Arithmetic

Arithmetic is the fundamental branch of mathematics dealing with numbers and the basic operations: addition, subtraction, multiplication, and division. Questions involve performing these operations, understanding number properties (like integers, fractions, decimals), and solving related word problems.

• 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?

  Show hint

  Log in here to receive the hint
  Topics:

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

• 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.

  Show hint

  Log in here to receive the hint
  Topics:

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

• Question

  Prove that any real number can be written as the sum of 9 numbers, each of which is composed only of the digits 0 and 7.

  Topics:

  Number Theory Arithmetic Proof and Example Algebra -> Sequences
  Sources:
  • Tournament of Towns, 1980-1981, Spring, Main Version, Grades 11-12 Question 3

• A mistake in the exercise

  Prove that there is an error in the following multiplication problem:

  \(\begin{array}& & & * & * & * & 2 & 7 \\ \times & & & & & * & * \\ \hline & * & * & * & * & * & 6 \\ + & * & * & * & * & * & \\ \hline & * & * & * & * & 4 & 6 \end{array}\)

  Topics:

  Arithmetic Logic -> Reasoning / Logic Proof and Example -> Proof by Contradiction Number Theory -> Division Puzzles and Rebuses -> Reconstruct the Exercise / Cryptarithmetic
  Sources:
  • Beno Arbel Olympiad, 2013, Grade 7 Question 4

• Question

  A box contains pencils in three colors: red, green, and blue, totaling `20` pencils. There are `6` times more blue pencils than green pencils. There are fewer red pencils than green pencils. How many red pencils are in the box?

  Topics:

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

• 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

• 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

  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