Young Mathematician Olympiad, 2017-2018, Stage A, Grades 3-4

Question 1 - 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
Additional sources:
- Young Mathematician Olympiad, 2017-2018, Stage A, Grades 5-6 Question 1
Question 2 - 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In the following exercise, one of the digits has been replaced by a smiley face.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
Question 3 - Triangles in a Hexagon

How many triangles are in the picture? Count all the triangles formed by the lines.

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Topics:
Combinatorics Geometry -> Plane Geometry -> Triangles
Question 4 - Cats for Grandma

Grandma Hannah has a number of cats. One day, her three grandchildren, Avi, Benny, and Gili, came to visit and tried to guess how many cats she has.

Avi said: "Grandma has at least 7 cats",

Benny said: "But less than 10, I think",

Then Gili said: "Grandma Hannah has either 9, or 10, or 11 cats".

"You are all correct!" replied the grandma

So how many cats does she have?

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Topics:
Logic -> Reasoning / Logic Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
Question 5 - Palindromic Number

Find a four-digit palindromic number that is divisible by 25 and not divisible by 3.

Note: A palindromic number is a number that does not change if its digits are read in reverse order. For example, the number 5775 is a palindromic number, and the number 5778 is not a palindromic number.

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Topics:
Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 3 and 9 Number Theory -> Division -> Parity (Even/Odd) Number Theory -> Modular Arithmetic / Remainder Arithmetic -> Divisibility Rules -> Divisibility Rules by 5 and 25 Combinatorics -> Case Analysis / Checking Cases -> Processes / Procedures
Question 6 - 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
Question 7 - Area of the Shape

Given a grid paper where the area of each square is one unit area. Find the area of the shape (in unit areas)

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Topics:
Geometry -> Plane Geometry Geometry -> Area Calculation Combinatorics -> Combinatorial Geometry -> Cut a Shape / Dissection Problems Combinatorics -> Combinatorial Geometry -> Grid Paper Geometry / Lattice Geometry

Question 8 - 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

Additional sources:

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

	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