AMC 10 · 2004 · #23
Grade 11 algebraPick an answer.
The zeros carry all the information, so name them first and let Vieta's formulas convert the coefficient -2004 into a statement about the zeros. That collapses the whole family of polynomials down to a single unknown number, the product of the two non-integer zeros. Counting values of n then becomes counting allowed values of that one integer, which is an interval of integers minus a short list of forbidden ones — a job for the complement.
Name the three zeros
Naming the hidden zeros makes the third one their sum.
A monic cubic is completely determined by its three zeros, so working with the zeros loses nothing.
11.A-APR.B.3Introduce A VariableRead off the sum of the zeros
The leading coefficients fix that integer zero at 1002.
The integer zero is counted twice inside the total, because it is the sum of the other two.
9.A-REI.B.3Introduce A VariableReduce everything to one number
A single product then controls both unknown coefficients.
Two zeros with a known sum are pinned down by their product alone.
Two zeros whose sum is already known are pinned down completely by their product alone.
▸ Why?
A polynomial's coefficients record both the sum and the product of its roots, so one of them plus the other fixes the pair.
▸ Why?
With the sum fixed, the two zeros sit symmetrically about their midpoint, so one offset describes both.
Write the two hidden zeros explicitly
The quadratic formula writes both hidden zeros using that one number.
Fixing the sum at 1002 makes the two zeros symmetric about 501, so one square root describes both.
9.A-REI.B.4Solve An Easier Related ProblemFind the range of t
Real, distinct and positive squeeze it to 251000 candidates.
The product of the two zeros is largest when they are equal, and that extreme is exactly the case the distinctness rule forbids.
9.A-CED.A.3Extreme PrincipleSpot which t are forbidden
The forbidden ones are exactly those making a perfect square.
A square root of a whole number cannot land on a fraction, so it is either exact or irrational.
8.NS.A.1Change Focus Count The ComplementCount the forbidden ones
Counting those squares gives exactly 500 to remove.
Perfect squares up to a bound are counted by taking the square root of the bound.
8.EE.A.2Make A Systematic ListSubtract and translate back to n
Subtracting leaves 250,500 values, choice (C).
Count everything in range, then delete the short list of cases that break the rules.
4.NBT.B.4Change Focus Count The ComplementWhen a polynomial's zeros are tied together by a rule, describe the whole family with one number, then count which values of that number obey every rule.
- Name the three zeros
- Read off the sum of the zeros
- Reduce everything to one number
- Write the two hidden zeros explicitly
- Find the range of t
- Spot which t are forbidden
- Count the forbidden ones
- Subtract and translate back to n