Grade 11 / Olympiad competition 38 exercises (including 36 corrected)

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1. Understanding a new definition E.5940 A non-zero natural number is a Harshad number if it is divisible by the sum of its digits. For example, n =24 is a Harshad number because the sum of its digits is 2+4=6 , and 24 is indeed divisible by 6 . 1 a Show that 364 is a Harshad number. b What is the smallest integer that is not a Harshad number? 2 a Give a Harshad number of 4 digits. b Let n be a non-zero integer. Give a Harshad number of n digits. E.5941 From two positive integers, we construct a list of numbers où each number is the sum of the previous two. 1 Choose two positive integers less than 10 and determine the first ten numbers from the list defined above. 2 A mathemagician claims to be able to quickly and ex-actly determine the sum of the first ten numbers of any list constructed in this way. Show that, whatever the starting numbers, this sum is a multiple of one of the numbers in the list whose position will be determined. E.5982 A faulty calculator only allows you to : enter positive numbers or zero; perform the following operation: given three numbers entered in succession ( x ; y ; z ) , it displays 0 if x = y and the result of z x − y otherwise. ( x ; y ; z ) −→ z x − y if x = y 0 if x = y the use of parentheses to compose calculations. 1 By detailing the calculations, check the following results given by the calculator: 0 ; 1 ; 2 ↦−→ − 2 ; 2 ; 0 ; 1 ; 1 ; 1 ↦−→ − 2 2 What are the results of 2 ; 0 ; 1 , 0 ; 2 ; 1 , and 2 ; 1 ; 2 ; 1 ; 2 ? 3 Give a calculation that gives − 1 . 4 Check that the calculation a ; 0 ; 1 gives the inverse of a for all a> 0 . 5 Propose a calculation that allows you to divide two pos-itive numbers : a b by a 0 and b> 0 . 6 Propose a calculation that allows two positive numbers to be multiplied: a × b with a 0 and b 0 . E.8128 Angle measurements ap-proximately A triangle ABC is said to be approximately right-angled at a vertex A if the measure of the angle at A is in the inter-val 75 o ; 105 o . A triangle ABC is said to be approximately isosceles at a vertex A if the measures of the angles at B and C differ by 15 o at most. 1 a Is a right-angled triangle approximately right-angled? Is an isosceles triangle approximately isosce-les. b Can a triangle be right-angled in two vertices? Approx-imately right-angled in two vertices? If so, when it is additionally acutangle (i.e. all its angles are acute) , is it approximately isosceles? 2 Is there an acutangular triangle that is neither approxi-mately right-angled nor approximately isosceles? 3 Write a program (in natural language or calculator) , to be copied onto your copy, testing whether a triangle ABC whose three angles A , B and C are known is approxi-mately isosceles. Measurements of lengths approximately In this part, we assume that a unit of length has been given in the plane, and adopt the following definitions : Two points are approximately equal if their distance is less than or equal to 0.1 ; Two segments are of approximately the same length if their lengths differ by 0.1 or less ; A triangle is approximately equilateral if the lengths of its sides differ, two by two, by 0.1 or less. 4 a Can a right-angled triangle whose hypotenuse mea-sures (exactly) 1 be approximately equilateral? b Can a right-angled triangle be approximately equilat-eral? 5 Consider a circle, center O radius (exactly) 2 and two points on this circle: A , fixed, and B , movable. We call I the middle of the segment OA ] and H the orthogonal project of B onto the line ( OA ) . a Show on a figure the set of points B for which H and I are approximately equal. Calculate the length (the result will be given rounded to the hundredth) . b If H and I are approximately equal, is the triangle AOB approximately equilateral? 2. Arithmetic E.5942 1 a Starting from 12 589 and counting from 29 to 29 , can we reach the number 12 705 ? b Starting from 1 485 and counting from 29 in 29 , can we reach the number 310 190 ? Explain your approach. 2 What is the smallest positive integer from which, count-ing from 29 to 29 , we can reach 2013 ? https://chingmath.fr chapExoCorrec/5940 sacados/5940 chapExoCorrec/5941 sacados/5941 chapExoCorrec/5982 sacados/5982 chapExoCorrec/8128 sacados/8128 chapExoCorrec/5942 sacados/5942
