AMC 10 · 2016 · #23

Grade 8 algebra

Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity

function-evaluationfraction-arithmeticgcd logical-deductionconvert-to-algebra ↑ Prerequisites: function-evaluation

📏 Long solution 💡 4 insights

Problem

A mystery operation ◆ on nonzero real numbers obeys two rules: a◆(b◆ c)=(a◆ b) · c and a◆ a=1. Using only these rules, find x from 2016◆(6◆ x)=100, write it as a reduced fraction p/q, and report p+q.

Pick an answer.

(A)

109

(B)

201

(C)

301

(D)

3049

(E)

33,601

AMC 10 2016 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Introduce a Variable

The operation is never told to us directly, so treat the result a◆ b as an unknown expression and feed clever inputs into the two rules until the expression is forced. Substituting equal arguments collapses Rule 1 using Rule 2, which pins the operation down as ordinary division. After that the equation is routine: plug in, simplify, and solve a linear equation.

1STEP 1

Make two arguments equal

Put c=b in Rule 1; by Rule 2 the inner b◇b=1, so it collapses to a◇1=(a◇b) · b.

a◆(b◆ b)=(a◆ b) · b → a◆ 1=(a◆ b) · b

2STEP 2

Pin down a ◇ 1

Now set b=a; the right side is (a◇a) · a and a◇a=1, so a◇1=a.

a◆ 1=(a◆ a) · a=1 · a=a

3STEP 3

Identify the operation

Combine a◇1=(a◇b) · b with a◇1=a: then (a◇b) · b=a, so a◇b= $\frac{a}{b}$ — the operation is division.

(a◆ b) · b=a → a◆ b=a/b

4STEP 4

Check division fits both rules

Test a◇b= $\frac{a}{b}$ : Rule 2 gives $\frac{a}{a}$ =1, and Rule 1 gives a÷ $\frac{b}{c}$ = $\frac{ac}{b}$ = $\frac{a}{b}$ · c. Both hold.

a÷b/c=ac/b=a/b · c

5STEP 5

Plug into the equation

Swap ◇ for division: 6◇x= $\frac{6}{x}$ , then 2016÷ $\frac{6}{x}$ =2016 · $\frac{x}{6}$ =336x, so 336x=100.

2016◆(6/x)=2016·x/6=336x=100

6STEP 6

Solve and reduce

From 336x=100, x= $\frac{100}{336}$ = $\frac{25}{84}$ in lowest terms, so p=25, q=84 and p+q=109. Answer (A).

x=100/336=25/84, p+q=25+84=109

Answer

109

Check the chain forward with division: 6◆

\frac{25}{84}

=6÷

\frac{25}{84}

\frac{504}{25}

, then 2016÷

\frac{504}{25}

=2016·

\frac{25}{504}

=4·25=100, exactly the target. The reduced fraction

\frac{25}{84}

is between 0 and 1, a sensible nonzero real, and p+q=109 matches choice (A).

💡Key takeaway

When an operation is defined only by rules, feed it equal inputs to collapse the rules until you recognize a familiar operation hiding underneath.

Make two arguments equal
Pin down a ◇ 1
Identify the operation
Check division fits both rules
Plug into the equation
Solve and reduce