MathDB

3

Part of 1999 National High School Mathematics League

Problems(2)

Logarithm

Source: 1999 National High School Mathematics League, Exam One, Problem 3

3/9/2020
If (log⁡23)x−(log⁡53)x≥(log⁡23)−y−(log⁡53)−y(\log_2 3)^x-(\log_5 3)^x\geq (\log_2 3)^{-y}-(\log_5 3)^{-y}, then (A)x−y≥0(B)x+y≥0(C)x−y≤0(D)x+y≤0\text{(A)}x-y\geq0\qquad\text{(B)}x+y\geq0\qquad\text{(C)}x-y\leq0\qquad\text{(D)}x+y\leq0
logarithms
Counter Balance

Source: 1999 National High School Mathematics League, Exam Two, Problem 3

3/10/2020
nn is a given positive integer, such that it’s possible to weigh out the mass of any product weighing 1,2,3,⋯ ,ng1,2,3,\cdots ,n\text{g} with a counter balance and kk counterweights, whose weights are positive integers. (a) Find f(n)f(n): the minumum value of kk. (b) Find all possible number of n,n, such that the mass of f(n)f(n) counterweights is uniquely determined.