R00123456R10123456R20123456 directionAdirectionBdirectionCdirectionDdirectionEdirectionFdirectionGdirectionH01234567891011121314151617 3 Are there positive integers less than 2 013 from which it is possible to reach this number both by counting from 29 in 29 and by counting from 31 in 31 ? If so, find them all. E.5943 A counter consists of three toothed wheels, named R 0 , R 1 and R 2 each comprising 7 notches, numbered from 0 to 6 . This counter is designed so that : The wheels always turn from one notch to the next, in this order : 0 −→ 1 −→ 2 −→ 3 −→ 4 −→ 5 −→ 6 −→ 0 When the wheel R 0 makes a full turn, i.e. when it turns 7 notches then the wheel R 1 turns one notch. When the wheel R 1 makes one complete turn, i.e. when it turns 7 notches, then the wheel R 2 turns one notch. Initially, the wheels R 0 , R 1 and R 2 all display 0 . Between each question, the counter is reset, i.e. each wheel displays 0 again 1 We turn the wheel R 0 by 15 notches. What are the num-bers displayed by the wheels? 2 We turn the wheel R 0 by 100 notches. What are the numbers displayed by the wheels? 3 We turn the wheel R 0 until the wheel R 2 displays 5 for the first time. How many notches has R 0 been turned? 4 How many notches must R 0 be turned for the wheels to return to 0 at the same time for the first time? 5 We turn the wheel R 0 by 3580 notches. What are the numbers displayed by the wheels then? E.5968 Consider regular octagons, of the same center O . On the vertices of the central octagon, note the first eight non-zero integers. On the vertices of the second octagon, we inscribe the next 8 first integers, with a rotation of 45 degrees around the point O . And so on. . . Each integer is said to have a direction ( A , B , C , D , E , F , G or H relative to the origin O ) . For example, 1 has direction A , 2 has direction B . . . Here’s a figure representing the first four octagons : 1 What will be the first integer inscribed on the fourth octagon? Specify its direction. 2 Determine the first integer inscribed on the eighth oc-tagon. Specify its direction. https://chingmath.fr chapExoCorrec/5943 sacados/5943 R00123456R10123456R20123456 chapExoCorrec/5968 sacados/5968 directionAdirectionBdirectionCdirectionDdirectionEdirectionFdirectionGdirectionH01234567891011121314151617
E.5984 We start with a strictly posi-tive integer n : If n is even, we transform it into n 2 If n is odd ( n> 1 ) , we turn it into 3 n +1 . If n =1 , we stop. Examples : If n =6 , we get the sequence : 6 ↦−→ 3 ↦−→ 10 ↦−→ 5 ↦−→ 16 ↦−→ 8 ↦−→ 4 ↦−→ 2 ↦−→ 1 If n =13 , we get the sequence : 13 ↦→ 40 ↦→ 20 ↦→ 10 ↦→ 5 ↦→ 16 ↦→ 8 ↦→ 4 ↦→ 2 ↦→ 1 It has been observed using a computer program, that for ev-ery integer tested, the sequence always results in 1 . But this result has not yet been demonstrated. We can also be interested in the length of this sequence, which we’ll denote L ( n ) . For example: L (6)=9 and L (13)=10 . 1 Determine L ( n ) for integers from 1 to 12 . 2 Let p be an integer, consider the integer n =2 p . Express L ( n ) as a function of p . 3 Find an integer n between 2 2008 and 2 2009 such that : L ( n )=2012 . Hint : You could look for a number of the form 2 p × q . 4 Let k be a non-zero integer. a Show that : L (8 k +4)= L (6 k +4)+3 . b Similarly, show that : L (8 k +5)= L (6 k +4)+3 . c Show that : L (16 k +2)= L (16 k +3) E.8129 A set S of rationals is an arithmetic set (abbreviated EA) if for any pair ( a ; b ) with a and b belonging to S , there exists an element c of S such that one of the numbers a , b or c is the arithmetic mean (i.e. the half-sum) of the other two. We wish to determine all the strictly positive integers n for which there exists an EA with n elements. 1 a Are the following sets EA? Justify. S 1 = 0 ; 1 ; 2 S 2 = 0 ; 1 ; 2 ; 3 S 3 = 0 ; 1 ; 2 ; 4 S 4 = 1 2 ; 3 2 ; 2 ; 5 2 ; 7 2 b Show that there is no 2-element EA. What about sin-gletons (single-element set) ? c Give an EA having 5 elements, included in the interval 0 ; 2 , and containing 0 , 1 , 2 . 2 a Besides a + b 2 , what are the other two rationals to consider to verify that a pair ( a ; b ) of elements of S does not defeat the definition of an EA? b We wish to write an algorithm that tests whether a set is an EA. The set S is decoded as a list S = S [1] ;:::;S [ n ] of size n . For example, the arithmetic mean of i ième and j ième element of S is written as S [ i ]+ S [ j ] = 2 . In addition, we have a function Belongs(r,S) which returns True when the rational r belongs to the list S and Faux otherwise. Complete the skeleton of the function below (to be copied onto its composition sheet) so that it returns True if, and only if, S = S [1] ;:::;S [ n ] is an arithmetic set of length n . function TesterEA(S=[S[1],...,S[n]],n) Result ← True For i from 1 to n For j from 1 to n [...] End For End For Resend(Result) 3. Equations and algebra E.5971 Calculate the following sum without using the calculator: 1 2 − 2 2 − 3 2 + 4 2 + 5 2 − 6 2 − 7 2 + 8 2 + 9 2 − 10 2 − 11 2 + 12 2 + · · · + 2009 2 − 2010 2 − 2011 2 + 2012 2 E.5976 1 L , S and V being three positive real numbers, show that the triplets ( a ; b ; c ) solutions of the system : a + b + c = L ab + ac + bc = S abc = V are such that a , b and c are solutions of the equation : X 3 − L · X 2 + S · X − V = 0 2 Determine the dimensions of a right block whose sum of the lengths of all its edges is 20 cm , the sum of the areas of its six faces is 14 cm 2 and whose volume is 3 cm 3 . 