# Mathematical Food for Thought

Serves a Daily Special and an All-You-Can-Eat Course in Problem Solving. Courtesy of me, Jeffrey Wang.

• ## Meta

Mission Accomplished! Topic: Algebra. Level: AIME. June 16th, 2006

Problem: (1991 India National Olympiad – #7) Solve the following system for real

.

Solution: Well, we simply go about solving this system the regular way – look for something to eliminate. We try using the first and second equations and find

,

which we subtract from the second equation to get

.

Using the third equation, we have so we substitute, multiply through by (it can’t be zero), and divide by to get

.

So . But from the second equation, we see that gives but they are real so this is impossible. Hence . It remains to solve for and using

.

Substitute into the first one and again multiply through by to find

.

We then have with corresponding . Checking, we see that both solutions work, so our final solution set is

; .

QED.

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Comment: Not bad for an olympiad question – just requires basic algebraic manipulation and solving quadratics. Looking for the right things to square and substitute made this problem considerably easier.

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Practice Problem: (1994 India National Olympiad – #2) If , prove that .

Posted in AIME, Algebra || 1 Comment »

### One Response to “Mission Accomplished! Topic: Algebra. Level: AIME.”

1. QC Says:

We wish to show

x^6 – 2x^5 + 2x^3 – 2x + 1 \ge 0

This factors as

(x-1)^2 (x^4 – x^2 + 1) \ge (x-1)^2 (x^4 – 2x^2 + 1) \ge (x-1)^2 (x^2 – 1)^2 \ge 0

QED.

(India MO problems are a little on the easy side, as far as MO problems go…)