AMC 10 · 2014 · #25
Grade 10 geometry-2dnumber-theoryPick an answer.
The parabola is tilted, so its equation in x and y is a messy conic and hunting lattice points on it directly is hopeless. Tool #13 (Convert to Algebra) turns the focus-directrix definition into a polynomial equation, but the real move is tool #15 (Organize Information in More Ways): rename the plane by m=4x+3y and n=3x-4y. These are not arbitrary — 4x+3y is the quantity the bound measures, 3x-4y is the quantity the directrix measures, and the identity (4x+3y)²+(3x-4y)²=25(x²+y²) says they are the two halves of the squared distance. In the (m,n) names the conic collapses to the single relation m²=50n+625. Tool #7 (Identify Subproblems) then splits the work into two independent questions — which m satisfy the curve equation, and which of those return integer x,y — and tool #4 (Introduce a Variable) packages the survivors as one integer parameter k. Tool #2 (Make a Systematic List) finishes by listing the admissible k inside the bound.
Find the directrix from the definition
The two given points fix the directrix.
Two points the same distance from the focus must be the same distance from the directrix, and only one line direction can do that for a pair placed symmetrically about the focus.
Two points the same distance from the focus must be the same distance from the directrix, and only one direction does that.
▸ Why?
Every point of the curve is as far from the focus as from the directrix, so the two distances travel together.
▸ Why?
Points equally far from two given points lie on the line that folds one onto the other, which fixes the direction.
Turn equal distances into an equation
Equal distances give one equation.
Two equal distances stay equal after squaring, and squaring is what turns a distance statement into a polynomial one.
10.G-GPE.A.2Convert To AlgebraRename the plane with m and n
Renaming the plane makes it simple.
Naming the plane after the two expressions the problem already talks about makes the curve say something simple about them.
9.A-SSE.A.2Organize Information In More WaysForce x and y back to integers
Whole coordinates force a divisibility chain.
t and t+5 are multiples of 5 together or not at all, so their product can only hold 25 when 5 already divides t.
6.NS.B.4Identify SubproblemsOne odd k per lattice point
Each odd parameter gives exactly one grid point.
Once each lattice point wears a single integer label, counting points is the same as counting labels.
9.A-REI.C.6Introduce A VariableCount the odd k inside the bound
Counting them gives 40, choice (B).
The odd numbers from 1 to 39 come in 20 steps of two, and the negative side doubles the tally.
6.NS.C.7Make A Systematic ListRename the plane after the two expressions the problem already mentions — 4x+3y and 3x-4y — and the tilted parabola turns into one odd number per lattice point, so counting points becomes counting the odd numbers from -39 to 39.
- Find the directrix from the definition
- Turn equal distances into an equation
- Rename the plane with m and n
- Force x and y back to integers
- One odd k per lattice point
- Count the odd k inside the bound