AMC 10 · 2023 · #23
Grade 11 algebraPick an answer.
Expanding the left side gives a two-variable cubic with seven terms and no obvious factorisation, so brute force is a dead end. The shape of the equation points somewhere else. Every factor on the left is a sum of two positive terms, and the right side is a bare product of the variables. Sum on one side, product on the other, is the signature of the arithmetic-mean-geometric-mean inequality, which says a sum of two positive numbers is at least twice the square root of their product, with equality exactly when the two numbers are equal. So the plan is to stop treating the equation as something to solve and start treating it as something to bound. Prove the two-number inequality once from scratch, apply it separately to each of the three factors, then multiply the three results. If the combined lower bound turns out to be exactly 32ab, the given equation is not asking for a solution at all: it is asserting that the left side sits precisely on its own floor. An extreme like that is rigid. A product of three quantities can equal the product of three smaller-or-equal quantities only if nothing was lost anywhere, so all three separate equalities must hold at once. That converts one hard equation in two unknowns into three easy ones, and the count of solutions is just the count of ways that little system can be satisfied. Finally, check the system is consistent rather than contradictory, because a contradiction would have made the answer zero instead.
Sums on the left, product on the right
The left is a sum, the right a product.
When one side is built from sums and the other from products of the same letters, the equation is probably about an inequality wearing an equals sign.
9.A-SSE.A.1Look For A PatternOne fact about two positives
A sum is at least twice the square root of the product.
A sum of two positive numbers can only get down to twice the square root of their product by making them equal; any imbalance between them costs you.
A sum of two positive numbers can only get down to twice the square root of their product when they are equal.
▸ Why?
For a fixed product the parts add to the least when they are equal, and any imbalance costs.
▸ Why?
So every unequal pair sits strictly above that floor, and only the balanced pair reaches it.
Aim it at each factor
Apply it to each of the three.
Three factors, three separate applications of the same one-line fact; the hard part is only bookkeeping.
9.A-SSE.A.2Identify SubproblemsMultiplying lands exactly on the target
Multiplying makes the floor equal the right side.
The three separate floors multiply into one floor, and that floor is sitting exactly where the problem's right-hand side is standing.
11.N-RN.A.2Extreme PrincipleRead it as an equality case
Equality needs all three conditions at once.
If a product of things has already been pushed down as far as it can go, then nothing inside it can still have any room left.
9.A-CED.A.3Work BackwardsSolve and verify
One solution exists, so the answer is 1.
Three demands on two numbers usually clash; here they happen to agree, so one pair survives instead of none.
9.A-REI.C.6Guess And CheckIf one side of an equation can never dip below the other, then the equation is not asking you to solve anything — it is telling you that you are standing exactly on the floor, and every reason the floor could have been missed has to be switched off at once.
- Sums on the left, product on the right
- One fact about two positive numbers
- Point it at each of the three factors
- Multiply, and the floor lands exactly on 32ab
- The equation is an equality case, not a solve
- Solve the system and check it is not empty