The Open Math Problems series is designed to help students think through complex exercises with individual or group work, and to develop initiative and scientific reasoning for middle and high school students.
A series of exercises to develop initiative and scientific reasoning in the student or in the Internet user.
The base is the set of knowledge and skills that every student must have acquired by the end of secondary school and which will be validated in the personal skills booklet (L.P.C) in three levels.
These numerous open-ended math problems and complex tasks will allow you to assess many skills and will also allow students to begin a scientific process and encourage initiative.
The shadow
| It is assumed that the sun’s rays are parallel.
AB = 120 cm; AD = 210 cm; AE = 518 cm. Calculate BC |
Geometric and arithmetic mean
Consider a semicircle of diameter [AB].
M is any point on the semicircle and the point H is its orthogonal projection on [AB].
The point I is the middle of [HB].
Show that AI > AM.
THE FINGER:
A fir tree is drawn on a sheet of graph paper: the trunk is a rectangle made up of two squares, while the rest of the tree is a square.
of the fir tree is formed by five equal triangles, partially superimposed, and a smaller triangle which constitutes the point.
Mary looks at the drawing and is convinced that the part of the leaf occupied by the tree is larger than the remaining part.
Do you think Mary is right?
THE GARDEN TABLE
Luke’s dad built a rectangular garden table using 7 identical wooden boards, each with a perimeter of 3 m.
Here is the drawing of the table top, as it looks at the end of the construction.
What is the length and width of this garden table?
PARK BENCHES
In a large park, there are two kinds of benches: two-seater benches and three-seater benches.
There are 15 more two-seat benches than three-seat benches.
There is a total of 185 seats on the park benches.
How many benches does this park have in total?
Patterns of containers
In a cardboard box factory there are rectangular plates of length 6 dm and width 4 dm.
With such plates we want to make boxes without a lid whose shape is a cube whose volume is 
.
To do this, four identical squares are cut from each plate.
Problem :Determine the length of the sides of the squares to be cut out ?
The age of the teacher
The math teacher offers her students a subtle question:
Calculate my age knowing that :
if I double the age I will be in 4 years and subtract 20 from the age I was 4 years ago, the
The difference between the two numbers obtained is twice the age I am today!
Now it’s up to you to find my age!
How old is the teacher?
Surface to be painted
Two painters Yoann and Benoit have to paint a fresco.
Yoann has to paint the area Aire1.
Benoit paints the area Aire 2.
Which one has the biggest area to paint?
THE LENGTH OF THE CHALLENGE
A rectangular DEFI plot is divided into six plots of the same shape and area.
On the plan below, the layout of the parcels is respected, but the distances and proportions are not correct.
We only know that AB = BC = 1.
THE FATHER’S FIELD : open problem
Pierre Méable has a square field of 100.
NOT ALLOWED TO SEE IT
An elephant tusk is represented below by two semicircles tangent at A and centered on (AB), with point O being the center of the large semicircle.
We know that OA = 9 dm and DE = 3 dm.
Determine the length AC.
APPLES AND PEARS
The apples all have the same mass and the pears all have the same mass.
What is the mass of an apple?
THE THREE HIKERS
Three hikers move on the pedestrian circuit represented here, each one always walking in the same direction, as shown on the figure, and at a constant speed. Albert and Beatrice walk at the same speed, while Camille walks twice as fast. Albert and Beatrice left at 10 o’clock from the fountain, and Camille at 11 o’clock from the old oak tree, just as Albert was passing by.
What time will Beatrice and Camille meet for the first time?
THE HERITAGE IS IN THE LAKE
THE DIAMOND
The circles of this diamond must contain the numbers from 1 to 14, so that the difference between two numbers connected by a segment, taken in absolute value,
- is always a number less than or equal to 5
- is never equal to 3.
Complete the diamond.
THE AZTEC MASK
Recent excavations have brought to light an Aztec mask made of pure gold. The plan of this mask is shown below.
