1 – Placing four numbers ---------------------------
**
**
Place the numbers 1, 2, 3 and 4 in the circles so that each number in a small square equals the sum of the two numbers connected to it by a line.
2 – The squares --------------
Young Mathis declares, "There are eighteen squares in this figure." His sister Mathilde replies, "Yes, if you only count the small squares… but there are medium-sized and large squares too!"
How many complete squares are there in the figure altogether?
3 – At preschool -------------------
The teacher gave each child in a group a sheet bearing these three symbols, along with three felt-tip pens in different colors: one blue, one red and one yellow. She told them to color the inside of each symbol, making sure that no two symbols on the same sheet were the same color.
The children took great care and followed the instructions. Only two children in the group produced identical pictures; all the others were different.
What is the largest possible number of children in the group?
4 – The ladybird -----------------
A ladybird moves around the thirteen shaded squares of a circuit. Its first move is in the direction of the arrow. It moves one square per second and, whenever it has a choice, it may go one way or the other, but it never turns back.
Mark with a cross every square it could be on after exactly eleven seconds.
5 – The birthday -------------------
"The day before yesterday I was still only 8, but by the end of the year I'll already be 10," says young Mathis. When is Mathis's birthday?
6 – Sequence ----------
The first term of a sequence is 718. Each subsequent term is 13 times the sum of the digits of the preceding term. What is the 2,018th term of this sequence?
7 – Mathilde's book -------------------------
For her birthday, Mathilde received a 225-page book with three chapters. The sum of the digits in the page numbers of the first two pages of the second chapter is 18.
By a curious coincidence, the sum of the digits in the page numbers of the last two pages of that same second chapter, which has more than two pages, is also 18. How many pages does the second chapter of this book have?
8 – The trominoes -----------------
A tromino is made up of three small squares. Rectangular trominoes are placed on a four-by-four square grid; the figure shows the first one already in place. Each tromino must cover exactly three squares of the grid, at least one of which is still uncovered. It may therefore overlap a previously placed tromino by one or two squares. What is the maximum number of rectangular trominoes that can be placed while following this rule?
9 – The basketball club ----------------------
Exactly 40% of the members of this basketball club were boys. Six new boys joined, and there are now as many boys as girls. How many members—girls and boys—does the club now have?
10 – Lottery -------------
Ten thousand tickets, numbered from 0000 to 9999, were sold in a lottery. The draw works as follows:
• a three-digit number is drawn at random;
• every ticket whose number contains all the digits of the number drawn wins.
The number drawn was 116. The winning tickets will therefore be precisely those containing at least two 1s and at least one 6. How many winning tickets will there be?
11 – The boat --------------
After completing half its journey, a boat increased its speed by 25% because of the threat of a storm. It then reached port half an hour earlier than expected. How long did the boat spend sailing?
12 – Motor race ----------------------
Two motorists set off at the same time, one from Arithméville towards Géocity and the other from Géocity towards Arithméville. The two towns are 200 km apart. They drove at constant integer speeds in km/h that differed by a multiple of 7. After two hours, the faster car's distance from Géocity was one-fifth of the slower car's distance from Arithméville. What was the speed of the faster car?
13 – An unusual multiple --------------------------
What is the smallest multiple of 2,018 whose decimal representation begins with 1111…? Answer 0 if you think no such multiple exists.
14 – The little wood -------------------
---------------------------
The little wood behind my house is a square measuring 90 m on each side. Two paths cross it: one follows a diagonal of the square, while the other joins a vertex to a point two-thirds of the way along one side, as shown in the figure. These two paths divide the wood into four plots. What is the area of the largest of these four plots?
Give your answer in square metres, rounding to the nearest m2 if necessary.
15 – Mick Erinos's pyramid -------------------------------
In Mick Erinos's pyramid, each brick contains a positive integer; each brick above the bottom tier contains the sum of the numbers in the two bricks supporting it. The sum of all the numbers in the pyramid is 2,018. What number is written in the grey brick?
16 – Father Fide's plot ----------------------------
Father Fide owns a quadrilateral plot of land. It has two perpendicular sides, each 100 m long, two opposite right angles, and two other angles, one twice the size of the other. What is the area of Father Fide's plot?
Give your answer in square metres, rounding to the nearest m2 if necessary.
17 – Successor and double -------------------------
The number 2,018 is twice the prime number 1,009 and the successor of another prime number, 2,017. What will be the next year whose number is both twice a prime number and the successor of a prime number?
18 – Sum of cubes --------------------
Mathias loves playing with numbers. He chooses a starting number, calculates the sum of the cubes of its digits, and writes down the result as his second number. He repeats the process with this second number, then carries on until he reaches a number he has already written down. Thus, starting from 1,012, he writes 1,012; 10; 1; 1 and stops. What is the smallest number greater than 2,018 that will lead him to 1?