MathDB

9

Part of 2004 AIME Problems

Problems(2)

Similar figures.

Source:

11/27/2005
Let ABCABC be a triangle with sides 3, 4, and 5, and DEFGDEFG be a 6-by-7 rectangle. A segment is drawn to divide triangle ABCABC into a triangle U1U_1 and a trapezoid V1V_1 and another segment is drawn to divide rectangle DEFGDEFG into a triangle U2U_2 and a trapezoid V2V_2 such that U1U_1 is similar to U2U_2 and V1V_1 is similar to V2V_2. The minimum value of the area of U1U_1 can be written in the form m/nm/n, where mm and nn are relatively prime positive integers. Find m+nm+n.
geometrytrapezoidrectanglenumber theoryrelatively prime
Arithmetic and Geometric

Source:

11/17/2005
A sequence of positive integers with a1=1a_1=1 and a9+a10=646a_9+a_{10}=646 is formed so that the first three terms are in geometric progression, the second, third, and fourth terms are in arithmetic progression, and, in general, for all n≥1n\ge1, the terms a2n−1a_{2n-1}, a2na_{2n}, a2n+1a_{2n+1} are in geometric progression, and the terms a2na_{2n}, a2n+1a_{2n+1}, and a2n+2a_{2n+2} are in arithmetic progression. Let ana_n be the greatest term in this sequence that is less than 1000. Find n+ann+a_n.
geometric sequencearithmetic sequence