proyaop

We can use similar triangles here. First of all, ABC is an isosceles triangle because the altitude bisects the base, and thus forms two congruent triangles on each side by SAS congruency...

Angle BAD = angle FCB because both angles = 180 - 90 - angle B through angle-chasing. Ang le BAD = angle DAC, so we now can form a general similarity statement:

Triangle HCD ~ Triangle CAD. This means that AD / CD = CD / HD.

Let CD = s: \({16\over{s}} = {s\over{5}} \), and then cross multiplying yields \(80 = s^2\), square rooting both sides gives \(s = 4\sqrt{5}\).

Because the area of ABC = bh/2 = AD * BC/2 = AD*DC, we can substitute = 16 * 4sqrt(5) = 64sqrt(5), our final answer.

Our current goal is to find the value of xy, so we can get x^2 + y^2 = (x + y)^2 - 2xy.

x^3 + y^3 can be factored to (x + y)(x^2 - xy + y^2)

Additionally, x^2 - xy + y^2 = (x + y)^2 - 3xy 

So, x^3 + y^3 = (x + y)(x^2 - xy + y^2) = (x + y)[(x + y)^2 - 3xy], now substitute:

2 * (4 - 3xy) = 5  => 4 - 3xy = 2.5 => 1.5 = 3xy => xy = 0.5

Now, to get x^2 + y^2, just do (x + y)^2 - 2xy = 4 - 1 = 3

......Can you please plug it into latex and add the diagram for X?....

So basically we have a decagon inscribed in a unit circle, and the problem asks to find the perimeter.

Splitting the decagon up into 10 triangles and observing one of them, you will find that it is an isosceles triangle with vertex angle 360/10 = 36 degrees. Using the law of cosines, the side length of the decagon, s:

\(s^2=1^2+1^2-2*1*1\cos{36^o}\), if you know some trig, you will quickly find that cos(36 degrees) = (sqrt(5) + 1)/4, which can be substituted in to get \(s^2=2-{\sqrt{5}+1\over{2}}={3-\sqrt{5}\over{2}}\). 

Remember to square root both sides and simplify the radical to obtain s, and then multiply by 10 to get the perimeter: I don't want to directly spoil the answer for you Vxrtate- happy solving !

Floor 3: Room 3, 6, 9

Floor 5: Room 3, 6, 9

Floor 7: Room 3, 6, 9

9 total possible options, of which only 1 will be correct, so 1/9 (wait read below)

Also, doesn't it say that the hotel only has 5 floors, so there is no floor 7- and thus the answer could also be 1/6 if there were any typos in the question...

2^3 + 3^1 = 11, 2^3 + 3^6 = 737 = 11*67, 2^3 + 3^11 = 177155 = 16105*11.

2^13 + 3^1 is divisible by 11, same as 2^23 + 3^1 is also. Along with 2^13 + 3^6...

As you will notice with these calculations (that you can do brutally by hand or easily with a calculator (to find patterns, of course)), the values of m and n for 2^m + 3^n is divisible by 11 is m = 10a + 3, n = 5b + 1.

List for m: 3, 13, 23, ..., 593 = 60 values

List for n: 1, 6, 11, ..., 596 = 120 values

In total, there are 60 * 120 = 7,200 correct cases, along with 600 * 600 = 360,000 total possible cases: coming to a probability of 7200/360000 = 72/3600 = 2/100 = 1/50, or 2% chance.

Now, I leave you with this parmen... how can you prove that the generalizations for the patterns of m and n are correct?? 

We can do complimentary counting, where you have the total choices (1) minus the wrong choices.

The only way that the machine can generate four numbers where the product is NOT divisible by 2 is if and only if they are all odd numbers. From 1 through 9, there are 5 odd and 4 even numbers.

For the first number generated, there is a 5/9 chance it is odd, and for the next number 5/9, the third 5/9, and the last 5/9 chance again.

Thus, the probability is \({5^4\over{9^4}}={625\over{6561}}\) for generating four numbers with a product not divisible by 2. 

To calculate the probability that the machine will generate four numbers where their product is divisible by 2, you just subtract 625/6561 from 1, which equals to = \(5936/6561\) probability.