AMC 10 · 2009 · #11

Grade 6 algebra
Archetype Geometric Series Share └ Sub-archetype Cyclic Participant Share
sequences-geometricfraction-arithmeticpattern-recognition pattern-recognition ↑ Prerequisites: sequences-geometric
📏 Long solution 💡 3 insights
Problem
Each day a fixed mix is added while a fixed share of one ingredient survives and the rest is eaten. Find the first day on which that ingredient is more than half the total.

Pick an answer.

(A)
Tuesday
(B)
Wednesday
(C)
Thursday
(D)
Friday
(E)
Saturday
How to solve
Strategy Look for a Pattern

The other seeds are the easy half: birds eat all of them every day, so right after adding, there is always exactly 3/4 quart of other seeds. That turns the whole question into one moving quantity — the millet. Tool #4 (Introduce a Variable) names the millet on day n as m_n and captures the daily rule m_n+1=3/4 m_n+1/4. Tool #6 (Guess and Check) runs the rule day by day, and Tool #5 (Look for a Pattern) spots the clean closed form m_n = 1-(3/4)ⁿ, which makes the winning condition transparent.

1STEP 1

Fix the other seeds at 3/4

The other ingredient is a fixed amount every day.

other seeds = 3/4 (always) → need millet ＞ 3/4
2STEP 2

Write the daily millet rule

So the goal is a fixed threshold to beat.

m₁=1/4, m_n+1=3/4 m_n+1/4
3STEP 3

Run it day by day

The surviving ingredient follows a simple daily rule.

m₁=1/4, m₂=7/16, m₃=37/64, m₄=175/256 ＜ 3/4, m₅=781/1024 ＞ 3/4
4STEP 4

See the pattern behind the numbers

Running it day by day shows the crossing on the fifth day.

m_n=1-(3/4)ⁿ → m_n ＞ 3/4 ⇔ (3/4)ⁿ ＜ 1/4
5STEP 5

Find the first winning day

A closed form explains why, giving Friday, choice (D).

(3/4)⁴≈0.316 ＞ 1/4, (3/4)⁵≈0.237 ＜ 1/4 → n=5=Friday → (D)
Answer
Friday
On day 4 the millet is 175/256≈0.684 quart against 0.75 quart of other seeds — still losing, so Thursday is too early, which rules out (C) and everything before it. On day 5 the millet is 781/1024≈0.763 against 0.75, a slim but real win, so Friday is exactly the first day it crosses over. The margin being small and positive is a good sign we did not overshoot to Saturday (E). Everything is consistent with (D).
💡Key takeaway

When one part gets wiped out and refilled to the same size each day, it becomes a fixed target the other part just has to grow past.

  • Fix the other seeds at 3/4
  • Write the daily millet rule
  • Run it day by day
  • See the pattern behind the numbers
  • Find the first winning day

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