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

  At `12:00`, the two hands of a clock coincide. In exactly how much time will they coincide again? How many times within a 24-hour period will they coincide?

  Topics:

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

• Question

  Prove that the difference of squares of two consecutive odd numbers is divisible by `8`.

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Algebra -> Algebraic Techniques -> Short Multiplication Formulas / Algebraic Identities
  Sources:
  • problems.ru web site

• Question

  How long will it take a train traveling at a speed of `60` km/h and with a length of `50` meters to pass through a tunnel with a length of `50` meters?

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  Topics:

  Arithmetic Logic -> Reasoning / Logic Algebra -> Word Problems -> Motion Problems Units of Measurement

• Question

  Prove that for every prime number `p>3 ` the following holds: `p^2-1` is divisible by `6`.

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Algebra -> Algebraic Techniques -> Short Multiplication Formulas / Algebraic Identities Number Theory -> Division -> Parity (Even/Odd) Proof and Example Number Theory -> Prime Numbers -> Prime Factorization

• Question

  Find a two-digit number that is twice as large as the product of its digits.

  Topics:

  Number Theory Arithmetic Algebra -> Algebraic Techniques Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Algebra -> Equations -> Diophantine Equations
  Sources:
  • problems.ru web site

• Question

  Nina and Meir left their home at the same time and went to visit their grandmother. The route from their home to their grandmother's house is 3 km long, and benches are located along it.

  Nina sits on every bench along the way to eat one cookie. She eats each cookie for the same whole number of minutes. Meir also stops and sits on every bench to eat one cookie.

  It takes Meir twice as long as Nina to eat a cookie. It is also known that Nina walks at a speed of 3 km/h, and Meir at a speed of 4 km/h.

  It turned out that Meir and Nina arrived at their grandmother's house at the same time.

  How many benches were along the way? Find all the possibilities and justify your answer.

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  Topics:

  Arithmetic Algebra -> Equations Algebra -> Word Problems -> Motion Problems
  Sources:
  • Gillis Mathematical Olympiad, 2015-2016 Question 1

• Question

  Over the course of a year, the price of apples increased twice by `50%` each time, and before Rosh Hashanah, they were discounted by `50%`. What is the new price of a kilogram of apples if, at the beginning of the year, they cost `8` shekels per kilogram?

  Topics:

  Algebra -> Word Problems Arithmetic -> Percentages

• Question

  Given the polynomial `P(n)=n^2+n+41`. Is it true that this polynomial yields prime numbers for all natural numbers `n`?

  Topics:

  Number Theory -> Prime Numbers Algebra Proof and Example -> Constructing an Example / Counterexample
  Sources:
  • problems.ru web site

• Question

  Given a natural number `A`. When it is increased by `1`, its square increases by `1001`. Find `A`.

  Topics:

  Algebra -> Algebraic Techniques -> Short Multiplication Formulas / Algebraic Identities Algebra -> Equations

• Question

  There are chairs with `4` legs and with `3` legs in a room. When people sat on all the chairs, there were `39` legs in the room (no one remained standing). How many chairs of each type are there in the room?

  Topics:

  Algebra -> Word Problems Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM) -> Euclidean Algorithm Algebra -> Equations -> Diophantine Equations