3 What are the minimum and maximum volumes of a right block whose sum of the lengths of all edges is 20 cm , the sum of the areas of the six faces is 14 cm 2 . 4. Geometry E.5969 A and B are two points on a circle with center O and radius 5 such that AB =6 . The square PQRS is inscribed in the angular sector OAB so that : https://chingmath.fr chapExoCorrec/5984 sacados/5984 chapExoCorrec/8129 sacados/8129 chapExoCorrec/5971 sacados/5971 chapExoCorrec/5976 sacados/5976 chapExoCorrec/5969 sacados/5969
ABC ABCDOMNPC ABCDEABCDEF hBb ABCDFGH P is on radius [ OA ] ; S is on radius [ OB ] ; Q and R are two points on the arc of a circle connecting A and B . 1 Make a figure corresponding to the proposed situation. 2 Calculate the area of the square PQRS . E.5975 Five circles of radius 1 cm have been placed in a square as shown in the drawing. The circles are tangent to each other and tangent to the sides of the square. Determine the length of one side of the square. E.5977 A farmer has a large plot of land along a wall and a fence. Along the same wall, he wants to build a chicken coop in the shape of an isosceles triangle with vertex C . The chicken wire will not be placed against the wall. What is the maximum area of the chicken coop given that the wire mesh is 88 m long? (We can use the angle at the vertex and the fact that sin „ · cos „ = sin(2 „ ) 2 ) E.5980 We give a circle C with center O and two perpendicular diameters [ AB ] and [ CD ] . M being a point on segment [ AB ] , we trace ( CM ) which intersects the circle at N . The tangent at N to the circle and the perpendicular at M to ( AB ) in-tersect at P . Show that : OP = CM . E.8167 The metal workshop of a ship-yard cuts parts of various shapes from square steel plates that it orders from the rolling mill. To limit material losses and therefore production costs, the shop foreman must determine in advance the size of the square plates he needs to order according to the parts to be cut. For some orders, only the shape and surface of the parts to be cut are transmitted to them. In each of the following three parts, the cutting of certain types of parts is studied. These parts can be treated independently of each other. Where necessary, lengths should be rounded to the nearest mm , and areas to the nearest cm 2 . Part 1: Cutting triangular parts The workshop needs to produce a part in the shape of an equilateral triangle with a surface area of 20 m 2 . The workshop manager is considering two cutting solutions as illustrated in the following diagrams : Note a the side of the triangle and c the side of the square. 1 Schematic n o 1: a Express the height h as a function of a . b Deduce the side a of the square to be constructed to meet the constraints. c Calculate the area of steel lost with this method. 2 Diagram n o 2: a Justify that the angle ∠ BAE measures 15 o . b Deduce the side c of the square it must order. c Calculate the area of steel lost with this method. d What percentage of steel is gained compared to the first cutting prooposition? 5. Geometry and algebra E.5967 Reminder : Area of a trapezoid A = ( B + b ) × h 2 A rectangular ABCD pizza has crust on two consecutive sides, [ DA ] and [ AB ] . We’re looking at how to divide the pizza into three fair pieces : each slice must have the same crust length and the same area. In each situation, we fix the length of the short side AD =1 . 1 In the particular case opposite, it is assumed that the division made is fair. What is the length AB ? Determine lengths : DF , FH and HC . 2 We generalize the situation by posing AB = L (and always assuming that AD =1 ) . https://chingmath.fr chapExoCorrec/5975 sacados/5975 chapExoCorrec/5977 sacados/5977 ABC chapExoCorrec/5980 sacados/5980 ABCDOMNPC chapExoCorrec/8167 sacados/8167 ABCDEABCDEF chapExoCorrec/5967 sacados/5967 hBb ABCDFGH
