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.
Fractions Percentages Division with Remainder-
Numbers Squared
Ayala took all the numbers divisible by 3 from 1 to 99, inclusive, squared each of them, summed the results, and multiplied the resulting number by 2.
Sources:
The duck took all the numbers not divisible by 3 from 1 to 100, inclusive, squared each of them, and summed the results.
What is the difference between the number the duck obtained and the number Ayala obtained? -
The 1224th Digit
We write the natural numbers in order, one after the other from left to right:
1234567891011...
Note, for example, that the digit in the 10th place is 1 and the digit in the 11th place is 0.
Continuing with this writing as much as needed...
Which digit will be in the 1224th place in the sequence?
Sources: -
The Number with the Most Divisors
Among the positive integers less than 1000, which number has the most divisors?
Sources: -
How a Wheel Turns
All 6 wheels in the diagram rotate as they touch each other without slipping. The diameter of the leftmost wheel is 15.7 cm and it makes 12 revolutions per minute.
It is known that the smallest wheel makes one revolution per second.
What is the diameter of the smallest wheel?
Sources:
-
The Disguised Digits
In the following division exercise, (almost) all the digits are disguised!
What is the dividend?
\(\begin{align*} &\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }**\text{ }8**\\ &\overline{* * * * * * *\text{ } *}| * *\text{ } *\\ &\underline{\text{ }\text{ }* *\text{ } *}\\ &\text{ }\text{ }\text{ }\text{ }\text{ }* * * *\\ &\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\underline{\text{ }\text{ }* *\text{ } *}\\ &\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }* * * *\\ &\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\text{ }\underline{* * *\text{ } *}\\ \end{align*}\)
Sources: -
Question
Eighth-grade students threw rubber balls into a box and then tried to guess how many balls had accumulated there. Five students tried to guess: 45, 41, 55, 50, 43, but no one guessed the exact amount. The guesses differed from the truth by 3, 7, 5, 7, and 2 balls (not necessarily in the same order as the guesses). How many balls were in the box?
Sources:Topics:Number Theory Arithmetic Algebra -> Word Problems Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures- Beno Arbel Olympiad, 2017, Grade 8 Question 1
-
Question
The numbers from 1 to `10^9` (inclusive) are written on the board. The numbers divisible by 3 are written in red, and the rest of the numbers are written in blue. The sum of all the red numbers is equal to `X`, and the sum of all the blue numbers is equal to `Y`. Which number is larger, `2X` or `Y`, and by how much?
Sources:Topics:Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Logic -> Reasoning / Logic Algebra -> Sequences Algebra -> Inequalities -> Averages / Means Number Theory -> Division- Beno Arbel Olympiad, 2017, Grade 8 Question 2
-
Question
In this letter exercise, identical letters represent identical digits, different letters represent different digits, and asterisks represent any digit. Find all the digits.
`(ABCD)^2 = A B ** ** ** C D `Sources:Topics:Number Theory Arithmetic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Puzzles and Rebuses -> Reconstruct the Exercise / Cryptarithmetic- Beno Arbel Olympiad, 2017, Grade 8 Question 4
-
A POST-OFFICE PERPLEXITY
In every business of life we are occasionally perplexed by some chance question that for the moment staggers us. I quite pitied a young lady in a branch post-office when a gentleman entered and deposited a crown on the counter with this request: "Please give me some twopenny stamps, six times as many penny stamps, and make up the rest of the money in twopence-halfpenny stamps." For a moment she seemed bewildered, then her brain cleared, and with a smile she handed over stamps in exact fulfilment of the order. How long would it have taken you to think it out? Sources: -
YOUTHFUL PRECOCITY
The precocity of some youths is surprising. One is disposed to say on occasion, "That boy of yours is a genius, and he is certain to do great things when he grows up;" but past experience has taught us that he invariably becomes quite an ordinary citizen. It is so often the case, on the contrary, that the dull boy becomes a great man. You never can tell. Nature loves to present to us these queer paradoxes. It is well known that those wonderful "lightning calculators," who now and again surprise the world by their feats, lose all their mysterious powers directly they are taught the elementary rules of arithmetic.
A boy who was demolishing a choice banana was approached by a young friend, who, regarding him with envious eyes, asked, "How much did you pay for that banana, Fred?" The prompt answer was quite remarkable in its way: "The man what I bought it of receives just half as many sixpences for sixteen dozen dozen bananas as he gives bananas for a fiver."
Now, how long will it take the reader to say correctly just how much Fred paid for his rare and refreshing fruit?
Sources: