By Titu Andreescu, Razvan Gelca

1000s of gorgeous, hard, and instructive difficulties from algebra, geometry, trigonometry, combinatorics, and quantity conception historic insights and asides are provided to stimulate extra inquiry Emphasis is on artistic recommendations to open-ended difficulties Many examples, difficulties and strategies, with a undemanding and available variety more desirable motivatio References

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**Sample text**

And { } denote respectively the greatest integer and fractional part func- 12. For a given complex number a, find the complex solutions to the system (x1 + x2 + x3 )x4 = a, (x1 + x2 + x4 )x3 = a, (x1 + x3 + x4 )x2 = a, (x2 + x3 + x4 )x1 = a. 13. Find all real numbers a for which there exist nonnegative real numbers x1 , x2 , x3 , x4 , x5 satisfying the system 5 ∑ kxk = a, k=1 5 ∑ k 3 xk = a 2 , k=1 5 ∑ k 5 xk = a 3 . k=1 14. Solve the system of equations x3 − 9(y2 − 3y + 3) = 0, y3 − 9(z2 − 3z + 3) = 0, z3 − 9(x2 − 3x + 3) = 0.

Solve the system x2 − |x| = |yz|, y2 − |y| = |zx|, z2 − |z| = |xy|. Chapter 2. Algebra and Analysis 54 11. 3, where tions. and { } denote respectively the greatest integer and fractional part func- 12. For a given complex number a, find the complex solutions to the system (x1 + x2 + x3 )x4 = a, (x1 + x2 + x4 )x3 = a, (x1 + x3 + x4 )x2 = a, (x2 + x3 + x4 )x1 = a. 13. Find all real numbers a for which there exist nonnegative real numbers x1 , x2 , x3 , x4 , x5 satisfying the system 5 ∑ kxk = a, k=1 5 ∑ k 3 xk = a 2 , k=1 5 ∑ k 5 xk = a 3 .

The restriction on the size of the triangle implies that there is a point P at distance a from B and C. Construct P as one of the intersections of the circles centered at B and C with 26 Chapter 1. Geometry and Trigonometry − → radii a. 2). 2 If AB has arbitrary length, construct Q such that QA = QB = a. Then translate Q to − → −→ R by the vector BC and finally translate A to D by the vector QR. Here are more problems of this kind. 1. Given a polygon in the plane and M a point in its interior, construct two points A and B on the sides of the polygon such that M is the midpoint of the segment AB.