AMC 10 · 2010 · #4

Grade 6 algebra
Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation
function-evaluationformula-substitutionmean-median-mode-range identify-subproblems ↑ Prerequisites: function-evaluation
📏 Medium solution 💡 1 insight
Problem
For a real number x, the rule ♥(x) is defined to be the average of x and x². Compute ♥(1)+♥(2)+♥(3).

Pick an answer.

(A)
3
(B)
6
(C)
10
(D)
12
(E)
20

AMC 10 2010 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Identify Subproblems

The question is really three small identical problems glued together, so Tool #7 (Identify Subproblems) fits: find ♥(1), ♥(2), ♥(3) one at a time, then add. Tool #4 (Introduce a Variable) turns the unfamiliar ♥ symbol into a formula (x+x²)/2 so plugging in is mechanical. Tool #3 (Eliminate Possibilities) catches the built-in trap: forgetting to divide by 2 gives 20, which is exactly choice (E).

1STEP 1

Turn the heart rule into a formula

♥ is just a recipe: average x and x². Averaging means sum divided by 2, so write that with x as a formula.

♥(x)=(x+x²)/2
2STEP 2

Compute the value at 1

Substitute x=1: since 1²=1, average 1 and 1 — sum 2, halved, so ♥(1)=1.

♥(1)=(1+1²)/2=(1+1)/2=1
3STEP 3

Compute the value at 2

Substitute x=2: since 2²=4, average 2 and 4 — sum 6, halved, so ♥(2)=3.

♥(2)=(2+2²)/2=(2+4)/2=3
4STEP 4

Compute the value at 3

Substitute x=3: since 3²=9, average 3 and 9 — sum 12, halved, so ♥(3)=6.

♥(3)=(3+3²)/2=(3+9)/2=6
5STEP 5

Add the three values

Add them: 1+3+6=10, which is choice (C); skipping the halving gives 20, the trap in (E).

♥(1)+♥(2)+♥(3)=1+3+6=10 → (C)
Answer
10
A genuine average of two numbers always lands between them, so ♥(1)=1, ♥(2)=3 (between 2 and 4), and ♥(3)=6 (between 3 and 9) all look right. Their sum 10 is a modest number in the middle of the choices. The tempting 20 is exactly the sum of the x+x² pieces with no halving, which matches choice (E) and confirms it is the forget-to-average trap, not the true answer.
💡Key takeaway

The average of two numbers is their sum divided by 2 — skip the divide-by-2 and you land on double the real answer.

  • Turn the heart rule into a formula
  • Compute the value at 1
  • Compute the value at 2
  • Compute the value at 3
  • Add the three values

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