Grade 8 / Olympiad 45 exercises (including 41 corrected)

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123456789101112 3cm1cm1cm1cm1cm2cm2cm1cm 3773×7×71471×4×7282×8161×66 1. Logic E.8803 The frieze A frieze is made by copying to the right the pattern made up of the twelve columns shown below : Each column of the frieze is coded with three symbols, 000 if all three boxes (of the column) are white, 001 if the top two squares are white and the third noire. . . and finally 111 if all three squares are black. 1 How is the 10 column of the frieze coded? Same question for the 14 column. 2 In the above pattern, give the colonne (s) coded 100 ? 3 We are given an integer n . How is the column 12 n +5 coded? 4 How is the column 2 017 coded? E.8849 Peter is a numbers enthusiast. He has a four-digit digital clock in his car that tells the time from 00:00 to 23:59 . As he leaves for a long trip, Peter observes his clock and no-tices that the two numbers shown, the minutes and the hours, are squares of whole numbers (which, on a digital clock, are written as:00, 01, 04, 09, 25,. . . ) After returning from his trip, Peter finds that his clock is once again displaying squares of two whole numbers. His onboard computer tells him that he has traveled 352 km in 4 hours and 20 minutes. When can Peter be home from his trip? E.8709 Secret code Participants in a game are trying to get out of a room with a keypad and must figure out the code to do so. They have the following two clues : First clue The code is an ordered combination of four different digits that can make up a number. This number must be strictly less than 2 018 . For example, 0 6 2 7 is a code corresponding to the number 627 . How many different codes can be dialed? Second index Of all the possible codes obtained with the first index, one is such that : the number formed by the thousands digit and the hun-dreds digit is double the number formed by the tens digit and the units digit; e.g., for 1809 , 18 is the double of 09 ; the sum of the four digits of the code is even and not divisible by 9 What is this number? The code to dial to exit the room is the number obtained as the difference between the two previously obtained answers (first and second clues) What is this code? E.8787 Christophe cut out forty shapes identical to the one shown below (figure 1 ) . He began assem-bling them into a regular frieze (figure 2) . When he finishes laying out the fortieth shape, what will be the perimeter of the frieze thus created? Figure 1. Figure 2. 2. Arithmetic E.8702 The persistence of a number In this exercise, we consider integers greater than or equal to 10 , written in the decimal system. When we multiply the digits that make up the writing of an integer, we get a new number. We repeat this calculation with this new number and so on. For example, for the number 377 : The process stops when we get a number written with only one digit. It took 4 steps in all: we say that the persistence of 377 is 4 . 1 What is the persistence of each of the numbers a 77 b 28 534 c 6 785 791 2 The persistence of each of the numbers 2 019 , 4 806 , and 13 970 875 is equal to 1 . What general result do these https://chingmath.fr chapExoCorrec/8803 sacados/8803 123456789101112 chapExoCorrec/8849 sacados/8849 chapExoCorrec/8709 sacados/8709 chapExoCorrec/8787 sacados/8787 3cm1cm1cm1cm1cm2cm2cm1cm chapExoCorrec/8702 sacados/8702 3773×7×71471×4×7282×8161×66
a1a2a3a4a5a6a7a8a9a10a11a12C results seem to illustrate? Justify. 3 Is there a digit that could be inserted into the writing of a number without changing its persistence? 4 Find a number written with 20 digits whose persistence is 4 . 5 What are the possible persistences of a number whose writing includes an even number and 5? E.8704 The EAN-13 (EEuropean Arti-cle Numbering) is a barcode used by trade and industry to uniquely identify objects and be read by a scanner. This bar-code consists of 13 digits (integers between 0 and 9 ) , the last one being a control key calculated from the previous 12 digits. A barcode is symbolized by the following table : . . . where the thirteen digits making up the barcode have been carried. The number C is the verification key. To determine this, we calculate: S = a 1 + a 3 + a 5 + a 7 + a 9 + a 11 +3 × a 2 + a 4 + a 6 + a 8 + a 10 + a 12 . C is then the number such that S + C is a multiple of 10 . 1 is code 4971850187820 valid? 2 Determine C for the code 978204732850 C to be valid. 3 A digit has been replaced with x in the code 32525 x 7041767 . What value should be given to x for this code to be valid? 