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

  Three hedgehogs have three pieces of cheese weighing `5`, `8`, and `11` grams. A fox offers to help the hedgehogs divide the cheese equally. The fox can bite off one gram from each of two cheese pieces of its choice. Can the fox, using these actions, reach a state where it leaves the three hedgehogs with equal pieces of cheese?

  Topics:

  Arithmetic Algebra -> Word Problems Proof and Example -> Constructing an Example / Counterexample Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Number Theory -> Division
  Sources:
  • problems.ru web site

• Question

  Can you divide `24` kilograms of nails into two parts of `15` and `9` kilograms using a balance scale without weights?

  Topics:

  Arithmetic Proof and Example -> Constructing an Example / Counterexample Algorithm Theory -> Weighing

• 5 = 6?

  Here is a "proof" that `5` is equal to `6`:

  `35+10-41=42+12-50 `

  `35+10-45=42+12-54 `

  `5*(7+2-9)=6*(7+2-9) `

  ׂ`5=6`

  Where is the mistake?

  Topics:

  Arithmetic Algebra -> Equations Number Theory -> Division

• Question

  There are two coins in a wallet, and their sum is `15` shekels. One of the coins is not a `5` shekel coin. What are the coins in the wallet?

  Topics:

  Arithmetic Algebra -> Word Problems Logic -> Reasoning / Logic

• Question

  a. The apple is heavier than the peach, and the peach is heavier than the pear. Which is heavier – the apple or the pear?

  b. The orange is heavier than the pear, and the plum is lighter than the orange. Which is heavier – the pear or the plum?

  Topics:

  Arithmetic Algebra -> Inequalities Logic -> Reasoning / Logic
  Sources:
  • problems.ru web site

• Question

  The sum of six distinct natural numbers is equal to `22`. Find these numbers.

  Topics:

  Number Theory Arithmetic
  Sources:
  • problems.ru web site

• Question

  Given that the number `A` is positive, the number `B` is negative, and the number `C` is equal to `0`. What is the sign of the number `A*B+A*C+B*C`?

  Topics:

  Arithmetic Algebra -> Inequalities

• Question

  A cube with a side length of one meter is cut into cubes with a side length of one centimeter. If all the resulting cubes are placed in a row, what will be the length of the row?

  Topics:

  Geometry -> Solid Geometry / Geometry in Space Arithmetic Algebra -> Word Problems Units of Measurement

• Question

  In a foreign country, every phone number has `7` digits, and the first digit is always different from `0`. What is the maximum number of phone numbers that can exist in that country?

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  Topics:

  Arithmetic Combinatorics -> Product Rule / Rule of Product
  Sources:
  • problems.ru web site

• Question

  We call a number nice if it consists only of odd digits. How many four-digit nice numbers exist?

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  Topics:

  Arithmetic Combinatorics -> Product Rule / Rule of Product
  Sources:
  • problems.ru web site