Mink Exercises and Additional Competition Materials

• Question from sources: 2018-2019, Exercise 1

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

• Question from sources: 2018-2019, Exercise 1

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

• Question from sources: 2018-2019, Exercise 1

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

• Question from sources: 2018-2019, Exercise 3(1)

  The numbers a, b, c are different from 0, and the numbers ab, ac, bc are rational.

  a. Show that `a^2+b^2+c^2` is a rational number.

  b. Show that if `a^3+b^3+c^3` is a rational number, then `a+b+c` is a rational number.

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  Topics:

  Number Theory Algebra
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 3 Question 1

• Question from sources: 2018-2019, Exercise 3(2) - 50 to the Power of

  Show that in the rightmost 504 digits of `1+50+50^2+...+50^1000`

  Each digit appears a number of times divisible by 12

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Algebra -> Sequences -> Arithmetic Progression / Arithmetic Sequence Algebra -> Algebraic Techniques -> Telescoping Sums
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 3 Question 2

• Question from sources: 2018-2019, Exercise 3(3)

  Given distinct rational numbers a, b, c, prove that `sqrt{1/(a-b)^2+1/(b-c)^2 +1 /(c-a)^2}`

  is rational.

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  Topics:

  Number Theory Algebra -> Algebraic Techniques -> Roots / Radicals
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 3 Question 3

• Question from sources: 2018-2019, Exercise 3(4)

  Given natural numbers m, n such that `m/n <= sqrt 23`, prove that `m/n+3/{mn} <= sqrt 23`

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic Algebra -> Inequalities Proof and Example -> Proof by Contradiction Algebra -> Algebraic Techniques -> Roots / Radicals
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 3 Question 4

• Question from sources: 2018-2019, Exercise 3(5)

  Given natural numbers n, a, b such that `3n+1=a^2` and `4n+1=b^2`, prove that:

  a. n is divisible by 8 (easier)

  b. n is divisible by 56

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Algebra -> Equations -> Diophantine Equations
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 3 Question 5

• Question from sources: 2018-2019, Exercise 3(6)

  Is there a solution in natural numbers to the equation `x^2 + 12 = y^3` such that

  a. x is even (easier)

  b. x is odd

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  Topics:

  Number Theory -> Prime Numbers Number Theory -> Modular Arithmetic / Remainder Arithmetic Number Theory -> Division -> Parity (Even/Odd) Proof and Example -> Proof by Contradiction Arithmetic -> Division with Remainder Algebra -> Equations -> Diophantine Equations
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 3 Question 6

• Question from sources: 2018-2019, Exercise 4

  The plane is divided by n lines and circles.

  Prove that the resulting map can be colored with two colors such that any two adjacent regions (separated by a segment or an arc) are colored with different colors.

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  Topics:

  Combinatorics -> Combinatorial Geometry Combinatorics -> Induction (Mathematical Induction) Geometry -> Plane Geometry Proof and Example Combinatorics -> Colorings
  Sources:
  • Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 4