4 The code 3742278085985 is not valid. What can be the values of the first two digits on the left in other valid codes with the 11 same digits on the right? E.8829 Game of darts We play darts at a dartboard with three areas : one at 5 points, one at 7 points, and one at 11 points. We are inter-ested in the different possible scores, as the number of darts is not limited. For example, 30 is a possible score since : 30 = 11 + 7 + 7 + 5 or 30 = 5 + 5 + 5 + 5 + 5 + 5 1 Check that 26 , 43 , 220 012 are possible scores. 2 Two games are said to be identical if, for each game, each area of the target has the same number of darts. For example, the games corresponding to the scores : 7+5+5+11 and 5 + 7 + 11 + 5 are identical. a Find four different games giving the score 40 . b Show that there are two different games and only two matching the score 34 . 3 Find all the scores that can be obtained with a throw of three darts that all hit the target. Present the results in an organized manner. 4 a Show that 14 and the following four integers are pos-sible scores. b Determine the list of positive nonzero integers that cor-respond to no score. E.8841 A calculation program Here’s a calculation program : Let N be a natural number written in the decimal system : Step n o 1: Locate the X digit of the N units. Step n o 2: Calculate N − X . Let M be the number ob-tained. Step n o 3: Divide M by 10 . Let D be the number ob-tained. Step n o 4: Calculate D +2 X . Let R be the number ob-tained. Step n o 5: If R is different from N , then go to step n o 6. Otherwise, stop and display R . Step n o 6: If R is written with a single digit, then stop and display R . Otherwise ,give N the value R and re-sume the calculation program at step n o 1. 1 What result is displayed when N =15 is entered into the calculation program? 2 What happens when N =2015 is entered in the calcula-tion program? 3 Which numbers are likely to appear in the final display? E.8844 String Numbers 1 → 2 9 → 10 25 → ↓ ↑ ↓ ↑ 4 ← 3 8 11 24 ↓ ↑ ↓ ↑ 5 → 6 → 7 12 23 ↓ ↑ 16 ← 15 ← 14 ← 13 22 ↓ ↑ 17 → 18 → 19 → 20 → 21 The consecutive integers 1 , 2 , 3 , 4 , etc. . . are arranged in the cells of an array accord-ing to the diagram at right. The number 8 is at the in-tersection of the second row and the third column. The number 13 is at the intersection of the fourth row and the fourth column. What number is at the intersection of the twenty-fifth row and the twenty-fifth column? E.8845 Of one with nine 1 Calculate the sums : a = 99 + 999 ; b = 99 + 999 + 9999 2 Consider the number N defined as the sum : N = 99 + 999 + 9 999 + · · · + 9999 : : : 999 The first term of this sum is written with two digits 9 ; we add the numbers written with three and then four digits 9 , etc. . . The last term of the sum is written with one hundred digits 9 . We perform the sum and write N in ordinary decimal form. How many times does the number 1 appear in this writing? https://chingmath.fr chapExoCorrec/8704 sacados/8704 a1a2a3a4a5a6a7a8a9a10a11a12C chapExoCorrec/8829 sacados/8829 chapExoCorrec/8841 sacados/8841 chapExoCorrec/8844 sacados/8844 chapExoCorrec/8845 sacados/8845
7777722222Exemple de pyramide de Pascale avecm7etn2On n’a représenté ici que les cinq premières lignes 346 E.8846 We are interested in the various ways to write a natural number as the sum of other natural numbers (to avoid repetition, they are written in ascending order) . For example: 5 = 1+1+1+1+1 5 = 1 + 1 + 1 + 2 5 = 1 + 1 + 3 5 = 1 + 2 + 2 5 = 1 + 4 5 = 2 + 3 are the six possible decompositions of the number 5 . To each of the sums thus written, we associate the product of its terms. The results for 5 are 1 , 2 , 3 , 4 and 6 . 1 What are the possible decompositions of the number 7 ? What are the corresponding products? Which one is the largest? 2 We are now interested in the decompositions of the num-ber 28 , which we will not try to write down, and the corresponding products. a We consider any decomposition of the number 28 where the number 1 appears. We call P the associ-ated product. Find a decomposition whose associated product is greater than P . b Consider any decomposition of the number 28 where the number 5 appears. We call R the associated prod-uct. Find a decomposition whose associated product is greater than R . c Which decomposition of 28 gives the largest associated product? E.8847 A stack of boxes completed in the following manner is called a Pascale pyramid: we choose two positive integers m and n ; the first number in each row is equal to m (in the example below m =7 the last number in each row equals n (in the example below n =2 ) ; a number in one box is equal to the sum of the numbers in the two boxes directly above it. 1 a Reproduce and complete the Pascale pyramid exam-ple above. b For each line, calculate the sum of the numbers on it. What do you notice? 