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

Baobab

In the following exercise, identical digits have been replaced with identical letters, and different digits have been replaced with different letters. Reconstruct the exercise.

`BAOxxBAxxB = 2002`

Topics:
Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Logic -> Reasoning / Logic Number Theory -> Prime Numbers -> Prime Factorization Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures Puzzles and Rebuses -> Reconstruct the Exercise / Cryptarithmetic
Signs

A cyclist is riding on a straight road without turns and sees the following sign by the side of the road:

He continues moving and sees another sign:

Find the distance between cities B and C.

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Topics:
Arithmetic Algebra -> Word Problems
Sources:
- Young Mathematician Olympiad, 2017-2018, Stage A, Grades 3-4 Question 1
- Young Mathematician Olympiad, 2017-2018, Stage A, Grades 5-6 Question 1
What is the Smiley?

In the following exercise, one of the digits has been replaced by a smiley face. Find the digit.

`1☺xx6-☺=70`

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Topics:
Arithmetic Algebra -> Equations
Sources:
- Young Mathematician Olympiad, 2017-2018, Stage A, Grades 3-4 Question 2
5 but not 7

How many numbers from 1 to 100 (inclusive) are divisible by 5 but not divisible by 7?

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Topics:
Arithmetic Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules Number Theory -> Division
Sources:
- Young Mathematician Olympiad, 2017-2018, Stage A, Grades 3-4 Question 6

Differences in the Multiplication Table

Color the `10xx10` multiplication table with a black and white chessboard coloring, such that the cell of `1xx1` is colored black.
Find the difference between the sum of all the numbers in the black cells and the sum of all the numbers in the white cells.

	1	2	3	4	5	6	7	8	9	10
1	1	2	3	4	5	6	7	8	9	10
2	2	4	6	8	10	12	14	16	18	20
3	3	6	9	12	15	18	21	24	27	30
4	4	8	12	16	20	24	28	32	36	40
5	5	10	15	20	25	30	35	40	45	50
6	6	12	18	24	30	36	42	48	54	60
7	7	14	21	28	35	42	49	56	63	70
8	8	16	24	32	40	48	56	64	72	80
9	9	18	27	36	45	54	63	72	81	90
10	10	20	30	40	50	60	70	80	90	100

Arithmetic Combinatorics -> Number Tables Combinatorics -> Colorings -> Chessboard Coloring

Sources:

Young Mathematician Olympiad, 2017-2018, Stage A, Grades 3-4 Question 8
Young Mathematician Olympiad, 2017-2018, Stage A, Grades 5-6 Question 8

Digits in a circle

The digits from 1 to 5 are written in a circle in some order. Danny summed five two-digit numbers formed from pairs
of adjacent digits on the circle (clockwise). Find all the possible values for this sum.

Topics:
Arithmetic
Sources:
- Young Mathematician Olympiad, 2017-2018, Final, Grades 3-4 Question 3
Calculations on the Multiplication Table

Danny performed the following calculations on the multiplication table of numbers from 1 to 10:
He calculated the sum of all the numbers in the upper left square of size `9 times 9` within the multiplication table,
Added to it the sum of all the numbers in the lower right square `9 times 9`,
Subtracted from the result the sum of all the numbers in the upper right square `9 times 9`,
And then subtracted the sum of all the numbers in the lower left square `9 times 9`.
What result did he get?

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Topics:
Arithmetic
Sources:
- Young Mathematician Olympiad, 2017-2018, Final, Grades 5-6 Question 7
Digit Sum - 9

How many numbers in the range from 1 to 500 have a digit sum of 9?

Topics:
Number Theory Arithmetic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
Sources:
- Young Mathematician Olympiad, 2018-2019, Stage A, Grades 3-4 Question 2
Sum

In the following problem, different digits have been replaced with different letters, and the sum is a three-digit number.
What is the sum if the number "BA" is as small as possible?
\(\begin{align} AB& \\ \underline{+\quad BA}& \\ ***& \end{align}\)

Topics:
Arithmetic Algebra
Sources:
- Young Mathematician Olympiad, 2018-2019, Stage A, Grades 3-4 Question 5
Puzzle

You have a puzzle where each piece looks like this:

The following shape is made up of 12 pieces. What is its perimeter?

Topics:
Arithmetic Geometry -> Plane Geometry Geometry -> Area Calculation Algebra -> Word Problems
Sources:
- Young Mathematician Olympiad, 2018-2019, Stage A, Grades 3-4 Question 6

	1	2	3	4	5	6	7	8	9	10
1	1	2	3	4	5	6	7	8	9	10
2	2	4	6	8	10	12	14	16	18	20
3	3	6	9	12	15	18	21	24	27	30
4	4	8	12	16	20	24	28	32	36	40
5	5	10	15	20	25	30	35	40	45	50
6	6	12	18	24	30	36	42	48	54	60
7	7	14	21	28	35	42	49	56	63	70
8	8	16	24	32	40	48	56	64	72	80
9	9	18	27	36	45	54	63	72	81	90
10	10	20	30	40	50	60	70	80	90	100

	1	2	3	4	5	6	7	8	9	10
1	1	2	3	4	5	6	7	8	9	10
2	2	4	6	8	10	12	14	16	18	20
3	3	6	9	12	15	18	21	24	27	30
4	4	8	12	16	20	24	28	32	36	40
5	5	10	15	20	25	30	35	40	45	50
6	6	12	18	24	30	36	42	48	54	60
7	7	14	21	28	35	42	49	56	63	70
8	8	16	24	32	40	48	56	64	72	80
9	9	18	27	36	45	54	63	72	81	90
10	10	20	30	40	50	60	70	80	90	100

	1	2	3	4	5	6	7	8	9	10
1	1	2	3	4	5	6	7	8	9	10
2	2	4	6	8	10	12	14	16	18	20
3	3	6	9	12	15	18	21	24	27	30
4	4	8	12	16	20	24	28	32	36	40
5	5	10	15	20	25	30	35	40	45	50
6	6	12	18	24	30	36	42	48	54	60
7	7	14	21	28	35	42	49	56	63	70
8	8	16	24	32	40	48	56	64	72	80
9	9	18	27	36	45	54	63	72	81	90
10	10	20	30	40	50	60	70	80	90	100