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

• Camels and Horses

  In a herd, there are dromedary camels (one-humped), Bactrian camels (two-humped), and horses. It is known that the total number of humps in this herd is `200`. In addition, the number of horses is equal to the number of Bactrian camels. How many animals are there in the herd in total?

  Topics:

  Algebra -> Word Problems Logic -> Reasoning / Logic

• The Clubs

  In the grade, there are `70` children. Of these, `27` go to theater club, `32` sing in the choir, and `22` do Judo. In the theater club, there are `10` children from the choir, in the choir there are `6` children from Judo, and in Judo there are `8` children who also study theater. Three children also go to theater club, Judo, and sing in the choir. How many children from the grade do not participate in any of these three clubs?

  Topics:

  Arithmetic Combinatorics Algebra -> Word Problems Set Theory

• Watering Potted Plants

  Hannah and Rachel watered potted plants at home. It is known that each of them watered exactly half the number of potted plants in the house, and it is also known that three potted plants were accidentally watered twice. How many potted plants remain to be watered?

  Topics:

  Algebra -> Word Problems Logic -> Reasoning / Logic Set Theory

• Question

  Calculate the sum:

  `1/(1*2)+1/(2*3)+1/(3*4)+...+1/(99*100)`

  Topics:

  Algebra -> Sequences Arithmetic -> Fractions Algebra -> Algebraic Techniques -> Telescoping Sums

• Question

  Calculate the product:

  `(1-1/2)(1-1/3)(1-1/4)*...*(1-1/100)`

  Topics:

  Algebra -> Sequences Arithmetic -> Fractions Algebra -> Algebraic Techniques -> Telescoping Sums

• Question

  Calculate the product:

  `(1-1/4)(1-1/9)(1-1/16)*...*(1-1/225)`

  Topics:

  Algebra -> Algebraic Techniques -> Short Multiplication Formulas / Algebraic Identities Algebra -> Sequences Arithmetic -> Fractions Algebra -> Algebraic Techniques -> Telescoping Sums

• Question

  It is known that `a+b-c=6`, `ac+bc-ab=2`. Find `a^2+b^2+c^2`.

  Topics:

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

• Question

  Given real numbers `a`, `b` and `c` distinct from `0` that satisfy: `a+b/c=b+c/a=c+a/b=1`. Prove that `ab+bc+ca=0`.

  Topics:

  Algebra -> Algebraic Techniques Algebra -> Equations

• Question

  Calculate the value of the expression (without a calculator):

  `(1+1/(2^2-1))(1+1/(3^2-1))(1+1/(4^2-1))*...*(1+1/(99^2-1))`

  Topics:

  Arithmetic Algebra -> Sequences Algebra -> Algebraic Techniques -> Telescoping Sums

• Question

  Calculate the value of the expression (without a calculator):

  `(2^3-1)/(2^3+1)*(3^3-1)/(3^3+1)*(4^3-1)/(4^3+1)*...*(100^3-1)/(100^3+1)`

  Topics:

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