ABCDFGHEsituation1:L>2 ABCDFGHEsituation2:L<2 IJKO25%35%40% ABOMNCC LABCDEFG xyBAMHJDC xyBAMHJEC xyABCDHJKMNP Determine, for each situation below, the useful lengths for cutting the pizza equally. E.5970 In a square of side 10 cm , we want to make a statistical graph in which the areas of the 3 parts must be pro-portional to the frequencies they represent ( O is the center of the square) . The point I is 2 cm from the near-est vertex. Calculate the distances of J and K to the nearest vertices of the square. E.5978 Consider a triangle OAB equilat-eral. Let R be the measure of the sides of the triangle OAB . Note C the circle with center O passing through point A . Let x be a real number such that 0 <x< R 2 . We place the point M on the segment [ AB ] verifying: AM = x . Consider the circle C with center M and tangent to the circle C at point N . Express the radius of the circle C as a function of R and x . E.8351 Disk suite footprint A series of wooden discs of the same thickness, whose radii can be different, is placed on a shelf. The slight slope given to the shelf ensures contact each disc is tangent to one or more neighbors. The aim of the problem is to study arrangements that minimize the L footprint. 1 Case of two disques Two disks (centered at A and B , of radii R and r , such that r R ) are tangent at H and J respectively to the line ( xy ) , see figure below : a Express as a function of R the maximum radius of the small disk r for which the footprint created by the two disks is the same as that created by the large one alone (see figure belowbelow) : Hint: we may have to solve a quadratic equation with unknown x = x b If r is greater than this maximum, establish that the footprint created by the two disks is : L = R + r + 2 · R · 2 Case of three disks : a Three disks, with centers A , B and C and radii p , q and r are tangent to the same straight line and tangent two to two (see figure below) . The space requirement is therefore equal to that created by the two outer disks alone. Express the radius q of the inner disk in terms of the radii p and r of the outer disks. b In this question, all three disks are tangent to the same line, and, from left to right (in the order of their cen-ters) , the circle with center A is tangent to the circle with center B , itself tangent to the circle with center C , which has no common point with the circle with center A . Express the crowding created by these three disks as a function of their radii p , q and r . c We swap the places of the disks of center B and C and assume as in b that the central disk separates its two neighbors. What is the new footprint? Still under the assumption made in b and c , we further assume that p q r . The disks are placed in the order A - B - C , A - C - B and B - A - C (arrangement in descending order of radii, or smallest in middle or largest in middle) . d Show that the layout A - B - C creates the maximum footprint. e Show that the layout B - A - C creates the minimum foot-print if and only if : q − r p − q . https://chingmath.fr ABCDFGHEsituation1:L>2 ABCDFGHEsituation2:L<2 chapExoCorrec/5970 sacados/5970 IJKO25%35%40% chapExoCorrec/5978 sacados/5978 ABOMNCC sacados/8351 Extrait Asie Pacifique 2019 LABCDEFG xyBAMHJDC xyBAMHJEC xyABCDHJKMNP
431432 6. Probability E.5983 A tetrahedral die has four faces like the one shown opposite. When such a die is rolled, the result is the number inscribed closest to the tetrahedral base. In our example, the tetrahedral die fell on face 4 . Antoine, Baptiste, Cyril and Diane play with four regular, balanced tetrahedral dice, but which are not numbered in the usual way. Thus, Antoine’s die has four sides numbered 1 , 6 , 6 and 6 . With this die, the number 1 is obtained with probability 1 4 and the number 6 with probability 3 4 . Baptiste’s die is numbered 4 , 4 , 5 and 5 ; Cyril’s 3 , 3 , 3 and 8 ; and finally, dianne’s 2 , 2 , 7 and 7 . 1 Each player rolls this tetrahedral die once. Who has the best chance of getting a number greater than or equal to 6 ? 2 Players begin a series of duels : Antoine plays Baptiste, Baptiste plays Cyril, Cyril plays Diane, Diane plays An-toine. The winner of each duel is the player with the highest score. a Show that in the first duel Antoine wins against Bap-tiste with probability 3 4 . b Give the players’ winning probabilities in the other three duels. 3 Antoine, Baptiste, Cyril and Diane simultaneously throw their dice. The player with the highest number wins. a Show that the probability of Baptiste winning is equal to 3 32 . b Who has the best chance of winning this game? E.8130 Let n be a natural number greater than or equal to 2 . There is an urn containing n balls that can be of different colors. The game consists of randomly extracting a ball from the urn, then without returning it to the urn extracting a second ball from the urn. The player has won when the two balls drawn are the same color. It is assumed that on each draw, all the balls in the urn have the same probability of being drawn. The game is said to be fair when the probability P G that the player wins is equal to 1 2 . 