2 If we consider the Pascale pyramid associated with the integers m =1 and n =1 , on which line will the sum of the numbers equal 1024 ? 3 Prove that, for any integers m and n , the sum of the numbers in the fourth row is equal to twice the sum of the numbers in the third row. 4 Complete the Pascale pyramid below : 3. Around the quotients E.8708 Decimal development When we perform the division of 28 by 27 , we find : 1.037 037 037 037 : : : Posed division yields an unlimited periodic decimal writing of the quotient 28 27 . The pediod of this writing is three repeating digits (here 037) . The 5 e decimal is 3 . 1 What is the 52 e decimal of 28 27 ? 2 When we perform the division of 19 by 13 , we find : 1.461 538 461 538 461 538 : : : How many digits does the period consist of? What is the 100 e decimal of 19 13 ? 3 When we perform the division of 9 533 by 270 , we find : 35.307 407 407 40 : : : How many digits does the period consist of? What is the 1 000 e decimal of 9 533 270 ? 4 The decimal writing of 1 97 shows a period of 96 digits that begins with 0309... What is the 96 e digit in this period? https://chingmath.fr chapExoCorrec/8846 sacados/8846 chapExoCorrec/8847 sacados/8847 7777722222Exemple de pyramide de Pascale avecm7etn2On n’a représenté ici que les cinq premières lignes 346 chapExoCorrec/8708 sacados/8708
53157428961811666138521041510...17...5168 345121314282930 E.8756 The Egyptians used only 1 numer-ator fractions, with the exception of the 2 3 fraction. To find the double of their numerator fractions 1 , tables were available whose use is described below : For example, by reading the first row, we have : The double of 1 5 is 1 3 + 1 15 1 Interpret the third row of the table. 2 There is one number missing from the 15 row and one from the 17 row. Find these two numbers. 3 Why are there only odd numbers in the first column? E.8850 And in the end, what is left? We write the list of one hundred numbers : 1 , 1 2 , 1 3 , 1 4 ,. . . , 1 98 , 1 99 , 1 100 to which we apply the following procedure : we choose num-bers a and b from the list and replace them with the single a + b + ab , and then continue in the same way. At each step, the headcount loses one unit. At the end, we cannot continue , there is only one number. 1 If we proceed systematically and starting from the left : a What list do you get after the first step? After the second? b What number do we get after the 99 steps? 2 What if we start on the right? 3 If we proceed randomly, what results can we get? 4. Use of power E.8851 1 What is the number of digits in the number: N =10 2006 − 2006 2 What is the sum of the digits of N ? 5. Algebra and equation E.8806 Lotto new In a game of Lotto, each player is dealt cards with whole numbers on them. Numbered chips are drawn and the num-ber worn by each is announced aloud. One wins when all the numbers on a line on his or her card have been announced. Tonight, the rules have changed a bit! The organizer passes out cards with nine numbers on them di-vided into three rows of three consecutive numbers (the chart above is an example of such a card) . 1 The organizer announces : ˇ a prize is offered to whoever completes a number line whose sum is 41 ı . Can we win? 2 The organizer announces : ˇ a consolation prize is offered to whoever completes a line of numbers whose sum is 57 ı. What are the numbers entered on a winning line? 3 From now on : ˇ a consolation prize is offered to whoever presents a line of numbers whose product is a multiple of 6 ı. Who wins? In this question, the cards have three lines of four consecutive numbers. 4 Organizer announces : ˇ if adding 1 to the product of the four numbers in a row gives you a square parfait ∗ , then you win a lot ı. Who wins? ∗ Recall that a perfect square is the square of an integer https://chingmath.fr chapExoCorrec/8756 sacados/8756 53157428961811666138521041510...17...5168 chapExoCorrec/8850 sacados/8850 chapExoCorrec/8851 sacados/8851 chapExoCorrec/8806 sacados/8806 345121314282930
