Algebra, Equations

An equation is a statement that two mathematical expressions are equal. Solving an equation involves finding the values of variables that make the statement true. Questions cover various types: linear, quadratic, polynomial, rational, radical, and systems of 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

• Finite Division

  Find all integers x, y, z, w that satisfy `x^2+y^2=3z^2+3w^2`.

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Division -> Parity (Even/Odd) Proof and Example Algebra -> Equations -> Diophantine Equations

• Finite Division

  Find all integers x, y, z, w that satisfy `x^2+y^2=3z^2+3w^2 `.

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic Proof and Example -> Proof by Contradiction Algebra -> Equations -> Diophantine Equations
  Sources:
  • Junior Science Team Exercises, 2024-2025, Assignment 1 Question 5

• A CHARITABLE BEQUEST

  A man left instructions to his executors to distribute once a year exactly fifty-five shillings among the poor of his parish; but they were only to continue the gift so long as they could make it in different ways, always giving eighteenpence each to a number of women and half a crown each to men. During how many years could the charity be administered? Of course, by "different ways" is meant a different number of men and women every time.
  Topics:

  Arithmetic Algebra -> Word Problems Algebra -> Equations -> Diophantine Equations
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 6