AMC 10 · 2023 · #7
Grade 11 algebraPick an answer.
Nothing here has to be computed — the whole problem is deciding when the expression is even allowed to exist. Tool #7 (Identify Subproblems) splits that into three independent demands: the logarithms must be defined, the denominator must not vanish, and the radicand must not be negative. Tool #4 (Introduce a Variable) does the heavy lifting: the symbol log n shows up three times, so naming it t turns a logarithm question into the rational inequality (t(2-t))/(t-3) ≥ 0, which is ordinary algebra. Tool #2 (Make a Systematic List) handles that inequality with a sign chart — the roots cut the line into four intervals and one test value settles each. Tool #14 (Extreme Principle) is what separates the right answer from the near misses: every wrong choice in this problem comes from mishandling a boundary point, so t = 0, t = 2, and t = 3 each get checked by hand.
List every condition for realness
List every condition for realness.
Before asking whether a value is non-negative, make sure every piece of it is defined at all.
9.F-IF.A.1Identify SubproblemsName the logarithm t
Give the logarithm a name.
One name for the repeated log n turns a logarithm problem into an ordinary rational inequality.
11.F-LE.A.4Introduce A VariableBuild a sign chart
Build a sign chart.
A product or quotient can only flip between positive and negative where one of its factors hits zero, so those points are the only borders worth checking.
A product or quotient can only flip sign where one of its factors hits zero, so those points are the only borders.
▸ Why?
A product is zero exactly when a factor is zero, and nowhere else can the value cross the axis.
▸ Why?
Between two neighbouring borders nothing changes sign, so one test value settles the whole stretch.
Decide the three border points
Remove where the denominator vanishes.
Each endpoint is worth exactly one value of n, so check them one at a time instead of guessing which way the inequality leans.
9.A-CED.A.1Extreme PrincipleTranslate t back to n
Translate back to integers.
Raising 10 to both sides never reverses an inequality, because the base is larger than 1.
11.F-LE.A.4Introduce A VariableCount the integers
Counting gives 901.
A block of consecutive integers has last minus first plus one members, and the extra 1 is what counts both ends.
4.OA.A.3Identify SubproblemsGive the repeated log n a single name, then remember a square root only needs its inside to be ≥ 0 — so the spots where the top is zero stay in, and the spot where the bottom is zero drops out.
- List every condition for realness
- Name the logarithm t
- Build a sign chart
- Decide the three border points
- Translate t back to n
- Count the integers