816 23-2332818833813 E.9472 We’re interested in magic squares 3 × 3 : these are tables with three rows and three columns (des-ignated respectively by L 1 , L 2 , L 3 and C 1 , C 2 , C 3 ) in which 9 numbers are written, so that the sums of the numbers written in each row, in each column and in each diagonal (the diago-nals are denoted by D 1 and D 2 ) be equal. This common sum is noted S , it is the constant of the magic square. 1 In the square opposite, write the inte-gers between 1 and 9 . Complete it to make a magic square. 2 Of the two arrays below, only one is magic. Which is it? 2 We denote by x any number. We wonder if it is possible to create a magic square 3 × 3 in which the nine numbers would appear: 16 x − 10 ; 2 x − 3 ; − 2 ; 4 x − 4 ; 12 x − 8 10 x − 7 ; 6 x − 5 ; 8 x − 6 ; 14 x − 9 a What would be the constant of this magic square? b Propose a magic square 3 × 3 using these nine numbers. 3 Finally, propose two magic squares 3 × 3 : a A square of 9 numbers all negative ; b A square of constant S 30 6. Geometry E.8705 Six semicircles of radius 1 , and the diameters of three of them, determine the domain shown below. What is the area of this domain? E.8842 A spiral Such a spiral is obtained by successively drawing quarter cir-cles of increasingly larger radii. The first quarter circle has a radius of 1 , as does the second. The third quarter circle has a radius of 2 , the fourth of 3 , the https://chingmath.fr sacados/9472 816 23-2332818833813 23-2332818833813 chapExoCorrec/8705 sacados/8705 chapExoCorrec/8842 sacados/8842
étape 0 étape 1 étape 2 étape 3 étape 0 étape 1 étape 2 next of 5 . 1 What is the radius of the sixth quarter circle? the sev-enth? the tenth? 2 The radius of the fifteenth quarter circle is 610 and that of the sixteenth is 987 . What is the radius of the seven-teenth? 3 What is the radius of the twentieth quarter circle? E.8843 The best packaging Seven cylindrical pipes of diameter 20 cm are wrapped with adhesive tape. They can be arranged flat or in a bundle, as shown in the figures below. In each case, what is the length of one turn of the tape, rounded to the nearest millimeter? Hint: we’ll use : ı ≈ 3 ; 1416 7. To the suites E.8828 Dig more and more but less and less 1 Sierpinski’s rug We consider a square of side 27 . In each step, we remove the center square from each gray square. At what stage does the area of the rug become less than half the original area? 2 Sierpinski’s sponge We now consider a cube with edge a . In each step, we hollow out each cube with seven small cubes inside as below. At what stage does the volume of the sponge become less than half the initial volume? https://chingmath.fr chapExoCorrec/8843 sacados/8843 chapExoCorrec/8828 sacados/8828 étape 0 étape 1 étape 2 étape 3 étape 0 étape 1 étape 2
Etape no1Etape no2Etape no3 AUBMNZPODWCV E.9471 On the merchant’s stall, all the fruits are arranged. The oranges are arranged in a pyramid with a square base. The top tier has a single orange. This tier will be noted E 1 . The floor below it will be noted E 2 , and so on. Each orange is placed on top of four others. 1 a How many oranges are on the floor E 2 ? b How many oranges are on the E 3 floor? c How many oranges are on floor E 10 ? d Is there a floor with exactly 64 oranges? e Is there a floor with exactly 200 oranges? 2 How many oranges has : a a pyramid with 2 floors? b a pyramid with 3 floors? c a pyramid with 10 floors? 3 The oranges in the stall were arranged to form a seven-story pyramid. Unfortunately, the pyramid proves to be too unstable and smaller ones need to be formed. Pro-pose a square-based pyramid organization that allows all the oranges to be stored. All pyramids will be complete and each will have at least 3 floors. E.9473 A sequence of patterns is con- structed according to a process whose first three steps are shown below : The unit of area is the grid square 1 For each of the steps 2 and 3 : a Determine the total area of the light interior parts ; b Determine the total area of the black parts. 2 a What is, in step 4 , the total area of the light interior parts? What about those of the black parts? b And in step 20 ? E.9474 You can climb a staircase one or two steps at a time. The figure on the right shows an example. 1 In how many ways can you climb a staircase with one step? two steps? three steps? four steps? five steps? 2 In how many different ways can you climb a staircase of 20 steps? 8. Geometry with the Pythagorean theorem E.8703 In the city of Nîmes, work has led to the discovery of a magnificent Roman mosaic from the early third century, known as ˇmosaic of Penthéeı. This mo-saic consists of several figured panels of various geometric shapes, separated by ˇbraidsı (torsdes) . This is the work of a master craftsman with a strong knowledge of ˇpractical geom-etryı and the ability to assemble arcs of circles into various shapes, ultimately composing a harmonious whole. In the figure above, we mentally extracted from the mosaic an oval enclosed in a square ABCD of side 3 units, itself cut into 9 squares. We know that the oval consists of 4 quarter circles. https://chingmath.fr sacados/9471 sacados/9473 Etape no1Etape no2Etape no3 sacados/9474 chapExoCorrec/8703 sacados/8703 AUBMNZPODWCV