1 a Demonstrate that if the urn contains 10 balls of which 4 are white and 6 are red then P G = 7 15 . b Calculate P G when the urn contains 12 balls includ-ing 4 white, 6 red and 2 black. 2 In this question, the urn contains 6 red balls and other balls that are all white. a Let x be the number of white balls contained in the urn. Show that : P G = x x − 1 +30 x +6 x +5 b How many white balls would be needed to make the game fair? 3 In this question, the urn contains only balls of two differ-ent colors. a It is assumed that the urn has the configuration ( a ; b ) , i.e. it contains, for example, a red balls and b white balls. Show that the game is fair when n = a − b 2 b Reciprocally, show that if n is the square of an integer p then there exist two natural numbers a and b with a b which we will express in terms of p such that the configuration ( a ; b ) leads to a fair game. c Give six ordered pairs ( a ; b ) leading to a fair game. 7. Yearbooks for all series E.5929 An integer is said to be digis-ible when the following three conditions are verified : none of its numbers is zero; it is written with all different digits ; it is divisible by each of them. For example, 24 is digisible because it is divisible by 2 and by 4 . 324 is digisible because it is divisible by 3 , by 2 and by 4 . 32 is not digisible because it is not divisible by 3 . Recall that an integer is divisible by 3 if, and only if, the sum of its digits is divisible by 3 . 1 Suggest another number digisible with two digits. 2 a Give all single-digit factors of the number 1000 . b Deduce a four-digit digisible number. 3 Let n be an integer digisible written with a 5 . a Demonstrate that 5 is the digit of its units. b Demonstrate that all the digits of n are odd. c Demonstrate that n is written with at most four digits. d Determine the largest integer digisible written with one 5 . https://chingmath.fr chapExoCorrec/5983 sacados/5983 431432 chapExoCorrec/8130 sacados/8130 chapExoCorrec/5929 sacados/5929
MH(D R¸ OA EABCDO E.5930 Definition: The distance between a point M and a line ( D ) is called the distance MH , where H is the point of intersection of ( D ) with the line perpendicular to ( D ) passing through M . In the figure opposite, if the radius of the disk is R , and if the angle of the shaded sector measures ¸ (in degrees) , then the area of the shaded portion of the disk is : ı · ¸ · R 2 360 . In part 2 of the exercise, we will con-sider the distance from point M to segment [ BC ] to be the distance from point M to line ( BC ) . Part 1 Let C be a circle with center O , A a point on this circle, and D the disk bounded by this circle. 1 Reproduce the figure and represent the set of points on the disk that are equidistant from O and A . 2 Shade the set of points on the disk that are closer to O than to A . 3 Let M be a point chosen at random with equal probabil-ity on the surface of the disk D . What is the probability that M is closer to O than to A ? Part 2 Let ABCD be a rectangle with length AB =20 cm and width BC =12 cm , centered at O Let E be a point located inside the rectangle, close to A , at 2 cm from each edge (as shown in the figure below, which is not to scale) . Let M be a point determined randomly and equally likely within the rectangle ABCD . 1 What is the probability that M is closer to side [ BC ] than to side [ AD ] ? 2 a Reproduce the rectangle and represent all the points inside the rectangle that are equidistant from sides [ AB ] and [ BC ] . b Shade the set of points inside the rectangle that are closer to side [ BC ] than to side [ AB ] . c What is the probability that M is closer to side [ BC ] than to side [ AB ] ? 3 What is the probability that M is closer to side [ AB ] than to sides [ BC ] , [ CD ] , and [ DA ] ? 4 What is the probability that M is closer to O than to E ? 5 What is the probability that M is closer to O than to the four vertices A , B , C , and D ? E.5931 I. A first algorithm Here’s an algorithm applicable to three-digit integers where the hundreds digit is not equal to the units digit: Step 1 : Inverser order of digits (e.g. 275 becomes 572 ) Step 2 : Calculate the difference between the larger and smaller of these two numbers. Step 3 : Reiterate step 1 on the number obtained. Step 4 : Adding these last two nombres 1 a Apply the algorithm to numbers 123 , 448 and 946 . b What can we conjecture? 