14

Part of 2010 AIME Problems

Problems(2)

Source: AIME 2010I Problem 14

For each positive integer n, let f(n) \equal{} \sum_{k \equal{} 1}^{100} \lfloor \log_{10} (kn) \rfloor. Find the largest value of n for which

f(n) \le 300

\lfloor x \rfloor

is the greatest integer less than or equal to

x

floor functionlogarithmssearchinequalitiesfunctionalgebraAMC

The Cevian With Length One

Source: AIME II 2010 Problem #14

In right triangle

ABC

with right angle at

C

\angle BAC < 45

degrees and AB \equal{} 4. Point

P

AB

is chosen such that \angle APC \equal{} 2\angle ACP and CP \equal{} 1. The ratio

\frac{AP}{BP}

can be represented in the form p \plus{} q\sqrt{r}, where

p,q,r

are positive integers and

r

is not divisible by the square of any prime. Find p\plus{}q\plus{}r.

ratiotrigonometrygeometrycircumcirclequadraticsAwesomeMathsummer program