AMC 10 · 2010 · #24

Grade 8 algebra

Archetype Small-Domain Constrained Search └ Sub-archetype Sum-Target Search

sequences-geometricsequences-arithmeticsystems-of-equations guess-and-check ↑ Prerequisites: sequences-geometricsequences-arithmetic

📏 Long solution 💡 3 insights

Problem

Two teams tie in the first quarter of a game. Over the four quarters the Raiders' points form an increasing geometric sequence and the Wildcats' points form an increasing arithmetic sequence. The Raiders' four-quarter total is exactly one more than the Wildcats', and neither team's total is above 100. Find the sum of all points both teams scored in the first half (quarters one and two combined).

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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

How to solve

Strategy Introduce a Variable

The problem hides three unknowns inside two number patterns. Naming the shared first-quarter score a, the geometric ratio r, and the arithmetic step d turns the story into expressions we can compare. The single equation 'won by one point' then links them, and a short bounded search finds the one whole-number solution.

1STEP 1

Model both teams as sequences

The tie means both teams scored the same in quarter one; call it a. Raiders then multiply by a fixed ratio, Wildcats add a fixed step.

Raiders: a, ar, ar², ar³ Wildcats: a, a+d, a+2d, a+3d

2STEP 2

Write each team's total

Add each team's four quarters. The Raiders' total factors neatly, while the Wildcats' total collects six copies of the step d.

Raiders total = a(1+r+r²+r³), Wildcats total = 4a+6d

3STEP 3

Pin down the ratio

The ratio must top 1, keep every score whole, and hold the total to 100. Ratios 3 and 4 and fractional ones all fail, leaving r=2.

r=2: Raiders = a, 2a, 4a, 8a, total = 15a ≤ 100 → a ≤ 6

4STEP 4

Use 'won by one point'

Now the totals are 15a and 4a+6d, and the one-point margin ties them together as 11a - 1 = 6d.

15a = (4a+6d) + 1 ⟹ 11a - 1 = 6d

5STEP 5

Search for the whole-number fit

The cap 15a ≤ 100 allows a from 1 to 6, and 11a-1 is a multiple of 6 only at a=5, where 54 = 6d gives d=9.

a=5: d = (11 · 5 - 1)/6 = 54/6 = 9

6STEP 6

Check and total the first half

Raiders 5,10,20,40 (75) and Wildcats 5,14,23,32 (74) meet every condition, so the two first halves give 15 + 19 = 34.

(5+10) + (5+14) = 15 + 19 = 34

Answer

The final scores 5,10,20,40 and 5,14,23,32 satisfy every stated fact: the first quarters match (tie), the Raiders double each quarter (increasing geometric), the Wildcats add 9 each quarter (increasing arithmetic), the totals 75 and 74 differ by exactly one with the Raiders ahead, and both stay under 100. The half-total 34 is one of the offered choices, confirming consistency.

💡Key takeaway

Name the hidden numbers with letters, turn each clue into one equation, then check the few values that fit.

Model both teams as sequences
Write each team's total
Pin down the ratio
Use 'won by one point'
Search for the whole-number fit
Check and total the first half