Calculate the area of this mask, the unit of area being the area of a small square. Do not forget to deduct the area of the eyes and mouth.
For future calculations, we will take 3.14 for pi.
THE FRIEZE THAT MAKES YOUR HAIR STAND ON END
Thomas cut out forty shapes identical to the one shown below.
He began to assemble them into a regular frieze.
When he has finished placing the 40th shape, what will be the perimeter of the frieze thus formed?
THE SURVEYOR ANTS
Two ants meet at point H.
1st ant: From B to A there are 125 units (of ant length), and from A to H there are 252 .
2nd ant: From D to C there are 76 units, and from C to H there are 156 units. Moreover, (AB) is perpendicular to (CD).
1st ant: (BD) and (AC) seem to be parallel.
2nd ant: Definitely not, because the entrance to my ant farm is at the intersection of these two trails!
1st ant: I was wrong, but your ant farm must be far away…
Calculate the distance as the crow flies from the second ant’s nest to the track (AB). The answer will be given in units of ants.
THE GABLED FIELD AND THE MEADOWS OF ILEXION
In the rural commune of Triangle, the cadastre contains only triangular plots (see extract from the cadastre below).
Mr. Ilexion owns three parcels of land whose surface areas he knows well, which are respectively equal to 420 m², 30 m², and 60 m².
But how big is the Champ Pignon?
Bricks:
Two identical bricks (projected dimensions 20 cm × 10 cm) are arranged as shown in the drawing.
The distance AB is 8 cm.
How far from the ground is point C?
Houses and matches:
How many matches are needed to build these houses in step 5? 16 ? 256 ?
How many steps can be performed with 1,465 matches?
Mrs. Tymar and her pool:
Mrs. Tymar decides to install an inground pool in her garden.
Here is a top view of his pelvis:
For safety reasons, she wants to cover the pool with a tarp.
A salesman offers him two rates:
– Rate A : 3€ per m² of tarpaulin and 150€ of installation;
– Rate B: a tarp + installation package at 399€.
He tells his client that the surface area of the cover must be 10% larger than the surface area of the pool.
Problem: what rate will be the most advantageous for Mrs. Tymar?
Download time
Jean launched the download of a free antivirus on the Internet: “Total antivirus”.
As he leaves to go jogging on the Pierre-Vernier promenade, he can see the window below:
A tennis court
A rectangular tennis court of 15 meters by 30 meters is surrounded by a driveway of constant width.
The outer perimeter of this driveway is double the size of the tennis court.
How wide is this driveway?
The ironing board
The height of an ironing board can be adjusted by opening, more or less, the angle formed by its legs.
Whatever its height, the table will always remain parallel to the ground.
How is this possible?
The following figure will help us to find out.
The bottles
In a 10 cm square box, 5 identical bottles have been placed that just fit in the box as shown in the drawing below.
What is the diameter of the bottles?
Literal calculation
The square ACFG and the equilateral triangle BDC have the same perimeter.
What is the measure of one side of the triangle?
Leonard and the crossbow
In the 15th century, Leonardo da Vinci became interested in lunula and completed the “collection” started by Hippocrates (5th century BC).
Among the 172 lunulae he described and drew, one could be called Leonardo’s crossbow.
We give you its design, its main dimensions and elements of its construction.
1. A circle of diameter [AB].
2. A circle with radius [AB] and center A.
3. A 45° angle.
4. A rectangle of width AC and length AB.
5. An axial symmetry.
Calculate the area of Leonard’s crossbow.
The bottle
The bottle shown here is filled to half its capacity with water.
How high in cm does the liquid reach?
Geometrically flowered
A flower bed is shaped like a 2m square STUV.
Man Jardin’tou, decides to plant hibiscus in the gray area, which is obtained from the two semicircles of diameter [ST] and [SV].
What is the area where Man Jardin’tou will plant the hibiscus?
Calculate the perimeter of a figure
Calculate the perimeter of this figure using the given dimensions.
