AMC 10 · 2004 · #22

Grade 8 geometry-2d
Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Integrality / Divisibility Screen
factorsprime-factorizationdivisibility-rules convert-to-algebrasystematic-enumeration ↑ Prerequisites: factors
📏 Medium solution 💡 3 insights
Problem
A three by three square of positive integers has 50 in the top-left cell and 2 in the bottom-right cell. Every row, every column and both diagonals have the same product. Find the sum of all values the bottom-left cell can take.

Pick an answer.

(A)
10
(B)
25
(C)
35
(D)
62
(E)
136
How to solve
Strategy Introduce a Variable

Tool #4 (Introduce a Variable) names the common product P and writes all eight line equations, which is the only way to see what is really being asked. Tool #15 (Organize Information in More Ways) supplies the key move: multiply four particular lines, the middle row, the middle column, and both diagonals, and compare with the product of all nine cells. That single regrouping pins the center down to e=10 and P=1000, after which every cell can be written in terms of g alone. Tool #2 (Make a Systematic List) finishes the job: with formulas in hand, ask which g leave every cell a whole number, then build each square to make sure it truly works instead of stopping at 'necessary'.

1STEP 1

Name the common product

Naming the common product turns the square into eight equations.

50bc = def = 2gh = 50dg = beh = 2cf = 100e = ceg = P
2STEP 2

Pin the center at e=10

Multiplying the four lines through the middle pins the centre at 10.

(def)(beh)(100e)(ceg) = P³ · e³ = P⁴ → e³ = P = 100e → e = 10, P = 1000
3STEP 3

Write every cell in terms of g

Every cell can then be written through the corner alone.

e=10, h=500/g, d=20/g, c=100/g, b=g/5, f=5g
4STEP 4

Ask which g keep every cell whole

Two divisibility demands leave only three surviving values.

g ∣ 20 and 5 ∣ g ⟹ g ∈ {5, 10, 20}
5STEP 5

Build the three squares and add

All three build real squares, so the sum is 35, choice (C).

5 + 10 + 20 = 35 (C)
Answer
35
Each square can be checked by hand: for g=10 the rows give 50 · 2 · 10 = 2 · 10 · 50 = 10 · 50 · 2 = 1000, the columns give 50 · 2 · 10 = 2 · 10 · 50 = 10 · 50 · 2 = 1000, and the diagonals give 50 · 10 · 2 = 10 · 10 · 10 = 1000. There is also a structural check: transposing a square (reflecting across the main diagonal) keeps 50 and 2 in place, since both sit on that diagonal, and swaps g with c=100/g. So solutions must pair up under g ⇔ 100/g, and they do: the g=5 square transposes into the g=20 square, while the g=10 square is its own transpose. That symmetry makes a missing solution unlikely. The total 35 is choice (C); the smaller choice (B) 25 would mean one value was lost and (D) or (E) would mean extra values that fail the whole-number test.
💡Key takeaway

Multiply the four lines through the middle cell to discover the center is 10 and every line multiplies to 1000; then every cell is a formula in g, and only g=5,10,20 keep all of them whole numbers, so the sum is 35.

  • Name the common product
  • Pin the center at e=10
  • Write every cell in terms of g
  • Ask which g keep every cell whole
  • Build the three squares and add

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