =  20
sum of counters earned in 3 rounds for player B   =   1 + 1 + 8  =  10
sum of counters earned in 3 rounds for player C   =   4 + 4 + 1  =  9

q  =  the middle number  =  4

On the first round, player C earned  4  counters.

I just got lucky by guessing those numbers for p, q, and r.....Maybe there is a way to do it without guessing!

5)

We can look at the diagram.....but....there is a way to solve this with some math

New Point  =  ( Old Point - Dilation Center)*(Scale Factor) + Dilation Center

Note that:    (a, b) - (c,d)  = ( a - c, b - d)

And  C(a, b)  = (Ca , Cb)     where C is a constant

SF  = Scale Factor

So   using the new point  (-8, - 3)  and  the old point (4, - 3)  we have

(-8, - 3)  =  [ (4, -3) - (-4, - 3)] * SF  +  ( -4, -3)  

Subtract  (-4, - 3)  from both sides

(-8, -3) - ( -4, -3)  =  [ (4, -3) - (-4, - 3)] * SF

(-8, -3) - ( -4, -3)  = [ (8, 0) ] * SF

(-4, 0)  =  (8, 0) * SF 

To find the scale factor, take the ratio  of the x coordinate on the left side and divide it by  the x coordinate on the right side

-4 / 8  =   - 1/2   =  scale factor

r  =     x                  ln(x)             ln(2)

        ____     -       ____     +

        3³√x                9

Note  that we can write this as :

r  = (1/3)x^(2/3) -  (1/9)ln(x)  +  ln (2)      so.....

We can use the Power Rule on the first term

And the definition of the derivative of ln(x)  on the second term

Note....the third term is a constant...its derivative is  0

So we have.....

dr

__    =   (2/9)x^(-1/3) - (1/9)(1/x)  =  

dx

   2                   1

 ____     -       ___        =

 9³√x               9x

[ 2 ³√x^2  -  1    ]

_____________

          9x

Angle C  =  ACB      Angle D  = ADB

BAD  = ADB         ABC  = ACB

DBA  + BAC   =  90    ⇒   DBA  = 90 - BAC 

ACB + DBC  = 90    ⇒      DBC =  90 - ACB 

DAC + ADB  = 90   ⇒       DAC  = 90 - ADB       

DAC + BAC  = BAD                      DBA + DBC = ABC

DAC + BAC  = ADB                      DBA + DBC =  ACB

[90 - ADB] + BAC  = ADB             DBA + [90 - ACB]  = ACB

90 + BAC = 2ADB   (1)                  90 + DBA  = 2ACB

                                                      90 + [90 - BAC]  = 2ACB

                                                     180  - BAC  = 2ACB

                                                      BAC  = 180 - 2ACB   (2)

Sub  (2)  into (1)

90 + [180 - 2ACB]  = 2ADB

270 =  2ACB + 2ADB

135  = ACB + ADB

135  =  ∠C +  ∠ D

Thank you!!!