Olympiad problems

1978 IMO Shortlist, 16

Determine all the triples $(a, b, c)$ of positive real numbers such that the system \[ax + by -cz = 0,\]\[a \sqrt{1-x^2}+b \sqrt{1-y^2}-c \sqrt{1-z^2}=0,\] is compatible in the set of real numbers, and then find all its real solutions.

2015 India Regional MathematicaI Olympiad, 7

Tags: trigonometric substitution , trig identities , inequalities

Let $x,y,z$ be real numbers such that $x^2+y^2+z^2-2xyz=1$. Prove that \[ (1+x)(1+y)(1+z)\le 4+4xyz. \]

1974 IMO Longlists, 8

Tags: equation , algebra , trigonometric substitution , trigonometric identities , trigonometry

Let $x, y, z$ be real numbers each of whose absolute value is different from $\frac{1}{\sqrt 3}$ such that $x + y + z = xyz$. Prove that \[\frac{3x - x^3}{1-3x^2} + \frac{3y - y^3}{1-3y^2} + \frac{3z -z^3}{1-3z^2} = \frac{3x - x^3}{1-3x^2} \cdot \frac{3y - y^3}{1-3y^2} \cdot \frac{3z - z^3}{1-3z^2}\]

1978 IMO Longlists, 53

Tags: trigonometric substitution , system of equations , algebra , trigonometry

Determine all the triples $(a, b, c)$ of positive real numbers such that the system \[ax + by -cz = 0,\]\[a \sqrt{1-x^2}+b \sqrt{1-y^2}-c \sqrt{1-z^2}=0,\] is compatible in the set of real numbers, and then find all its real solutions.

1974 IMO Shortlist, 9

Tags: trigonometry , equation , trigonometric identities , trigonometric substitution , algebra

Let $x, y, z$ be real numbers each of whose absolute value is different from $\frac{1}{\sqrt 3}$ such that $x + y + z = xyz$. Prove that \[\frac{3x - x^3}{1-3x^2} + \frac{3y - y^3}{1-3y^2} + \frac{3z -z^3}{1-3z^2} = \frac{3x - x^3}{1-3x^2} \cdot \frac{3y - y^3}{1-3y^2} \cdot \frac{3z - z^3}{1-3z^2}\]