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

  Simplify the expression and find its value:

  `1987/(198719871987^2-198719871988*198719871986)`

  Topics:

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

• Question

  Given a real number `a` such that `a+1/a` is an integer. Prove that `a^2+1/a^2` is also an integer.

  Topics:

  Algebra -> Algebraic Techniques Algebra -> Equations Proof and Example

• Question

  Given a real number `a` such that `a+1/a` is an integer. Prove that `a^n+1/a^n` is also an integer for every natural number `n`.

  Topics:

  Number Theory Combinatorics -> Induction (Mathematical Induction) Algebra -> Algebraic Techniques Proof and Example Algebra -> Sequences

• Question

  Simplify the expression:

  `1/(1-a)+1/(1+a)+2/(1+a^2)+4/(1+a^4)+8/(1+a^8)+16/(1+a^16)`

  Topics:

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

• Question

  Which number is larger: `sqrt(2016+2017)` or `sqrt(2016)+sqrt(2017)`?

  Topics:

  Algebra -> Algebraic Techniques -> Short Multiplication Formulas / Algebraic Identities Algebra -> Inequalities Algebra -> Algebraic Techniques -> Roots / Radicals

• Question

  This year, Yossi participated in the Mathematics Olympiad for the first time. Before the Olympiad, a survey was conducted in which each participant was asked what place they thought they would achieve. Yossi answered that he would probably be in last place. After the competition itself, it turned out that every participant, except for Yossi, achieved a worse place than they answered in the survey. What place did Yossi take in the Olympiad?

  Topics:

  Algebra -> Word Problems Logic -> Reasoning / Logic

• Question

  A two-digit number is written on the board. Avi claims that the units digit of the number is twice the tens digit. Beni claims that the number is divisible by `9`. Gal claims that the number is divisible by `4`. Dani claims that the number is divisible by `27`. It is known that one of them is wrong, and the rest are correct. What number is written on the board?

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 2, 4, and 8 Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Algebra -> Word Problems Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures

• Question

  The following numbers are written on the board: `1, 2, 3, …, 2016, 2017`. In one move, it is allowed to choose a pair of numbers written on the board, erase them, and write their (positive) difference in their place. After several such operations, a single number remains on the board. Is it possible that this is zero?

  Topics:

  Arithmetic Combinatorics -> Invariants Combinatorics -> Induction (Mathematical Induction) Number Theory -> Division -> Parity (Even/Odd) Algebra -> Sequences -> Arithmetic Progression / Arithmetic Sequence Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Proof and Example -> Proof by Contradiction

• Question

  Does there exist an infinite arithmetic progression consisting only of prime numbers?

  Note: We do not consider "trivial" arithmetic progressions, which are constant.

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  Topics:

  Number Theory -> Prime Numbers Algebra -> Sequences -> Arithmetic Progression / Arithmetic Sequence Proof and Example -> Proof by Contradiction

• Question

  By how much is the sum of all even numbers not exceeding `100` greater than the sum of all odd numbers not exceeding `100`?

  Topics:

  Arithmetic Combinatorics -> Double Counting Algebra -> Algebraic Techniques Logic -> Reasoning / Logic Algebra -> Sequences -> Arithmetic Progression / Arithmetic Sequence