Solution #14

Recall that, using the Binomial Theorem, the expansion of (x+y)ⁿ is the sum of all terms of the form nCk*x^n-ky^k.

We want a term in the expansion of  (a-1/ √ a  )⁷ in which the a term has an exponent of -1/2. Using the above general form, the k^th term will have an exponent of 7-k (from the a term) plus (-1/2)k (from the 1/ √ a  term). So we need 7 - k - k/2 = -1/2, or 15/2 = 3k/2, or k = 5. The coefficient is then (-1)⁵*7C5 = -21.

Problem #15

[This problem is worth 7 points.]

Let ABC be a triangle with AB = 125, AC = 117, and BC = 120. Let the angle bisector of A intersect BC at L, and the angle bisector of B intersect AC at K. Let M be the point of intersection of the perpendicular from C to BK, and let N be the point of intersection of the perpendicular from C to AL. Find MN.

Solution #13

Factoring 12! + 14! we have 12!(1 + 14*13) = 12!(183). The largest prime factor of 12! is clearly 11. 183 = 3*61. Thus 61 is the largest prime factor of 12! + 14!.

Solution #12

Note that triangle ABD and triangle AA'B have bases of the same length (since AD = AA' = 6). Their altitudes from B are also the same length, so they have the same area.

Because AB = BB', the altitude from B' to AD is twice the length of the altitude from B to AD. So triangle AA'B' as twice the area of triangle ABD.

By similar reasoning:

The area of triangle BB'C is twice the area of triangle BAC.
The area of triangle CC'D is twice the area of triangle CBD.
The area of triangle DD'A is twice the area of triangle DCA.

The area of A'B'C'D' is: [AA'B] + [BB'C] + [CC'D] + [DD'A] + [ABCD]

From the above, this is:

2[ABD] + 2[BAC] + 2[CBD] + 2[DCA] + 10

But [ABD] + [CBD] = [ABCD] = 10, and [BAC] + [DCA] = [ABCD] = 10. So:

[A'B'C'D'] = 2*10 + 2*10 + 10 = 50.

Problem #14

[This problem is worth 3 points.]

In the expansion of (a-1/ √ a  )⁷, what is the coefficient of a^-1/2?

Solution #11

Let x be the radius of the circle. Then the diameter is 2x, and the circumference is 2xπ. Thus 4/2xπ = 2x, so 4 = 4x²π, and x²π = 1. But x²π is the area of the circle, so the area is 1.

Problem #13

[This problem is worth 1 point.]

What is the largest prime factor of 12! + 14!?