O2cmDEFsocle0;5cm0;5cmABC5cm2cm4cm 2cmGH5cm 1cm3cmZIGZAGAméricain3cmNoeudPapillon3cmGavageABCDEF 1 What is the center of each of these quarter circles? 2 Calculate the perimeter of this oval. 3 Calculate the area of this oval. 4 a Construct such an oval (we will take 3 cm for unit) . b Derive a construction of a figure with the same perime-ter but with a smaller area. c Construct a figure with the same perimeter but with an area equal to 4 units of area. E.8711 Ants 1 Below are two solids : a straight paving stone and a ball on which two ants are moving. Ant n o 1 moves on the straight paving stone following the path formed by segments [ AB ] and then [ BC ] . Ant n o 2 moves on a sphere with center O and radius 2 cm , which rests on a base 1 cm high. It starts at point D , moves to E , following segment [ DE ] , then reaches point F following the semicircle with diameter [ EF ] . Which ant travels the shortest path? 2 Two ants are moving on a cylinder with a radius of 2 cm . Ant n o 1 starts at point G and travels around the upper circle of the cylinder several times in a row. Ant n o 2 , on the other hand, moves along the cylinder fol-lowing the arrowed path from G to H , taking the shortest route, then climbs back up to G along the same path. Both ants start their journey at the same time and move at the same speed, which is assumed to be constant. a Will ant n o 2 meet ant n o 1 on its return to G ? b Let’s imagine that the two ants continue to move in this way without stopping Will they be able to meet at some point in G ? Guidelines: All results will be rounded to the nearest millimeter. we will use : ı ≈ 3 ; 1416 E.8805 Do you know how to drive nails? Madame Briqueau wants to hang a picture frame in her liv-ing room. The frame is rectangular and measures 84 cm in length and 56 cm in height. Its hanging system consists of a 60 cm -long string, attached to the back of the frame at two points both 21 cm from the top edge of the frame and each 20 cm from one of the side edges. The wall intended for hanging is rectangular and measures 2.68 m high by 3.78 m wide. Madame Briqueau would like her frame, once installed, be cen-tered in width (same space to the left and right of the frame) and that its bottom edge be horizontal and 1.42 m from the ground. 1 How far from the left edge of the wall should Madame Briqueau plant the hook? 2 What shape will the string take when the frame is in place? 3 How far from the top of the wall should Madame Briqueau plant the hook (rounded to the nearest millime-tre) ? E.8808 Laces There are several ways to tie shoes. Here are three : What is the longest lacing? (We don’t consider the length of strands that are used to make a knot) . https://chingmath.fr chapExoCorrec/8711 sacados/8711 O2cmDEFsocle0;5cm0;5cmABC5cm2cm4cm 2cmGH5cm chapExoCorrec/8805 sacados/8805 chapExoCorrec/8808 sacados/8808 1cm3cmZIGZAGAméricain3cmNoeudPapillon3cmGavageABCDEF
ABCDE 4cm5cm5cm CBAFG 491681?936 E.8809 In the eyes We schematize the eyes as disks of radii 5 mm . Cat’s eye [ BC ] and [ DE ] are two per-pendicular diameters. The pupil is bounded by two arcs of circles with respec-tive centers B and C , passing through D . Rabbit’s eye The pupil is bounded by a circle with the same center as the eye and radius 3 mm Which of these two eyes has the larger pupil? 9. Geometry with Thales’ theorem E.8710 The Crown The vertices of the shaded polygon shown below lie on parallel lines spaced 5 cm apart. The ˇ base ı has length 4 cm . What is the area of this polygon? 10. Geometry with trigonometry E.8807 Hexagons gigognes Let H 1 be a regular hexagon in-scribed in a circle and H 2 a regular hexagon circumscribed in the same circle. In the figure below (which is not full size) , A , B and C denote three consecutive vertices of the hexagon H 2 and G and F , the respec-tive middles of segments [ AB ] and [ BC ] , are two vertices of hexagon H 1 . The area of the hexagon H 2 is 340 m 2 . How much is the area of the hexagon H 1 ? E.9014 In the figure below, the areas of six squares have been indicated. One of the vertices of the white oblique square coincides with a vertex of the square with area 1 . What is the area of this square? 11. Proportionality, percentage, unit https://chingmath.fr chapExoCorrec/8809 sacados/8809 ABCDE chapExoCorrec/8710 sacados/8710 4cm5cm5cm chapExoCorrec/8807 sacados/8807 CBAFG chapExoCorrec/9014 sacados/9014 491681?936
