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

• 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 square with side length 1, a finite number of segments parallel to the sides of the square were drawn, with a total length of 18 (they can intersect). Prove that among the parts into which the square is divided by the segments, there is a part with an area of at least 0.01. 
  A. Engenes, A. Brazins 

  Topics:

  Geometry -> Plane Geometry Geometry -> Area Calculation Algebra -> Inequalities Proof and Example -> Proof by Contradiction
  Sources:
  • Tournament of Towns, 1979-1980, Main, Spring Question 6

• Question

  Shuki walked for 3.5 hours. In each hour-long period, he traveled 5 km. Does it follow that Shuki's average speed during this entire time was 5 km per hour?

  Topics:

  Algebra -> Word Problems Algebra -> Inequalities -> Averages / Means

• Question

  Find all integer solutions `(k>1) y^k=x^2+x`

  Topics:

  Number Theory -> Prime Numbers -> Prime Factorization Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM) Algebra -> Equations -> Diophantine Equations
  Sources:
  • Tournament of Towns, 1980-1981, Spring, Main Version, Grades 9-10 Question 1 Points 3

• Continue the Sequence

  `2, 6, 12, 20, 30…`

  a. What is the next number (the sixth) in the sequence?

  b. What is the `2016`th number in the sequence?

  Topics:

  Arithmetic Logic -> Reasoning / Logic Algebra -> Sequences -> Arithmetic Progression / Arithmetic Sequence Algebra -> Sequences -> Complete/Continue the Sequence
  Sources:
  • problems.ru web site

• Question

  It is known that `a = b+1`. Is it possible that `a^4=b^4`?

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  Topics:

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

• Question

  The journey from home to school takes Yossi `20` minutes. One day, when Yossi was already on his way to school, he remembered that he had forgotten his pen at home. If he continues to school now, he will arrive `3` minutes before the bell. And if he goes back home to get the pen, he will be `7 ` minutes late for class. What fraction of the way to school had Yossi traveled when he remembered that he didn't have a pen?

  Note: Yossi walks at a constant speed the entire time.

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  Topics:

  Arithmetic -> Fractions Algebra -> Word Problems -> Motion Problems
  Sources:
  • problems.ru web site

• Question

  How can you divide `7` apples equally among `12` children, if you are not allowed to cut any apple into more than `5` pieces?

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  Topics:

  Algebra -> Word Problems Arithmetic -> Fractions Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Number Theory -> Division
  Sources:
  • problems.ru web site

• Question

  Can you find `35` integers whose average is equal to `6.35`?

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  Topics:

  Arithmetic Combinatorics -> Double Counting Algebra -> Inequalities -> Averages / Means Proof and Example -> Proof by Contradiction
  Sources:
  • problems.ru web site