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.

• Shlomi's Schedule

  Shlomi dedicates `1/3` of his time to playing games on the tablet, `1/5` of his time to studying at school, `1/6` to watching movies, `1/70` to solving mathematical problems, and `1/3` to sleeping.

  Is it possible to live like that?

  Topics:

  Algebra -> Inequalities Arithmetic -> Fractions
  Sources:
  • problems.ru web site

• The Cookies

  Avi, Benny, and Danny's mother left a plate of cookies on the table in the morning. Avi woke up, ate one-third of the total number of cookies, and went to school. After that, Benny woke up, unaware that Avi had already eaten some, so he ate one-third of the remaining cookies and also went to school. After that, Danny woke up and ate one-third of the cookies that were left. When their mother returned home, she saw that there were `8` cookies left. How many cookies were there at the beginning?

  Topics:

  Arithmetic -> Fractions Algebra -> Word Problems -> Solving Word Problems "From the End" / Working Backwards
  Sources:
  • problems.ru web site

• 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

• An Ancient Russian Riddle

  Two woodcutters, Ivan and Prokhor, were working together in the forest. They settled down to eat lunch. Ivan had `4` pitas, and Prokhor had `8`. At that moment, a hunter passed by them and asked to join the meal, so they divided the `12` pitas equally among the three of them. After the meal, the hunter really wanted to thank the woodcutters and gave them `15` kopecks. After he left, Ivan and Prokhor began to discuss how to divide the money.

  Prokhor suggested simply dividing it equally. But Ivan argued that this would not be fair, because he had twice as few pitas, therefore, in his opinion, Prokhor should receive `10` kopecks and Ivan himself - `5`. How do you think the money should be divided fairly?

  Topics:

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

• Camel Division (Ancient Question)

  An old Arab merchant had three sons. He bequeathed them 17 camels, and in his will, he requested that the eldest son receive half of the camels, the middle son receive a third, and the youngest a ninth. The sons could not divide the camels among themselves as stated in the will without slaughtering some of the camels – and they did not want to do that. So they turned to the Qadi for help.

  The Qadi added one of his own camels to the 17 camels, and divided the 18 camels as follows: the eldest son received 9 camels, which is half of the amount, the middle son received 6 camels, which is a third of the amount, and the youngest son received 2 camels, which is a ninth of the amount, for a total of 17 camels divided, and the extra camel was returned to the Qadi.

  The brothers were amazed by the wisdom of the Qadi and began to think: how did it happen that each received even more than he was supposed to receive according to the will?

   

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  Topics:

  Algebra -> Word Problems Arithmetic -> Fractions Logic -> Reasoning / Logic -> Paradoxes

• Question

  Given the sequence `1 , 1/2 ,1/3 ,1/4 ,1/5,...`, does there exist an arithmetic sequence composed of terms from the aforementioned sequence?

  1. Of length 5

  2. Of any length

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  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Proof and Example -> Constructing an Example / Counterexample Algebra -> Sequences -> Arithmetic Progression / Arithmetic Sequence Arithmetic -> Fractions Number Theory -> Greatest Common Divisor (GCD) and Least Common Multiple (LCM)
  Sources:
  • Tournament of Towns, 1981-1982, Spring

• Question

  Consider all the numbers from 1 to 4242. Find the difference between the number of odd numbers divisible by 3
  and the number of numbers divisible by 7 in this range.

  Topics:

  Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Arithmetic -> Fractions Number Theory -> Division
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Stage A, Grades 5-6 Question 5

• Long Division

  Calculate the value of the expression and write it as a decimal fraction:

  `1/(1+1/(1+1/(1+1/(1+1/(1))))`

  Topics:

  Arithmetic -> Fractions Number Theory -> Division
  Sources:
  • Young Mathematician Olympiad, 2017-2018, Stage A, Grades 5-6 Question 7