MathDB

11

Part of 2008 AIME Problems

Problems(2)

Binary Sequences

Source: AIME 2008I Problem 11

3/23/2008
Consider sequences that consist entirely of A A's and B B's and that have the property that every run of consecutive A A's has even length, and every run of consecutive B B's has odd length. Examples of such sequences are AA AA, B B, and AABAA AABAA, while BBAB BBAB is not such a sequence. How many such sequences have length 14?
floor functionAMCAIMEsearchratioalgebrapolynomial
Tangent Circles

Source: AIME 2008II Problem 11

4/3/2008
In triangle ABC ABC, AB \equal{} AC \equal{} 100, and BC \equal{} 56. Circle P P has radius 16 16 and is tangent to AC‾ \overline{AC} and BC‾ \overline{BC}. Circle Q Q is externally tangent to P P and is tangent to AB‾ \overline{AB} and BC‾ \overline{BC}. No point of circle Q Q lies outside of △ABC \triangle ABC. The radius of circle Q Q can be expressed in the form m \minus{} n\sqrt {k}, where m m, n n, and k k are positive integers and k k is the product of distinct primes. Find m \plus{} nk.
trigonometryquadraticsgeometryprobabilityAMCAIMEtrapezoid