2 To implement this algorithm, the 2 step, implicit when performing ˇ calculations at mainı , requires dissociating the integer entered : isolate the units digit, the tens digit and then the hundreds digit. Complete the following function, derived from an algo-rithm, whose argument n is a 3 -digit natural integer and whose role is to perform this dissociation. For this func-tion, a is the hundreds digit, b the tens digit and c the units digit of the number n we wish to decompose. Function f(n) a ← 0 b ← 0 c ← 0 As long as n 100 a ← a+1 n ← n − 100 End As long as As long as n ...... b ← ...... ...... ← ...... End As long as c ← ...... Renvoyer a ; b ; c 3 We now set out to prove the conjecture established in 1 b . To do this, we choose a three-digit number that we write abc où a , b and c are therefore integers between 0 and 9 and represent the hundreds digit, tens digit and units digit of n respectively. Without loss of generality, we can assume a<c a Decompose abc according to the powers of 10 . b Give the number obtained after step 1 in its decom-posed form. https://chingmath.fr chapExoCorrec/5930 sacados/5930 MH(D R¸ OA EABCDO chapExoCorrec/5931 sacados/5931
ABCD c Show that the number obtained after step 2 can be written : ( c − a − 1) × 100 + 9 × 10 + 10 + a − c d Apply steps 3 and 4 and conclude. II. Kaprekar’s algorithm In mathematics, the Kaprekar algorithm is an algorithm dis-covered in 1949 by the Indian mathematician D.R. Kaprekar for four-digit integers, but which can be generalized to all in-tegers. We will study it here for three-digit integers, all of which are distinct. Kaprekar’s algorithm consists in associating with any integer n another number K ( n ) generated as follows : Step 1: From the digits that make up n , form the largest number possible. This is noted G Step 2: From the digits that make up n , form the smallest number possible. This is noted P Step 3: K ( n ) is then equal to the difference G − P For example, starting from 539 , we have : G =953 and P = 359 . Therefore : K (539)=953 − 359=594 . 1 a Calculate K (198) , K (357) and K (495) . b Write an algorithm whose input is a number n with three digits, all distinct, and whose output is K ( n ) . For separating the digits of the units, tens and hun-dreds, we can repeat the algorithm from part I . c Apply Kaprekar’s algorithm, starting with the number 198 and iterating as many times as necessary. Repeat with other three-digit numbers, all distinct. What can be conjectured? 2 We propose to prove the conjecture made in question 1 . To do this, we choose an integer n consisting of three dig-its and write abc où a , b and c are therefore all distinct integers between 0 and 9 , which respectively represent the hundreds, tens and units digits of n . a Explain why we can, without loss of generality, assume that a<b<c . b Show that : K ( n )=99( c − a ) . c Demonstrate the conjecture and specify the maximum number of iterations required. E.5932 A ladybug moves along the sides of a square ABCD starting from point A . She can walk backwards if she wishes. Any path taken by the ladybug along a side of the square is called a displace-ment. A walk is made up of displace-ments, thus : A −→ B −→ A −→ D −→ C is a walk of four moves whose arrival is point C . Part A In this part, the ladybug moves randomly along the sides of a square ABCD and all its movements are considered equiprob-able. 1 a Can the ladybug reach the point B in three moves? b What are the possible arrivals for a three-move walk? c What are the possible arrivals if the walk has an even number of moves? d What are the possible arrivals if the walk has an odd number of moves? 2 In this question, the ladybird makes two moves. Possibly using a tree, calculate the probability of the event A 2 : ˇ the ladybug arrives in A by making two dé-placements ı. 3 Reproduce and complete the table below : Nombre de déplacements de la marche 1 2 3 4 5 Probabilité que la coccinelle arrive en A Part B In this part, the ladybug still moves along the sides of the square ABCD starting from the point A but is twice as likely to move vertically as horizontally. She can always walk back-wards if she wishes. On the other hand, she decides to stop as soon as she returns to A . 