AMC 10 · 2023 · #10
Grade 8 algebraPick an answer.
Tool #3 (Eliminate Possibilities) is the primary tool because the whole problem turns on a two-item finite set. Squaring is a two-to-one operation, so undoing it in (y-x)² = 4y² leaves exactly two candidate relations, y-x = 2y and y-x = -2y. Only one of them can coexist with "x and y are positive"; killing the other is the step that makes the answer unique, and skipping it is the trap the problem is built around. Tool #7 (Identify Subproblems) organizes that split: handle each branch as its own small problem, then feed the survivor into the other equation. Tool #9 (Solve an Easier Related Problem) is what makes the second equation worth attacking first — on its own (y-x)² = 4y² is a messy two-variable quadratic, but it collapses to the one-line relation x = 3y, and substituting that into y³ = x² turns a two-equation system into a single equation in y.
Make both sides squares
Make both sides squares.
Writing 4y² as (2y)² puts both sides in the same shape, and equal squares are far easier to compare than a general quadratic.
8.EE.A.1Solve An Easier Related ProblemUndo the square: two branches
Undoing gives two branches.
Squaring throws away sign information, so undoing it always hands back two possibilities — you must carry both until something else rules one out.
Undoing a square always hands back two possibilities, so both must be carried until something rules one out.
▸ Why?
A number and its opposite have the same square, so the square cannot tell them apart.
▸ Why?
The equation splits into two branches, and only where a factor vanishes does a solution appear.
Branch one contradicts positivity
The first contradicts positivity.
The opposite of a positive number is negative, so x = -y and x ＞ 0 cannot both hold — the word "positive" in the problem is doing real work here.
7.NS.A.2Eliminate PossibilitiesBranch two survives
The second branch survives.
Exactly one of the two branches keeps both numbers positive, so the constraint hands you a clean relation instead of a case split.
8.EE.C.7Eliminate PossibilitiesSubstitute into the cubic
Substitute into the cubic.
A relation like x = 3y lets you erase one variable entirely, and one equation in one unknown is always the easier problem.
8.EE.A.1Solve An Easier Related ProblemCancel to find y and x
Cancel to find both values.
Positivity guarantees y² is not zero, which is the licence you need to divide it away instead of losing solutions.
8.EE.A.1Identify SubproblemsCheck, then add
Checking and adding gives 36.
Substituting back is what turns "I solved it" into "I checked it" — both equations must come out true on the nose.
6.EE.B.5Eliminate PossibilitiesTaking the square root of both sides always leaves two branches, and here the words "positive real numbers" are what throw one of them away — the survivor x = 3y turns y³ = x² into y = 9, so x + y = 27 + 9 = 36.
- Make both sides squares
- Undo the square: two branches
- Branch one contradicts positivity
- Branch two survives
- Substitute into the cubic
- Cancel to find y and x
- Check, then add