MathDB

24

Part of 2003 AMC 12-AHSME

Problems(2)

Maximizing Logs

Source:

2/20/2008
If a≥b>1 a\ge b>1, what is the largest possible value of \log_a(a/b)\plus{}\log_b(b/a)? (A)\ \minus{}2 \qquad (B)\ 0 \qquad (C)\ 2 \qquad (D)\ 3 \qquad (E)\ 4
logarithmstrigonometry
Unique Solution

Source:

1/5/2009
Positive integers a a, b b, and c c are chosen so that a<b<c a<b<c, and the system of equations 2x\plus{}y\equal{}2003\text{ and }y\equal{}|x\minus{}a|\plus{}|x\minus{}b|\plus{}|x\minus{}c| has exactly one solution. What is the minimum value of c c? <spanclass=′latex−bold′>(A)</span> 668<spanclass=′latex−bold′>(B)</span> 669<spanclass=′latex−bold′>(C)</span> 1002<spanclass=′latex−bold′>(D)</span> 2003<spanclass=′latex−bold′>(E)</span> 2004 <span class='latex-bold'>(A)</span>\ 668 \qquad <span class='latex-bold'>(B)</span>\ 669 \qquad <span class='latex-bold'>(C)</span>\ 1002 \qquad <span class='latex-bold'>(D)</span>\ 2003 \qquad <span class='latex-bold'>(E)</span>\ 2004
ceiling functionanalytic geometrygraphing linesslopefunctionalgebrasystem of equations