Algebra, Equations, Diophantine Equations

Diophantine equations are polynomial equations, usually with integer coefficients, for which only integer solutions are sought. Questions involve finding these integer solutions, determining if solutions exist, or finding the number of solutions (e.g., linear Diophantine equations `ax+by=c`).

• Question

  Find all integer solutions `(k>1) y^k=x^2+x`

  Topics:

  Number Theory -> Prime Numbers -> Prime Factorization Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM) Algebra -> Equations -> Diophantine Equations
  Sources:
  • Tournament of Towns, 1980-1981, Spring, Main Version, Grades 9-10 Question 1 Points 3

• Question

  Find a two-digit number that is twice as large as the product of its digits.

  Topics:

  Number Theory Arithmetic Algebra -> Algebraic Techniques Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Algebra -> Equations -> Diophantine Equations
  Sources:
  • problems.ru web site

• Question

  There are chairs with `4` legs and with `3` legs in a room. When people sat on all the chairs, there were `39` legs in the room (no one remained standing). How many chairs of each type are there in the room?

  Topics:

  Algebra -> Word Problems Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM) -> Euclidean Algorithm Algebra -> Equations -> Diophantine Equations

• Question

  How many solutions in natural numbers are there to the equation `(2013 - x)(2013-y)=2013^2`?

  Topics:

  Number Theory -> Prime Numbers -> Prime Factorization Algebra -> Equations -> Diophantine Equations
  Sources:
  • Beno Arbel Olympiad, 2013, Grade 7 Question 6

• Question

  A grasshopper can jump `80` centimeters forward or `50` centimeters backward. Can the grasshopper move away from its starting point in fewer than `7` jumps to a distance of exactly one meter and `70` cm?

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  Topics:

  Arithmetic Number Theory -> Chinese Remainder Theorem Algebra -> Word Problems Proof and Example -> Constructing an Example / Counterexample Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Algebra -> Equations -> Diophantine Equations

• Question

  In the magical land, there are only two types of coins: `16` LC (Magical Pounds) and `27` LC. Is it possible to buy a notebook that costs one Magical Pound and receive exact change?

  Topics:

  Proof and Example -> Constructing an Example / Counterexample Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM) -> Euclidean Algorithm Number Theory -> Division Algebra -> Equations -> Diophantine Equations

• Question

  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

  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

• How many solutions does the equation have?

  Given the equation:
  `x^2+2xy+y^2-200x-200y+1900=0`
  How many solutions (x,y) are there, where x and y
  are integers from 1 to 100 (inclusive)?

  Topics:

  Algebra -> Algebraic Techniques Algebra -> Equations -> Diophantine Equations
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grade 9 Question 5

• Mathematical Conference

  202 participants from three countries attended a mathematical conference: Israel, Greece, and Japan.
  On the first day, every pair of participants from the same country shook hands. On the second day, every pair of participants
  where one was Israeli and the other was not Israeli shook hands. On the third day, every pair of participants where one
  was Israeli and the other was Greek shook hands. In total, 20200 handshakes occurred. How many
  Israeli participants were at the conference?

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  Topics:

  Number Theory Combinatorics Algebra -> Word Problems Algebra -> Equations -> Diophantine Equations
  Sources:
  • Gillis Mathematical Olympiad, 2019-2020 Question 2