Mink Exercises and Additional Competition Materials, 2018-2019, Exercise 3

• Question 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

• Question 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

• Question 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

• Question 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

• Question 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

• Question 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