Algebra

Algebra is a broad branch of mathematics that uses symbols (usually letters) to represent numbers and to state rules and relationships. It involves manipulating expressions, solving equations and inequalities, and studying functions and structures. Questions cover a wide range of these topics.

• Question

  There are `29` students in a class. The number of boys is `3` more than the number of girls. How many boys and how many girls are there in the class?

  Topics:

  Algebra -> Equations Algebra -> Word Problems

• Question

  In Danny's class there are `30` students. Danny claims that the number of boys is greater by `3` than the number of girls. Is it possible that Danny is right?

  Topics:

  Arithmetic Algebra -> Equations Algebra -> Word Problems Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd)

• Question

  On the circle, there are blue and red points. It is allowed to add a red point and change the colors of its neighboring points or remove a red point and change the colors of its neighboring points (it is not allowed to leave fewer than 2 points on the circle). Prove that it is impossible to move, using only these operations, from a circle with two red points to a circle with two blue points.
  K. Kaznvosky

  Topics:

  Combinatorics -> Combinatorial Geometry Combinatorics -> Invariants Algebra Logic -> Reasoning / Logic Number Theory -> Division -> Parity (Even/Odd) Set Theory Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Combinatorics -> Colorings -> Chessboard Coloring
  Sources:
  • Tournament of Towns, 1979-1980, Main, Spring Question 1

• Question

  On Alice's birthday, her friends are asked how old she is. The Mad Hatter says that Alice's age is greater than `11`, and the Cheshire Cat says that her age is greater than `10`. It is known that exactly one of them is wrong. How old is Alice now? Explain!

  Show hint

  Log in here to receive the hint
  Topics:

  Number Theory Algebra -> Inequalities Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures

• 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

• The broken chain

  There are five segments of a broken chain, each segment having three links. Moses wants to repair the chain. What is the minimum number of links he needs to open and close again to join all these segments together?

  Note: The chain is not circular!

  Topics:

  Algebra -> Word Problems Logic -> Reasoning / Logic Proof and Example -> Constructing an Example / Counterexample Minimum and Maximum Problems / Optimization Problems

• 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

  One-seventh of mathematicians are philosophers, and one-ninth of philosophers are mathematicians. Which are there more of – philosophers or mathematicians?

  Show hint

  Log in here to receive the hint
  Topics:

  Algebra -> Word Problems Logic -> Reasoning / Logic Arithmetic -> Fractions
  Sources:
  • problems.ru web site