The vegetable patch
Michao’s plot is triangular and its dimensions are 111 dm, 148 dm and 185 dm. It therefore has the form
of a right-angled triangle as you can verify by calculation. Michao knows that it is possible to
set up a square vegetable garden as shown in the figure opposite (one vertex on each side of the
the right angle and two vertices on the hypotenuse) but he would like to know the area of the garden thus obtained.
Can you help him determine this?
Michel, Michao’s gardener friend, advised him to calculate the height h from the vertex of the right angle of his land.
The rope
The point O is the middle of the segment [AB] and the point C is the middle of the segment [AO].
The line (MN) is parallel to the line (AB) and tangent at H to the circle of center C and radius CO.
We give MN = 2,012.
Calculate the radius of the great circle and round the result to the nearest unit.
Fraction of a square
Measurement of the side of a triangle
Area of a square :
The submerged ball (high school)
We wish to calculate the radius R of a steel ball by placing it at the bottom of a cylindrical container of 10 cm radius,
and pouring a volume V of oil into it, until the ball is covered.
The free surface of the oil is then flush with the top of the ball.
The height of the container exceeds 20 cm.
What must be the radius R for V to be equal to ?
The ball and the jack (high school)
The radius of the ball is four times that of the jack.
They are placed in a 27 cm square box.
What are their radii?
Aligned points (high school)
ABCD is a square, AEB and BCF are equilateral.
Are the points D, E and F aligned?
Two polygons (high school)
The figure opposite represents a rectangle ABCD and an isosceles triangle ABE having both 12 cm of perimeter.
Determine which of these two polygons has the larger area depending on the value of AB.
Maximum area (high school)
Consider a triangle ABC isosceles and rectangular in A such that AB=5 cm.
Let F be the middle of [AC].
Let (d) be the perpendicular to (AB) from M, it intersects (BC) at E.
We are interested in the area of the polygon EFAM.
The goal of the search is to find the position of the point M on [AB] for which the area is maximal.
The yin and yang (high school)
On a diameter [AB] of a circle of radius 4 cm, we mark a point M.
We denote by , with , the length of AM.
Two semicircles are drawn on either side of (AB), one with a diameter of [AM] and the other with a diameter of [BM].
Express the area of the hatched portion and determine for what value of x this area is maximum.
Disc fractions:
1. To what fraction of the large disk do the six small disks correspond?
2. To which fraction of the large disk does the area in brown correspond?
The string and the two squares (high school)
We cut a 32 cm long string in 2 pieces with which we form 2 squares.
Where should the string be cut so that the sum of the areas of the 2 squares is as small as possible?
Evaporation of a liquid (high school):
In a laboratory, to study the evaporation of a liquid, Professor Holè is in charge of measuring every day the
height of this liquid in a test tube.
It starts on Monday (day 1) and measures 8.2cm high.
The next day, the height of the liquid is 7.6cm.
Mr. Holè forgets to do the survey on Wednesday.
He realizes it on Thursday, the height of the liquid is then of 6,4 cm.
After how many days will there be no more liquid?
Ant problem (high school)
An ant moves along the edges of a cube.
If it goes from one vertex to the opposite vertex without passing twice through the same point,
what is the maximum length of its journey?
An ant ( M ) tries to reach a lump of sugar ( S ) by the shortest route. (the ant finds
always the shortest way! How about you?)
Building a box (high school):
Here, in bold, is the pattern of a box without a lid cut out of a sheet of cardboard.
Objective 1: Using an identical sheet of paper, build the box with the largest volume!
Objective 2: Build the lightest box with an identical sheet of paper!
Property Custodian (High School):
A guard is responsible for the surveillance of a rectangular property of 5 hm by 4 hm. It has a walkie-talkie.
to communicate with another guard located inside the property.
The quality of the communication depends on the distance between the two guards.
The diagram below illustrates this situation:
We note M the position of the first guard who moves from the point A in direction of the point B until completing the turn of the property.
The point O symbolizes the second guard.
The dimensions are shown on the drawing.
.
Describe the evolution of the OM distance according to the distance covered by the guard.
