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.

Algebraic Techniques Equations Inequalities Word Problems Sequences

• Apples and Pears

  There is a basket containing `30` fruits. It is known that among any `12` fruits we take from the basket, there is necessarily at least one apple, and among any `20` fruits there is necessarily one pear. How many apples and how many pears are there in the basket?

  Topics:

  Combinatorics -> Pigeonhole Principle Algebra -> Inequalities Logic -> Reasoning / Logic
  Sources:
  • problems.ru web site

• Corruption in Parliament

  The parliament of the magical country consists of `20` people. It is known that among the `20` members of parliament, there is at least one who is not corrupt. Additionally, it is known that for any two members of parliament we choose, one is necessarily corrupt. How many members of parliament of the magical country are corrupt?

  Topics:

  Algebra -> Inequalities Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
  Sources:
  • problems.ru web site

• Clock Tower

  A bell in a clock tower rings `3` times in `12` seconds. How long does it take to ring `6` times?

  Topics:

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

• Shlomi's Schedule

  Shlomi dedicates `1/3` of his time to playing games on the tablet, `1/5` of his time to studying at school, `1/6` to watching movies, `1/70` to solving mathematical problems, and `1/3` to sleeping.

  Is it possible to live like that?

  Topics:

  Algebra -> Inequalities Arithmetic -> Fractions
  Sources:
  • problems.ru web site

• Question

  There are three sacks of nuts. The first sack has `6` kilograms of nuts less than the other two sacks combined. The second sack has `10` kilograms of nuts less than the other two sacks combined.

  How many kilograms of nuts are in the third sack?

  Topics:

  Algebra -> Equations Algebra -> Word Problems
  Sources:
  • problems.ru web site

• Question

  Which number is larger:

  `2^300` or `3^200`?

  Topics:

  Arithmetic Algebra -> Inequalities

• The Restless Fly

  Cities A and B are 300 kilometers apart. Two cyclists start simultaneously from A and B, heading towards each other. Their speeds are constant and equal to `30` km/h and `20` km/h, respectively. At the same moment, a fly departs from city A, flying at a speed of `100` km/h. The fly overtakes the first cyclist and flies until it meets the cyclist who left from city B. The moment the fly meets the cyclist, it immediately turns around and flies back until it meets the first cyclist again, and then it turns around again, and so on, until the cyclists meet. How many kilometers did the fly travel?

  Topics:

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

• The Cookies

  Avi, Benny, and Danny's mother left a plate of cookies on the table in the morning. Avi woke up, ate one-third of the total number of cookies, and went to school. After that, Benny woke up, unaware that Avi had already eaten some, so he ate one-third of the remaining cookies and also went to school. After that, Danny woke up and ate one-third of the cookies that were left. When their mother returned home, she saw that there were `8` cookies left. How many cookies were there at the beginning?

  Topics:

  Arithmetic -> Fractions Algebra -> Word Problems -> Solving Word Problems "From the End" / Working Backwards
  Sources:
  • problems.ru web site

• What is more worthwhile?

  What is more worthwhile – to receive a discount of `20%`, or to buy a gift voucher worth `120%` of its price?

  Topics:

  Algebra -> Word Problems Arithmetic -> Percentages

• Successive Discounts

  What is more worthwhile – to receive a `15%` discount and then a `5%` discount, or to first receive a `5%` discount and then a `15%` discount?

  Topics:

  Algebra Arithmetic -> Percentages