Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 1

• Question

  Positive numbers a, b, c, d satisfy `a^3 + b^3 +c ^3 + d^3 >= 3` and also `a^5 + b^5 +c ^5 + d^5 <= 5`

  Prove that `a + b +c + d >= 3 / 2`

   

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  Topics:

  Algebra -> Algebraic Techniques Algebra -> Inequalities -> Averages / Means

• Question

  Let the sides of a triangle be a, b, c and the lengths of the corresponding medians be `m_a , m_b, m_c`. Show that

  `sum_{cyc} m_a / a >= {3( m_a + m_b + m_c)} /{a + b + c}`

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  Topics:

  Geometry -> Plane Geometry -> Triangles Algebra -> Inequalities Geometry -> Plane Geometry -> Triangle Inequality

• Question

  For all `a,b,c >=1 ` that satisfy `a+b+c= 2abc `

  Prove that `root (3) ((a+b+c)^2) >= sum_{cyc} root (3) (ab-1) `

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  Topics:

  Algebra -> Inequalities -> Averages / Means