Park and Bridge (High School):
ABCD is a square park of side 10 meters.
A watercourse of width 1 meter passes through this park, materialized by the rectangle EFGH with AE = 6 meters.
Where to cross the bridge to make the journey from A to C as short as possible?
Square and area (high school):
The square ABCD has a side length of 8 cm.
M is a point on the segment [AB].
We draw in the square ABCD :
– A square of side [AM]
– An isosceles triangle with base [MB] and whose height has the same measure as the side [AM] of the square.
Three drawings are proposed for three different positions of point M.
from this situation, several problems:
– Problem 1: In which situation does the area of the triangle have the largest area?
– Problem 2: In which situation is the area of the square equal to the area of the triangle?
– Problem 3: In which situation is the area of the pattern equal to half of ABCD ?
– Problem 4: In which situation is the area of the triangle greater than half that of the square?
– Problem 5: How does the area of the pattern change as a function of AM ? according to MB?
A DIAMOND FOR GUINNESS:
A precious diamond of size and brilliance
is exhibited in the LUX museum.
To protect it, we built a glass box in the shape of a
cube of 10 cm of edge which contains it exactly, so that
that each vertex of the diamond is in the center of a face.
To propose this diamond to the “Guinness”, you must give your
volume.
Calculate its volume (in ).
So the volume of the polyhedron is 1/6 of the volume of the cube:
V= 1000/6 = 500/3 ≈167 (in cm3).
FACTUAL :
Anne, Berthe and Claire observe this table of numbers, discovered in the last pages of a
old mathematics textbook :
1! = 1
2! = 1 x 2 = 2
3! = 1 x 2 x 3 = 6
4! = 1 x 2 x 3 x 4 = 24
5! = 1 x 2 x 3 x 4 x 5 = 120
6! = 1 x 2 x 3 x 4 x 5 x 6 = 720
7! = 1 x 2 x 3 x 4 x 5 x 6 x 7= 5,040
8! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 = 40 320
9! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 = 362,880
10! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 = 3,628,800
11! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 = 39,916,800
12! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 = 479,001,600
13! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 x 13 = 6,227,020,800
14! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 x 13 x 14 = 87 178 291 200
…
Anna says: I think the last number on line 22! will end with four zeros.
Berthe says: I think the last number on line 27! will end with five zeros.
Claire says: no, in my opinion, the last number on line 27! will end with six zeros.
And you, what do you think?
Say whether the statements of each of the three friends are true or false, and why.
There are 6 factors 5, hence 6 digits 0 at the end of 27!
GRANDFATHER’S FIELD:
A grandfather gives his five grandchildren a field of
square shape divided into five plots, one square and four
triangles, such that the length of the sides of the square located at
center is equal to that of the short sides of each of the four
triangles. (See figure below)
Indication:
In your opinion, are the five plots the same area?
SOCCER BALL:
A football is made of 12 regular pentagons and 20
regular hexagons maintained between them by seams.
Their sides are all 4.5 cm.
What is the total length of the seams?
length of the seam: 90 x 4.5 cm, i.e. 405 cm
THE BOX OF CUBES :
François has a box in the shape of a rectangular parallelepiped of
interior dimensions 13 cm, 8 cm and 7 cm.
It has many wooden blocks,
some with a 2 cm edge, others with a 1 cm edge.
François wants to fill the box completely with the least
possible cubes.
How many of each kind should he put on?
BISCUITS:
Here are the cookies that the baker prepared for five children and placed them very precisely on
a tray.
The cookies are all the same thickness, but some children are unhappy and say that their
cookie is smaller than the others.
Do you think all children will have the same amount of cookie to eat?
If not, put the cookies in order, from smallest to largest.
CANDY JARS:
In the first jar, Grandma puts 6 orange candies
and 10 with lemon.
In a second jar, she puts 8 orange candies and 14 orange
lemon.
The candies are of the same shape and wrapped in the same
how.
