MathDB

15

Part of 2014 AIME Problems

Problems(2)

Circle and 3-4-5

Source: 2014 AIME I Problem 15

3/14/2014
In △ABC \triangle ABC , AB=3 AB = 3 , BC=4 BC = 4 , and CA=5 CA = 5 . Circle ω\omega intersects AB‾\overline{AB} at EE and BB, BC‾\overline{BC} at BB and DD, and AC‾\overline{AC} at FF and GG. Given that EF=DFEF=DF and DGEG=34\tfrac{DG}{EG} = \tfrac{3}{4}, length DE=abcDE=\tfrac{a\sqrt{b}}{c}, where aa and cc are relatively prime positive integers, and bb is a positive integer not divisible by the square of any prime. Find a+b+ca+b+c.
trigonometryanalytic geometrygeometrycyclic quadrilateralPythagorean TheoremAIME
Function with Primes

Source: 2014 AIME II Problem 15

3/27/2014
For any integer k≥1k\ge1, let p(k)p(k) be the smallest prime which does not divide kk. Define the integer function X(k)X(k) to be the product of all primes less than p(k)p(k) if p(k)>2p(k)>2, and X(k)=1X(k)=1 if p(k)=2p(k)=2. Let {xn}\{x_n\} be the sequence defined by x0=1x_0=1, and xn+1X(xn)=xnp(xn)x_{n+1}X(x_n)=x_np(x_n) for n≥0n\ge0. Find the smallest positive integer, tt such that xt=2090x_t=2090.
functioninductionnumber theoryprime factorizationAMC2014 AIME IIAIME