nombre entiermodulonombre entier=0 66modulo11=0 E.8804 BavardaCar Mr. A has to make a 350 km (no tolls) car trip. He has to pick up four other people, Mrs. B , Mr. C , Mrs. D , and Mr. E , who are going to the same place as him. He picks up Mrs. B after 60 km of driving and then drives for another 60 km before picking up Mr. C and Mrs. D . He picks up Mr. E at 200 km of arrival. Fuel expense is estimated at 36.40 e . Different sharing arrangements are considered for distributing the expense among all occupants of the car . 1 First mode: everyone pays the same amount. What is this sum? 2 Second mode: each person’s share is proportional to the distance he or she has traveled in the car. What is everyone’s share? 3 Third mode: everyone’s share is calculated by taking into account, on each of the sections, the number of peo-ple in the car and the length of the section. What is everyone’s share? E.8840 Pursuit A new type of footrace was recently created. The runners all start at the same time and have no finish line to cross. A car starts after them half an hour later. Any run- ner passed by the car is eliminated. The last runner passed is declared the winner of the race. The objective of each runner is to cover as much distance as possible, before being caught by the car. Here is the organization of the race: Runners start at 10 in the morning. The car chasing them starts 30 minutes later. It gradu-ally increases its speed : For the first hour, it runs at 15 km = h The next hour, she drives at 16 km = h The next hour, she drives to 17 km = h The next two hours, she drives at 20 km = h She then stabilizes her speed at 35 km = h 1 Robert was caught by the car an hour after he left. How far did he travel? 2 Michele was caught by the car two hours after she left. At what average speed did she run? 3 Philip ran 30 km before he was caught up. At what time was he caught by the car? 4 The winner of last year’s race ran 78 km . How long did he run, and at what average speed? 5 Victoire thinks this year she can run for hours at 14 km = h . If she can do this, how far will she run? 12. Probability E.8848 True - False Ludovic has to answer a test of 25 questions. The answers are ˇ True ı or ˇ Faux ı. His math teacher gives the following indication : in any series of 5 consecutive answers, there are exactly three answers ˇ True ı. 1 Why isn’t the following list suitable? Question 1 V Question 2 F Question 3 V Question 4 V Question 5 F Question 6 F . . . . . . 2 How many answers are there ˇ Vrai ı in the list of 25 an-swers 3 The teacher indicates that the answer to the first ques-tion is ˇ False ı. Ludovic claims he knows the answer to sixth without having read the questions. How did he do that? 4 The teacher whispers to Ludovic that the answer to the last question is also ˇ False ı. Ludovic claims that he can now find all the answers without reading the questions. Is he right? 13. Algorithms E.8706 Under scratch, the command be-low tests whether an integer is divisible by an integer. Thus, the proposition is true be-cause 66=6 × 11 . In other words, 66 divisible by 11 . https://chingmath.fr chapExoCorrec/8804 sacados/8804 chapExoCorrec/8840 sacados/8840 chapExoCorrec/8848 sacados/8848 chapExoCorrec/8706 sacados/8706 nombre entiermodulonombre entier=0 66modulo11=0
64modulo11=0 quandest cliquémettre à50% de la taille initialebasculer sur le costumeballon bleumettreordonnéeà-155mettrepointsà0aller à x:0y:ordonnéesiordonnéemodulo11=0alorsmettrepointsàpoints*3basculer sur le costumeballon jaunesiordonnéemodulo7=0alorsmettrepointsàpoints*2basculer sur le costumeballon violetmettrepointsàpoints+10basculer sur le costumeballon bleusinonsinonattendre2secondesajouter àordonnée15aller à x:0y:ordonnéesiordonnée<150alorsrépéter indé∏nimentligne 0ligne 1ligne 2ligne 3ligne 4ligne 5ligne 6ligne 7ligne 8ligne 9ligne 10ligne 11ligne 12ligne 13ligne 14ligne 15ligne 16ligne 17ligne 18 12364758910111213141517161819202122 quandest cliquéeInitialisationavancer deVitessetournerde90degréssiCapteurtouché?alorsrépéter indé∏nimentde∏nir Initialisationaller àDépartmettreVitesseà5 On the other hand, the proposition is false because 64 is not divisible by 11 . The command below is used to program the vertical move-ment of a sprite representing a balloon changing color (blue, purple or