Since Grandma knows that Julien does not like the taste of
lemon, she said to him:
You can have a candy. I let you choose the pot in
which you can slip your hand into, without looking inside.
Julien thinks carefully and finally chooses the pot where he thinks he has the best chance of taking a
orange candy.
If you were Julien, which pot would you have chosen?
AT THE FOUNTAIN:
Two friends, Laure and Pauline, fetch water with a bucket from the Eauclaire fountain.
Their two buckets together hold 26 liters.
With the water contained in the bucket of Laure we can fill 3 times the bucket of Pauline
and there is still 2 liters of water in the bucket of Laure.
How many liters does Pauline’s bucket hold? And Laure’s?
CHINESE RESTAURANT:
The sign of the Chinese restaurant “Le serpent rouge” is a long red snake inside a
golden rectangle.
This figure is a faithful reproduction of the sign:
What is the measure of the area of the snake?
PROFESSOR SUNFLOWER:
Mr. Sunflower drives from his house to his office.
It’s only when he’s exactly halfway there that he realizes that the little fuel level light is flashing and that his tank is almost empty.
He then decides to turn around and go to the gas station which is located exactly at
in the middle of the route already travelled.
After refueling, he heads back to his office. When he gets there, he finds that
his meter indicates 24 km.
He had reset it in the morning when he left his house.
How far from the house is Mr. Sunflower’s office?
The karting track
What you see in the drawing is the plan of a circuit for Go-Kart races.
When the circuit is not used for competitions, it is possible to walk around.
Luigi and Enrico want to know if it is more advantageous to go on the circuit
clockwise or counterclockwise to the rest area from the entrance.
They decide to walk, at the same speed, from the entrance,
but going in two opposite directions,
Luigi clockwise, Enrico counterclockwise.
Who will be the first to arrive at the rest area?
Justify your answer and show your calculations.
The bouquet
In Sandra’s class, the students really appreciate their math teacher. They decided to give her a bouquet of flowers for the Christmas party.
Each student gave as many times 2 cents as there are students in the class.
Sandra collected the contributions and tallied what she received. Not including her own contribution, she has 22 euros and 44 cents.
How many students are in the class?
Explain how you found your answer.
The factorials
Anne, Berthe and Claire observe this table of numbers, discovered in the back pages of an old mathematics textbook:
1! = 1
2! = 1 x 2 = 2
3! = 1 x 2 x 3 = 6
4! = 1 x 2 x 3 x 4 = 24
5! = 1 x 2 x 3 x 4 x 5 = 120
6! = 1 x 2 x 3 x 4 x 5 x 6 = 720
7! = 1 x 2 x 3 x 4 x 5 x 6 x 7= 5,040
8! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 = 40 320
9! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 = 362,880
10! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 = 3,628,800
11! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 = 39,916,800
12! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 = 479,001,600
13! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 x 13 = 6,227,020,800
14! = 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 x 11 x 12 x 13 x 14 = 87 178 291 200
…
Anna says: I think the last number on line 22! will end with four zeros.
Berthe says: I think the last number on line 27! will end with five zeros.
Claire says: no, in my opinion, the last number on line 27! will end with six zeros.
And you, what do you think?
Say whether the statements of each of the three friends are true or false, and why.
The password
Marie-Thérèse Rococo has chosen a password for her computer, consisting of 6 numbers followed by 3 capital letters.
– the 6 numbers chosen are all different and the 0 is not among them,
– their sum is 23,
– the six digits form a number less than 420,000,
– the product of the first number and the last is 28,
– the third, fourth and fifth digits form a number that is a multiple of 59,
– the three letters of the code are the initials of Rococo Marie-Thérèse, in this order.
What is Marie-Thérèse’s password?
Explain your reasoning.
The French Fry Machine
In the Bellefrites factory, several identical machines were installed to cut potatoes into French fries.
On the first day, we ran three machines for two hours and got 300 kg of fries.
On the second day, we ran six machines for four hours.
How many kilograms of fries were obtained during these two days?
Explain how you found the answer.
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