1 In this question, the ladybug makes exactly two moves. a Calculate the probability of the event A 2 : ˇ the lady-bug arrives at A by performing two déplacements ı. b Calculate the probability of the event C 2 : ˇ the lady-bug arrives in C by performing two déplacements ı. 2 a Calculate the probability of the event A 4 : ˇ the lady-bug arrives at A by making exactly four déplacements ı. b Calculate the probability of the event A 6 : ˇ the lady-bug arrives at A performing exactly six déplacements ı. 3 Let n be a natural number greater than or equal to two. a Note A 2 n the event : ˇ the ladybug arrives in A by performing exactly 2 n déplacements ı and P ( A 2 n ) the probability of this event. Express P ( A 2 n ) as a function of n . Let q be a real number other than 1 and n a non-zero natural number. Recall that : 1 + q + q 2 + · · · + q n = 1 − q n +1 1 − q https://chingmath.fr chapExoCorrec/5932 sacados/5932 ABCD
„oER ER b We note G 2 n the event : ˇ the ladybug arrives in A making at most 2 n déplacements ı. Express as a func-tion of n the probability of G 2 n noted P ( G 2 n ) . c What is the smallest integer n such that : P ( G n ) 0.9999 ? E.5938 It is assumed that there exists a function f defined on the set of natural integers N verifying the property: ( E ) : for all x and y of N , f ( x + y )= f ( x ) · f ( y ) − x · y Preliminary Show that f (0)=1 . The results can be admitted in the follow-ing parts. A. Study of a first example : We assume here that : f (1)=3 . 1 Calculate f (2) then f (3) . 2 Show by two separate calculations that f (4)=60 and that f (4)=63 . Conclude. B. Study of a second example : We assume here that : f (1)=0 . 1 Calculate f (2) , f (3) and f (4) . 2 Conjecture the expression of f ( n ) as a function of n . 3 Demonstrate this conjecture. 4 Prove that for the function found at 2 and 3 the property ( E ) is indeed verified. C. General case First part : we note f (1) = a 1 Express f (2) and f (3) in terms of a . 2 Express f (4) as a function of a in two different ways. 3 Deduce that : a =0 or a =2 . Second part : we study the second case : It is assumed that : f (1)=2 . Express f ( n ) as a function of n . 8. Annales S series E.5934 Pierre and his daughter Eloise are walking along a horizontal road. At a point R , this road descends making an angle „ of 5 o with the horizontal (see figure) Eloise, whose eyes are 1.6 meter from the ground, stops at a point E , 24 meters from the R point. His father continues walking, passes the point R then enters the sloping part of the road. 1 When he is 86 meters from R , he disappears from his daughter’s view. Determine Pierre’s height. 2 On the sloping part of the road, poles 6.5 meters high are planted vertically every 28 meters, as in the diagram below The foot of the first post is 28 meters from the point R . It is assumed that the posts cannot hide from each other. How many poles can Eloise see from where she is? 3 What is, in reality, the measure of the angle „ , given that Eloise can only see 5 poles? We can use the formula : 1+ tan „ 2 = 1 cos „ 2 We’ll give an approximate value of „ to the nearest 10 − 3 . E.5936 Three distinct natural num-bers a , b , c ordered strictly croissant , a<b<c , are in arith-metic progression if : c − b = b − a We then say that ( a ; b ; c ) is an arithmetic triplet. 1 Complete the following arithmetic triplets : a (57 ; 101 ; : : : ) b 57 ; : : : ; 101 c : : : ; 57 ; 101 2 a Can we find an arithmetic triplet ( a ; b ; c ) whose sum is 2012 ? b How many arithmetic triplets are there ( a ; b ; c ) with sum 2013 ? 3 We randomly take three integers a , b , c in 1 , 2 , 3 , . . . , 10 with a<b<c . What is the probability that ( a ; b ; c ) is an arithmetic triplet? 4 Recall that a natural number p is prime if p 2 and if its only positive factors are 1 and p . a What are the five smallest prime integers? b Give an arithmetic triplet ( a ; b ; c ) consisting of prime integers. Is this the triplet for which the sum a + b + c is minimal? You are asked to justify your answer. If not, find the three prime integers a<b<c in arithmetic progression and of minimal sum. c Can-we find an arithmetic triplet ( a ; b ; c ) consisting solely of prime integers and whose sum a + b + c is 366 ? 5 Let n be a natural number greater than or equal to 3 . We are given a list [ a 1 ;a 2 ; : : : ;a n ] , of integers arranged in strictly ascending order. We want to know whether three of its consecutive terms form an arithmetic triplet. a In this question only , the list is [1.3.6.10.15.21.27.32.39.45] . Does it contain an arithmetic triplet formed by three consecutive terms? b We return to the general case of a list [ a 1 ;a 2 ; : : : ;a n ] , of https://chingmath.fr chapExoCorrec/5938 sacados/5938 chapExoCorrec/5934 sacados/5934 „oER ER chapExoCorrec/5936 sacados/5936
