Arithmetic

Arithmetic is the fundamental branch of mathematics dealing with numbers and the basic operations: addition, subtraction, multiplication, and division. Questions involve performing these operations, understanding number properties (like integers, fractions, decimals), and solving related word problems.

• Question

  Avi invited guests. They know he lives in apartment `333`, in entrance number `10`, but they don't know which floor. There are `9` floors in the building, and it is known that there are an equal number of apartments on each floor. The guests want to use the elevator. Can they calculate in advance which floor they need to press in the elevator?

  Topics:

  Algebra -> Inequalities Algebra -> Word Problems Arithmetic -> Division with Remainder

• Question

  Given two numbers such that one is obtained from the other by changing the order of its digits. Prove that their difference is divisible by `9`.

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Proof and Example Arithmetic -> Division with Remainder

• 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

  Show hint

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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

• Log of Wood

  You have a very long log of wood. Can you measure exactly one meter from it, if you have for this purpose:
  а. A stick with a length of one and a half meters and another stick with a length of 40 centimeters,
  б. A stick with a length of one and a half meters and another stick with a length of 30 centimeters,

  Assuming you have no other measuring tools? Explain!

  Topics:

  Combinatorics -> Invariants Algebra -> Word Problems Logic -> Reasoning / Logic Arithmetic -> Division with Remainder
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Final, Grades 3-4 Question 7
  • Young Mathematician Olympiad, 2017-2018, Final, Grades 5-6 Question 2

• Consecutive Numbers and Deletion

  Yossi wrote 10 consecutive natural numbers on the board. Danny erased one of the numbers. The sum of the remaining numbers on the board is 2020.
  Which number was erased?

  Topics:

  Algebra -> Word Problems Algebra -> Sequences Arithmetic -> Division with Remainder
  Sources:
  • Young Mathematician Olympiad, 2020-2021, Stage A, Grades 7-8 Question 5

• THE MILLIONAIRE'S PERPLEXITY

  Mr. Morgan G. Bloomgarten, the millionaire, known in the States as the Clam King, had, for his sins, more money than he knew what to do with. It bored him. So he determined to persecute some of his poor but happy friends with it. They had never done him any harm, but he resolved to inoculate them with the "source of all evil." He therefore proposed to distribute a million dollars among them and watch them go rapidly to the bad. But he was a man of strange fancies and superstitions, and it was an inviolable rule with him never to make a gift that was not either one dollar or some power of seven—such as `7, 49, 343, 2,401`, which numbers of dollars are produced by simply multiplying sevens together. Another rule of his was that he would never give more than six persons exactly the same sum. Now, how was he to distribute the `1,000,000` dollars? You may distribute the money among as many people as you like, under the conditions given.
  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic Arithmetic -> Division with Remainder
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 16

• THE CONVERTED MISER

  Mr. Jasper Bullyon was one of the very few misers who have ever been converted to a sense of their duty towards their less fortunate fellow-men. One eventful night he counted out his accumulated wealth, and resolved to distribute it amongst the deserving poor.

  He found that if he gave away the same number of pounds every day in the year, he could exactly spread it over a twelvemonth without there being anything left over; but if he rested on the Sundays, and only gave away a fixed number of pounds every weekday, there would be one sovereign left over on New Year's Eve. Now, putting it at the lowest possible, what was the exact number of pounds that he had to distribute?

  Could any question be simpler? A sum of pounds divided by one number of days leaves no remainder, but divided by another number of days leaves a sovereign over. That is all; and yet, when you come to tackle this little question, you will be surprised that it can become so puzzling.

  Topics:

  Algebra -> Word Problems Arithmetic -> Division with Remainder
  Sources:
  • Amusements in Mathematics, Henry Ernest Dudeney Question 116