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

  A grasshopper can jump `80` centimeters forward or `50` centimeters backward. Can the grasshopper move away from its starting point in fewer than `7` jumps to a distance of exactly one meter and `70` cm?

  Show hint

  Log in here to receive the hint
  Topics:

  Arithmetic Number Theory -> Chinese Remainder Theorem Algebra -> Word Problems Proof and Example -> Constructing an Example / Counterexample Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Algebra -> Equations -> Diophantine Equations

• Birds and Seeds

  Nine identical birds eat less than `1001` seeds for lunch, and ten such birds eat more than `1100` seeds for lunch. How many seeds does one bird eat for lunch?

  Topics:

  Algebra -> Inequalities Algebra -> Word Problems Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Number Theory -> Division

• Question

  Find all natural numbers with the following property: when divided by 7, their remainder is equal to their quotient.

  Topics:

  Number Theory Algebra -> Inequalities Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Arithmetic -> Division with Remainder

• Question

  Find all two-digit numbers `A` such that the square of the sum of its digits is equal to the sum of the digits of `A^2`.

  Topics:

  Number Theory Arithmetic Algebra -> Inequalities Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures

• Question

  Can the product of two consecutive natural numbers be equal to the product of two consecutive even numbers?

  Topics:

  Number Theory Algebra -> Inequalities Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Proof and Example -> Proof by Contradiction

• Question

  `10` identical books cost more than `11` dollars, and `9` books of the same type cost less than `10` dollars. How much does one book cost?

  Topics:

  Algebra -> Inequalities Algebra -> Word Problems Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures

• Question

  Given three real numbers `a`, `b`, and `c`. It is known that `a+b+c>0`, `ab+bc+ca>0`, and `abc>0`. Prove that `a>0`, `b>0`, and `c>0`.

  Topics:

  Algebra -> Inequalities Proof and Example -> Proof by Contradiction

• Question

  A `29×29` table contains all integers from `1` to `29`, each appearing exactly `29` times. The sum of all numbers above the main diagonal is exactly three times greater than the sum of all numbers below the main diagonal. What number is written in the central cell of the table?

  Topics:

  Arithmetic Logic -> Reasoning / Logic Algebra -> Sequences Algebra -> Inequalities -> Averages / Means Combinatorics -> Number Tables Minimum and Maximum Problems / Optimization Problems

• Bacteria in a Test Tube

  A scientist has a test tube containing bacteria. Every second, each bacterium splits into two. After two hours, the test tube was completely full of bacteria. How long before that was the test tube exactly half full?

  Topics:

  Arithmetic Logic -> Reasoning / Logic Algebra -> Sequences Algebra -> Word Problems -> Solving Word Problems "From the End" / Working Backwards

• Question

  Find the sum of all natural numbers from `1` to `100`.

  Topics:

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