AMC 10 · 2014 · #14
Grade 9 algebranumber-theoryPick an answer.
Both words in the problem are definitions in disguise, so Tool #13 (Convert to Algebra) turns "arithmetic" into 2b=a+c and "geometric" into c²=ab. Two equations in three letters leave one degree of freedom, so Tool #4 (Introduce a Variable) keeps b as the single free parameter and expresses a and c through it. Tool #3 (Eliminate Possibilities) throws away the branch of the factored equation that collides with a ＜ b ＜ c. Tool #15 (Organize Information in More Ways) rewrites the whole triple in one parameter, which is what makes the inequality a ＜ b ＜ c collapse to a single condition. Only then does Tool #14 (Extreme Principle) pick the minimum, and the minimum has to be exhibited, not just bounded.
Turn both words into equations
Both words become plain equations.
The middle term of an arithmetic run is the average of its neighbours, and the middle term of a geometric run squares to the product of its neighbours.
The middle term of an evenly stepped run is the average of its neighbours, and of a geometric run it squares to their product.
▸ Why?
Each term is the previous one plus the same fixed step, so the middle sits halfway between the outer two.
▸ Why?
Each term is the previous one times the same fixed factor, so the two neighbouring ratios have to agree.
Use one equation to delete a
One equation deletes a variable.
Two equations in three unknowns leave one free letter, so spending the easy linear equation to erase a costs nothing.
9.A-CED.A.2Introduce A VariableFactor and drop one branch
Factoring leaves two branches.
Factoring converts one hard equation into a short list of easy ones, and the strict inequality b ＜ c deletes an entry from that list for free.
9.A-REI.B.4Eliminate PossibilitiesRewrite the triple in one letter
The whole triple rides on one letter.
Once all three numbers ride on one parameter, checking a condition means checking a single expression in that parameter.
9.A-CED.A.3Organize Information In More WaysLet the ordering pin down b
The ordering makes that letter negative.
Because each inequality reduces to the same statement about b, the ordering costs one condition rather than two.
9.A-REI.B.3Eliminate PossibilitiesMinimize c over the family
The smallest largest value is 2, choice (C).
Multiplying by -2 flips order, so the largest allowed b produces the smallest c, and that c counts only because a real triple sits underneath it.
7.NS.A.2Extreme PrincipleWrite each progression word as an equation, squeeze the three numbers down to one letter, and then the ordering tells you exactly which values that letter may take.
- Turn both words into equations
- Use one equation to delete a
- Factor and drop one branch
- Rewrite the triple in one letter
- Let the ordering pin down b
- Minimize c over the family