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Looking at only the red triangle in the top 3 squares:  Area  =  ½·18·18  =  162.

Looking at only the triangle in the bottom row of squares:  Area  =  ½·27·9  =  121.5.

Adding them together ...

What is your AoPS username, tell me and I will tell you the answer, then we can friend each other.

Let's assume that it ws a valuable chicken, worth $100.00.

Discounting this chicken by 30% give us a chicken worth 70% of $100.00   

     --->     0.70 x $100.00  =  $70.00

Now we discount the $70.00 by another 10%

     --->     0.90 x $70.00  =  $63.00

Our final price of $63.00, when compared to our original price of $100.00:

     --->     $63.00 / $100.00  =  0.63,  or  63%.

[The final percentage would be the same no matter what we chose for an original value.]

Solve:  log₂(x² - 5x + 7)  >=  0

--->              x² - 5x + 7  >=  2⁰

--->              x² - 5x + 7  >=  1

--->              x² - 5x + 6  >=  0

--->              (x - 3)(x - 2)  >= 0

Areas tp check:  (-infinity, 2)     (2)     (2, 3)     (3)     (3, ininity)

Checking values from each area indicate that the values below 2, 2, 3, and above 3 work.

Only those values between 2 and 3 do not work.

--->     answer:  (-infinity, 2]  union  [3, infinity)

Call the points:  A = (-1,2)    B = (3,0)     and    C = (9,0).

The idea behind this problem is this:  the perpendicular bisectors of the chords intersect at the center.

For chord AB.

1)  Find the midpoint of this chord.

2)  Find the slope of this chord

3)  Use this slope to find the slope of the line perpendicular to this chord.

4)  Find the equation of the perpendicular bisector of this chord by using the point from step 1 and the slope from step 3.

Now, follow the same steps for chord BC or chord AC (your choice).

Then, use the two equations to find their point of intersection; this is the center of the circle.

There are 37 points.

Each point is connected to the other 36 points.

Find the number of connections and then divide by 2 because each connection has been doubled.

Its -120. Thank you!

If one root is  5 - 3i  then the other root is  5 + 3i.

An equation with these roots are:  [ x - (5 - 3i) ] · [ x - (5 + 3i) ]  =  0

Expand this equation.

However, the equation is supposed to be of the form:  2x² + rx + s  =  0,

so, you'll have to multiply your answer (above) by 2 to get the proper form.

Then, you'll have r and s.

hey dragonslayer, nice to see you again:)

nice expalanatio, geno I was stuck on this for like 20 minutes!

An equation for a circle is:  (x - h)² + (y - k)²  =  r²  where the center is (h, k) and the radius is r.

Center = (2, 3)   --->   h = 2   and   k = 3

The distance from the center to the point (-1, 6) is a radius:  r  =  sqrt( (-1 - 2)² + (6 - 3)² )  =  sqrt(18)

Equation:  (x - 2)² + (y - 3)²  =  18

Now expand this equation to get it into the proper form ...

521_b  =  5·b² + 2·b + 1

24_b  =  2·b + 4

(24_b)²  =  (2b + 4)²  =  4b² + 16b + 16

(24_b)²  =  521_b   --->   4b² + 16b + 16  =  5b² + 2b + 1

                                                         0  =  b² - 14b - 15

                                                         0  =  (b - 15)(b + 1)   --->   b  =  15

Hints --

Draw a 3 x 3 grid  --  How many lines did you draw?

What is the total length of the lines?

So, how long must each line be?

Draw a 6 x 6 grid  --  How many lines did you draw?

Is each line the same length as the line that you drew for the 3 x 3 grid?

So, how long is the total length?

For problem 1,  I placed the diagram on a coordinate axis, with P(0,0), C(4,0), and B(-14,0).

Since PA = 13, it is on a circle with center P and radius 13:  x2 + y2  =  13

-- I called the x-value of this point 'a', so the y-value became  sqrt(169 - a2).

I used the theorem that the center of a circle is on the perpendicular bisectors of the chords.

Since one chord is BP, its midpoint has x-value -7 and its y-value is on the line x = -7.

Since another chord is PC, its midpoint has x-value 2 and its y-value is on the line x = 2.

Another chord is AP: using its endpoints, I found its midpoint and its slope.

From these values, I could find the equation of its perpendicular bisector (not a lot of fun, for it had a lot of a-terms and square roots).

Then, I found the intersection of this perpendicular biscector with the line x = -7 and the intersection of this perpendicular bisector with the line x = 2 (there were still a lot of a-terms and square roots).

With these two points (the centers of the circumcircles), I could find the distance from one center to point P and the distance from the other center to point P (By this time, I was on a firt-name basis with the a-terms and square roots).

Since these are equal, by solving this equation, I could find the value of a, which was 8.

This is the x-value of point A, allowing me to find the y-value of point A, 12, which is the height of triangle ABC.

With the height 8 + 8 = 16 and the base 12, the area is 96.

Since the altitudes are inside the triangle, the triangle is acute.  This gives us the inequalities 12^2 + 14^2 > h^2, h^2 + 12^2 > 14^2, and h^2 + 14^2 > 12^2.  Then the largest h can be is 18.

Bro, how did you even get those values

g(4) = 17.

165 is incorrect, a hint is "We're told that the circles have the same radius. In your diagram, what other parts of the circles must be equal? What does that tell you about triangle ABC?"

volume of a cube= s^3

1=s^3

s=∛1

s=1

there are 6 sides on a cube. therefore, find the area of one side and multiply it by 6 to find the surface area:

s^2=1^2

     =1

multiply the side by 6 to get 6; hence, 6 is the surface area.

I placed the diagram on a coordinate axis, with P(0,0), C(4,0), and B(-14,0).

Since PA = 13, it is on a circle with center P and radius 13:  x2 + y2  =  13

-- I called the x-value of this point 'a', so the y-value became  sqrt(169 - a2).

I used the theorem that the center of a circle is on the perpendicular bisectors of the chords.

Since one chord is BP, its midpoint has x-value -7 and its y-value is on the line x = -7.

Since another chord is PC, its midpoint has x-value 2 and its y-value is on the line x = 2.

Another chord is AP: using its endpoints, I found its midpoint and its slope.

From these values, I could find the equation of its perpendicular bisector (not a lot of fun, for it had a lot of a-terms and square roots).

Then, I found the intersection of this perpendicular biscector with the line x = -7 and the intersection of this perpendicular bisector with the line x = 2 (there were still a lot of a-terms and square roots).

With these two points (the centers of the circumcircles), I could find the distance from one center to point P and the distance from the other center to point P (By this time, I was on a firt-name basis with the a-terms and square roots).

Since these are equal, by solving this equation, I could find the value of a, which was -14.

This is the x-value of point A, allowing me to find the y-value of point A, 15, which is the height of triangle ABC.

With the height 14 + 8 = 22 and the base 15, the area is 165.