AMC 10 · 2019 · #8

Grade 9 algebra

Archetype Pair-Sum Invariant └ Sub-archetype Uniform Pair-Sum Recovery

function-evaluationsymmetry-argumentparityfraction-arithmetic pattern-recognitionsymmetry-argumenteasier-related-problem ↑ Prerequisites: function-evaluation

📏 Medium solution 💡 2 insights

Problem

A function is defined as x squared times one minus x, squared. Feed it the 2018 fractions with denominator 2019 in order and add the results with alternating signs: plus when the numerator is odd, minus when it is even. Find the exact value of that total.

Pick an answer.

(A)

(B)

$\frac{1}{2019^{4}}$

(C)

$\frac{2018^{2}}{2019^{4}}$

(D)

$\frac{2020^{2}}{2019^{4}}$

(E)

How to solve

Strategy Look for a Pattern

Nothing here can be brute-forced, so the value must come from a pattern hidden in the formula. The expression x²(1-x)² treats x and 1-x the same way, which is the pattern worth chasing (Look for a Pattern). To be sure the hunch is right I first run the identical problem with 2019 replaced by a small odd number I can compute by hand (Solve an Easier Related Problem). Once the symmetry is confirmed I name a general term with an index k (Introduce a Variable) and turn the symmetry into a statement about numerators (Convert to Algebra). The last move is to stop reading the sum left to right and instead re-group it into matched pairs (Organize Information in More Ways), which is what makes it collapse.

1STEP 1

Name the general term

Name the general term.

S = Σ_k=1²⁰¹⁸ (-1)^k+1 f (k/2019), f(x) = x²(1-x)²

2STEP 2

Run the same problem with 5

Run the same problem with a small number.

f (1/5) = 16/625, f (2/5) = 36/625, f (3/5) = 36/625, f (4/5) = 16/625

3STEP 3

Prove the mirror symmetry

Confirm the function is mirror symmetric.

f(1-x) = (1-x)²(1-(1-x))² = (1-x)²x² = f(x)

4STEP 4

Say the symmetry in numerators

Say the symmetry in terms of numerators.

f (k/2019) = f (1 - k/2019) = f ((2019-k)/2019)

5STEP 5

Check the signs of a mirror pair

Paired terms carry opposite signs.

k odd ⇔ 2019 - k even

6STEP 6

Re-group into matched pairs

Everything pairs off to 0.

S = [f (1/2019) - f (2018/2019)] + [f (3/2019) - f (2016/2019)] + … + [f (2017/2019) - f (2/2019)] = 0 + 0 + … + 0₁₀₀₉ pairs = 0

Answer

Two independent checks. First, the hand computation with 5 in place of 2019 gave (16 - 36 + 36 - 16)/625 = 0, and the same pairing argument works for any odd denominator, so 0 is not an accident of small numbers. Second, count the terms: 1009 plus signs and 1009 minus signs, and the pairing matches them one to one with equal values, so nothing is left over — an odd number of terms would have left a stray term and a nonzero answer. The other four choices are all positive, and each would require the pairing to fail somewhere; they look like the sizes of individual terms such as f(1/2019) = 2018²/2019⁴, which is exactly the trap for someone who cancels all but one term. Choice (A) is the only value consistent with a complete cancellation.

💡Key takeaway

When a formula reads the same forwards and backwards, pair the first term with the last instead of adding left to right — equal values with opposite signs wipe each other out.

Name the general term
Run the same problem with 5
Prove the mirror symmetry
Say the symmetry in numerators
Check the signs of a mirror pair
Re-group into matched pairs