Two Equations, One Genius Move — And a Perfect Square Appears | Math Olympiad

Описание: Math Olympiad Challenge — A Symmetric System With a Stunning Shortcut!

x² + xy = 12
y² + xy = 13
Find ALL values of (x, y)

Most students stare at these two equations
and immediately try substitution — isolating
x or y and plugging into the other equation.

That path leads to a messy fourth-degree
polynomial with no clean solution in sight.

But competition mathematicians do something
completely different.

They ADD.

x² + 2xy + y² = 25
(x+y)² = 25
x+y = ±5

That one observation — spotting the perfect
square hiding inside the sum of both equations —
unlocks the entire problem in seconds.


Now with x+y already known, x-y follows
immediately. And from those two linear equations
both x and y emerge as clean fractions.

Two solution pairs. Both elegant. Both perfect.
(12/5, 13/5) and (-12/5, -13/5).

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🧠 What You'll Learn:
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✔️ Why adding both equations reveals (x+y)² = 25
✔️ How to spot the perfect square algebraic identity
✔️ Why x+y = ±5 gives two symmetric cases to explore
✔️ Why knowing x+y immediately unlocks y-x = ±1/5
✔️ How two linear equations give x and y as clean fractions
✔️ Why BOTH solution pairs (12/5, 13/5) and (-12/5, -13/5)
satisfy both original equations perfectly
✔️ A symmetric system strategy reusable across dozens of Olympiad problems


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📌 Try It Yourself First!
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Pause before the solution — can you
spot what happens when you simply
add both equations together?
That one step changes everything!
Drop your answers in the comments below!
Did you find BOTH solution pairs? 👇

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