Problem 24 · 2013 Math Kangaroo
Stretch
Algebra & Patterns
substitution
How many solutions \((x, y)\) with real x and y does the equation \(x^{2} + y^{2} = |x| + |y|\) have?
Show answer
Answer: E — infinitely many
Show hints
Hint 1 of 2
Let u=|x|, v=|y|; the equation becomes a relation between two non-negative numbers.
Still stuck? Show hint 2 →
Hint 2 of 2
Completing the square shows it traces a whole curve, not isolated points.
Show solution
Approach: reduce to a curve
- With u=|x|, v=|y|: u²+v² = u+v is a circle (u−½)²+(v−½)² = ½.
- Every point of this arc with u,v≥0 yields real (x,y), forming continuous curves.
- So there are infinitely many solutions: E.
Mark:
· log in to save