yellow) according to its ordinate and each step al-lowing the number of points to evolve. a At each pass through the loop ˇ If the ordinate is less than 150 ı, how much is the ordinate of the balloon increased? b If the ordinate of the balloon is equal to 55 , explain why the number of points is multiplied by 3 ? c If the ordinate of the balloon is equal to 100 , what hap-pens to the number of points? d For which ordinate(s) does the balloon pass through the color yellow? Why or why not? e What will be the final number of points awarded to the ball? f With what number must I replace the value ˇ10ı (line 14) so that the final number of points equals 955 ? g By what number must we replace the value 2 (line 11) for the number of points to equal 17 300 ? E.8707 Robot Gymkhana A robot is placed on a grid-patterned board with numbered obstacles arranged on it. It is positioned at the start, ready to move forward in the direction of the arrow. This robot is equipped with a presence sensor located at the front. This sensor allows it to detect obstacles in front of it. Finally, this robot is also equipped with a chip that allows its movements to be programmed using a block-based language such as ˇ Scratch ı or ˇ MBlock ı. The following program has been implemented in the robot’s chip. The robot is then started using the green flag. 1 List the first ten obstacles encountered. 2 What will be the 50 ème obstacle it encounters? 3 What will be the 2019 ème obstacle it encounters? https://chingmath.fr 64modulo11=0 quandest cliquémettre à50% de la taille initialebasculer sur le costumeballon bleumettreordonnéeà-155mettrepointsà0aller à x:0y:ordonnéesiordonnéemodulo11=0alorsmettrepointsàpoints*3basculer sur le costumeballon jaunesiordonnéemodulo7=0alorsmettrepointsàpoints*2basculer sur le costumeballon violetmettrepointsàpoints+10basculer sur le costumeballon bleusinonsinonattendre2secondesajouter àordonnée15aller à x:0y:ordonnéesiordonnée<150alorsrépéter indé∏nimentligne 0ligne 1ligne 2ligne 3ligne 4ligne 5ligne 6ligne 7ligne 8ligne 9ligne 10ligne 11ligne 12ligne 13ligne 14ligne 15ligne 16ligne 17ligne 18 chapExoCorrec/8707 sacados/8707 12364758910111213141517161819202122 quandest cliquéeInitialisationavancer deVitessetournerde90degréssiCapteurtouché?alorsrépéter indé∏nimentde∏nir Initialisationaller àDépartmettreVitesseà5
123456ABCDBonnesNerépondspasFaussesScore00=8-A2-B2=A2*5+B2*(-2)+C2*(-3)017-23026-22035-21044-2000=8-A2-B2=A2*5+B2*(-2)+C2*(-3)00=8-A2-B2=A2*5+B2*(-2)+C2*(-3) quandaestpressémettreà250%delatailleinitialemettrecyclesà0costumesuivantRépéter14foisajouteràcycles1stoptoutsicycles=40alorsrépéterindé∏nimentcycles:40chrono:12.7 E.8712 Louise, Nassim, Ilam, and Sophie participate in a game show with 2 parts. In the 1 ère game, these four candidates must each answer three questions. For each correct answer, the player earns 5 points, loses 2 points if he or she does not answer, and loses 3 points if his or her answer is wrong. The candidate with the fewest points is eliminated in the 1 er round. Louise only answers 2 questions, one of which is wrong. Nassim answers all the questions 2 are correct. Ilam answers the first 2 questions correctly but does not an-swer the last. Sophie answers only one question, but she is correct. 1 a Which candidate was eliminated in this 1 ère game? b List all the possible scores that a candidate playing this game can get in the 1 ère game. 2 In the 2 ème part of the game, the principle is the same but candidates must answer 8 questions. Zoe is watching this game at home. She takes out her tablet and opens a spreadsheet of calculations in a spread-sheet. Here is what she writes : How many correct answers does it take to get a positive score? E.8713 The twelve images below are taken from an animated image file (containing 15 successive im-ages) showing the development of a flower from birth to death. Only the first and last images are correctly positioned. 1 re Part : Indicate the order in which the ten other images should ap-pear during the animation. 2 ème Part : Preliminary questions: understanding the algorithm a What do you need to do to launch the program? b How many cycles of the flower can you observe before the program stops? c What is the ˇ duration ı of the animation? d Is the variable cycles displayed the number of cycles al-ready executed or the rank of the cycle currently being executed? Justify your choice. Question 1 What would be the duration of the animation for exactly 5 cycles? 