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13

Part of 2009 AIME Problems

Problems(2)

Integer sequence

Source: 2009 AIME I #13

3/18/2009
The terms of the sequence (ai) (a_i) defined by a_{n \plus{} 2} \equal{} \frac {a_n \plus{} 2009} {1 \plus{} a_{n \plus{} 1}} for n≥1 n \ge 1 are positive integers. Find the minimum possible value of a_1 \plus{} a_2.
inductionmodular arithmeticinequalitiesarticlespigeonhole principlefunctionAMC
Segment Product

Source: AIME 2009II Problem 13

4/2/2009
Let A A and B B be the endpoints of a semicircular arc of radius 2 2. The arc is divided into seven congruent arcs by six equally spaced points C1,C2,…,C6 C_1,C_2,\ldots,C_6. All chords of the form ACi‾ \overline{AC_i} or BCi‾ \overline{BC_i} are drawn. Let n n be the product of the lengths of these twelve chords. Find the remainder when n n is divided by 1000 1000.
trigonometrymodular arithmeticsearchalgebrapolynomialVietaARML