AMC 10 · 2018 · #25
Grade 8 number-theoryPick an answer.
Tool #13 (Convert to Algebra): a repdigit of k equal digits d equals d·(10^k-1)/9, so the word problem about strings of digits becomes one clean equation. Tool #4 (Introduce a Variable): set x=10ⁿ so the equation turns into a polynomial in x; 'two values of n' becomes 'two values of x'. The key leverage is that a degree-one (linear) equation in x can be satisfied by two different x only if it is an identity, which forces its coefficients to be zero. Tool #14 (Extreme Principle): once the digit relations are pinned down, push a, b, c as large as the digit rules allow to maximize the sum. Tool #3 (Eliminate Possibilities): test the few legal values of a and throw out the one whose b exceeds 9.
Write each repdigit with a repunit
Write each one with a repunit.
Every repdigit is just its digit times a string of ones, and a string of ones is a power of ten minus one over nine.
8.EE.A.1Convert To AlgebraLet x=10ⁿ
Let one letter be ten to the n.
Renaming 10ⁿ as x turns a statement about digit counts into ordinary algebra.
8.EE.A.1Introduce A VariableCancel the common factor x-1
Cancel the common factor.
The shared factor x-1 divides out cleanly because x=10ⁿ is never 1.
7.EE.A.1Convert To AlgebraCollect into a linear equation in x
What remains is linear.
Lining the equation up as (slope) · x+(constant)=0 exposes exactly two numbers that control everything.
8.EE.C.7Convert To AlgebraTwo values of n force both coefficients to zero
Two values of n force both coefficients to zero.
A straight line crosses zero only once, so hitting zero twice means the line is flat at zero.
A straight line crosses zero only once, so hitting zero twice means the line is flat at zero.
▸ Why?
When an identity holds for more inputs than its degree allows, every coefficient has to vanish.
▸ Why?
A line with a nonzero slope only ever passes a level once, so two crossings are impossible for it.
Solve the system for the digit relations
The system gives relations between the digits.
The two coefficient equations pin a and b directly to c.
8.EE.C.8Introduce A VariableUse digit limits to find legal triples
The single-digit limit leaves two triples.
a² must be nine times a digit, so a has to be divisible by 3, leaving only three cases to check.
7.EE.B.4Eliminate PossibilitiesMaximize the sum
Taking the bigger one, the sum is 18.
Among the surviving cases, the largest digits give the largest sum.
8.EE.C.7Extreme PrincipleTurn each block of repeated digits into (digit)×(10^k-1)/9; the equation becomes linear in 10ⁿ, and since a line hits zero only once, holding for two n forces a²=9c and b=2c, whose biggest legal digits 6,8,4 add to 18, choice (D).
- Write each repdigit with a repunit
- Let x=10ⁿ
- Cancel the common factor x-1
- Collect into a linear equation in x
- Two values of n force both coefficients to zero
- Solve the system for the digit relations
- Use digit limits to find legal triples
- Maximize the sum