9 cycles? Question 2 When the timer displays the end of the 9 ème ˇ second ı, , what is the number of cycles displayed? Which ˇ costume ı of the ˇ elf ı would be on the image at that moment? 14. End of the year: with remarkable identity E.8757 The triangle ABC opposite is such that : AC = 13 ; AB = 14 ; BC = 15 Let H , J , K be the feet of the heights from the vertices C , A , B , respectively. https://chingmath.fr chapExoCorrec/8712 sacados/8712 123456ABCDBonnesNerépondspasFaussesScore00=8-A2-B2=A2*5+B2*(-2)+C2*(-3)017-23026-22035-21044-2000=8-A2-B2=A2*5+B2*(-2)+C2*(-3)00=8-A2-B2=A2*5+B2*(-2)+C2*(-3) chapExoCorrec/8713 sacados/8713 quandaestpressémettreà250%delatailleinitialemettrecyclesà0costumesuivantRépéter14foisajouteràcycles1stoptoutsicycles=40alorsrépéterindé∏nimentcycles:40chrono:12.7 chapExoCorrec/8757 sacados/8757
ABCHKJ OABCD OABCDEFGH 1 Show that CH is an integer. 2 Show that AJ is a decimal number and that BK is a quotient of integers. 3 Propose a triangle whose three sides and three heights have lengths of integers. 15. Unclassified exercises E.9747 Sum of numbers In this exercise, the numbers considered are integers written in decimal numeration. For this exercise, we call weight of a number N the sum of its digits. 1 What is the weight of the number 29 ? What is the weight of the number 7 646 ? 2 Propose three different numbers of the same weight 42 . 3 Is it correct to say that ˇ the more digits a number has, the greater its weight ı? 4 What is the smallest number of weights 50 ? 5 What is the smallest number of weights 2 022 ? 6 Can we find a number written only with 5 and 7 and whose weight is 53 ? 7 Can we find a number written only with 3 and 6 and whose weight is 200 ? E.9748 Square inscribed in a circle inscribed in square. . . The unit of length is cm. Caution : figures are not to scale. All numerical results requested are expected to be exact val-ues. Shown in the figure above is the circle C 1 , with center O and radius 2 . The segments [ AC ] and [ BD ] are two perpendicular diameters of this circle. 1 What is the nature of the quadrilateral ABCD ? We say that the circle C 1 is the circumscribed circle of the square ABCD . 2 What is the area of the shaded portion of the figure? Consider the square EFGH whose sides are parallel to those of ABCD and tangent to the circle C 1 . The circle C 1 is said to be inscribed in the square EFGH . Similarly as before, consider the circle C 2 circumscribed by the square EFGH . The figure below represents this situation. 3 Calculate the area of the shaded portion on this new fig-ure. 4 Using the same principle, we can construct a new figure with a circle C 3 circumscribed by a new square IJKL whose sides would be tangent to C 2 and parallel to the sides of the square EFGH . What is the area on this new figure between the circle C 3 and the sides s of the square IJKL ? https://chingmath.fr ABCHKJ chapExoCorrec/9747 sacados/9747 Olympiades Mars 2022 chapExoCorrec/9748 sacados/9748 Olympiades Mars 2022 OABCD OABCDEFGH
Corde à13noeuds et triangle égyptien ABCDEF E.9749 Pythagorean triples A unit of length is given in the plane. A triangle ABC has sides : AB = 15 ; AC = 8 ; BC = 17 1 Show that this triangle is right-angled, indicating which point is the vertex of the right angle. More generally, we are interested in right-angled triangles whose sides have integer lengths. We set : AB = m ; AC = n ; BC = p . We assume that m<n<p and say that the triplet ( m ; n ; p ) is Pythagorean . 2 a If ( m ; n ; p ) is a Pythagorean triple, what point is the vertex of the right angle of the associated right triangle ABC ? b Show that 3 ; 4 ; 5 is a Pythagorean triple. The asso-ciated triangles are the ˇ Egyptian triangles ı. c Show that, if the triple m ; n ; 5 is Pythagorean, then m =3 and n =4 . The Plimpton 322 tablet (Columbia University, New York) bears witness to research conducted by the Babylonians. 4 We assume that the triplet (5 ; n ; p ) is Pythagorean. a Show that : p − n p + n =25 b Compare p + n and p − n and deduce their values, then finally the values of p and n . The associated triangle is called ˇ Babylonian ı. 5 Are there integers m and p such that the triplet ( m ; 5 ; p ) is Pythagorean? E.9750 Unknown angle The angle in C of triangle ABC measures 70 o . We placed on the [ BC ] side the point D and on the [ AC ] side the point E such that : BD = DE = EA . Segments [ BE ] and [ AD ] intersect in F . What is the measure of the angle ∠ AFB ? https://chingmath.fr chapExoCorrec/9749 sacados/9749 Olympiades Mars 2022 Corde à13noeuds et triangle égyptien chapExoCorrec/9750 sacados/9750 Olympiades Mars 2022 ABCDEF