parmen

We know that P(a) - P(b) is divisible by a - b.  This means if a - b is not a prime number, then P(a) - P(b) is not a prime number.  So by modular arithmetic, we can find a positive integer n such that P(n) is composite.

If P(a_1) and P(a_2) are composite, then P(a_1) - P(a_2) will be divisible by some prime number.  Similarly, if P(a_1) - P(a_2) is divisible by some prime number, and P(a_1) or P(a_2) is composite, then the other one is also composite.

We can then take a_1 = n.  So there exists a positive integer a_2 such that P(a_2) is composite.  But since P(a_2) is composite, we can apply the result above, to find a positive integer a_3 such that P(a_3) is composite.  In this way, we can generate an infinite number of positive integers n such that P(n) is composite.

By properties of the orthocenter, angle BHC is equal to 75 degrees.

Understanding the Problem:

We have triangle ABE with points C and D on side BE.

We know the lengths of BD and CE, as well as the areas of triangles ABC and ADE.

We need to find the area of triangle ACD.

Strategy:

Find the ratio of areas of triangles ABC and ADE.

Use this ratio to find the ratio of heights from A to BC and DE.

Use this ratio to find the ratio of bases BC and DE.

Use the given lengths of BD and CE to find the lengths of BC and DE.

Use the lengths of BC, DE, and the ratio of their areas to find the area of triangle ACD.

Solution:

Ratio of areas:

Area of ABC / Area of ADE = 10 / 24 = 5 / 12

Ratio of heights:

Since the triangles have the same base (AE), the ratio of their areas is equal to the ratio of their heights from A.

Height of ABC / Height of ADE = 5 / 12

Ratio of bases:

Since the triangles have the same height from A, the ratio of their areas is equal to the ratio of their bases.

BC / DE = 5 / 12

Lengths of BC and DE:

BD = 8 and CE = 12

BC = BD + DE = 8 + DE

Substituting BC in the ratio: (8 + DE) / DE = 5 / 12

Solving for DE: DE = 48/7

BC = 8 + 48/7 = 104/7

Area of ACD:

Area of ACD / Area of ADE = BC / DE = 5 / 12

Area of ACD = (5/12) * 24 = 10

Therefore, the area of triangle ACD is 10 square units.

The answer is 17n - 8.