24 integers arranged in strictly ascending order. Write an algorithm that displays, if it exists, the first arithmetic triplet formed by three consecutive terms. c With the calculator, program and then test this algo-rithm on the list [ a 1 ;a 2 ; : : : ;a 20 ] où, for 1 k 20 : a k = − k 3 + 36 k 2 + 9 k We don’t ask you to check that this list is made up of natural integers arranged in strictly ascending order. E.8131 We roll two dice D a and D b successively and independently; we consider the total points thus brought back and its probability of occurrence. For ex-ample, with two standard six-sided dice, if the first roll pro-vides 1 , and the second 1 as well, the total will be worth 1+1=2 , and its probability of occurrence 1 12 . The statistical study of these sums can be used in certain games of chance, such as Goose. The dice considered are tetrahedral, as in the sketch opposite. In question 1 and 2 , their four faces are standard, num-bered 1 , 2 , 3 , 4 . 1 Give the three ways of obtaining for total 6 , deduce that the probability of obtaining a total of 6 is 3 16 . 2 Give the various totals that can be reached in this way, and then their probabilities of occurrence. What do the coefficients of the polynomial expression : indicate? P ( x )= x + x 2 + x 3 + x 4 2 when expanded? Explain. For more originality, we now take non-standard dice : a D 1 die with faces numbered 1 , 1 , 2 , 5 and a D 2 die with faces numbered 1 , 4 , 4 , 4 . 3 What is the probability of getting a total of 6 ? Generally speaking, the die D a has four sides with values a 1 , a 2 , a 3 , a 4 check 1 a 1 a 2 a 3 a 4 and are stored in an ar-ray t a = a 1 ;a 2 ;a 3 ;a 4 . Similarly, the die D b has four faces b 1 , b 2 , b 3 , b 4 verifying 1 b 1 b 2 b 3 b 4 and stored in the table t b = b 1 ;b 2 ;b 3 ;b 4 . We define the polynomial quantities: A ( x ) = x a 1 + x a 2 + x a 3 + x a 4 et B ( x ) = x b 1 + x b 2 + x b 3 + x b 4 For example, the dice in question 3 give rise to : t a = [1.1.2.5] t b = [1.4.4.4] A ( x ) = 2 x + x 2 + x 5 B ( x ) = x + 3 x 4 4 Determine t a , t b , A ( x ) , B ( x ) dice attached D a and D b faces 1 , 2 , 2 , 3 and 1 , 3 , 3 , 5 . 5 The following algorithm (which it will be possible to ex-tend to large dice) returns the coefficient of x k in the product : x p x b 1 + x b 2 + x b 3 + x b 4 . Coef ← 0 For j ranging from 1 to 4 If p+t b [j]=k then Coef ← Coef+1 End If End For Return Coef Modify this algorithm so that it returns the coefficient of x k in the product A ( x ) · B ( x ) of two n -sided dice. Colonel George Sicherman (USA, XX e century) searched for pairs of nonstandard D a and D b dice whose face sums obey the same probability laws as those of two standard dice. Here’s how he was able to proceed, first on four-sided dice. 6 The notations are repeated : 1 a 1 a 2 a 3 a 4 , 1 b 1 b 2 b 3 b 4 and P ( x ) = x + x 2 + x 3 + x 4 2 a Justify that : A ( x ) · B ( x )= P ( x ) . b Factorize x + x 2 + x 3 + x 4 showing only quantities of de-grees 1 and 2 . c What is A (0) , A (1) , B (0) , B (1) worth? d Suggest a possible and viable distribution of P factors between A and B , defining a good pair of non-standard dice. 7 Determine a pair of non-standard dice with 6 faces whose sum of faces obeys the same probability law as that of two standard dice (with faces: 1 , 2 , 3 , 4 , 5 , 6 ) 9. Annales series other than S E.5933 Mr. and Mrs. Logic’s sextu-plets are in the same class at 2 nde . At the end of the day, after taking a math test, they go home and show their parents the answers they gave to the various questions : Alix Béa Carol Del-phine Émile Félix Question 1 150 700 150 100 700 150 Question 2 103 101 101 101 103 35 Question 3 101 732 107 101 101 107 Question 4 34 125 216 28 34 34 Question 5 216 216 27 55 25 103 ˇDad, can you tell us how many we each got?ı I’d be happy to, but you haven’t given me the questions! We don’t have them ; we had to hand in the test with the answers. I remember we had to find the smallest prime number after 100 , said Bea. We also had to calculate the volume of a cube with sides that were integers, I can’t remember which ones, added Felix. We also had to find the age of the captain of some boat or other, remembered Carol. Is that all you can remember, asked their father? https://chingmath.fr chapExoCorrec/8131 sacados/8131 24 chapExoCorrec/5933 sacados/5933
ABCIEF ABCK Figure 2Figure 1 Yes, but after quickly looking at the papers, the teacher told us that one of us had got everything right. . . and another had got everything wrong! After a while, their father tells them that he knows their grades What are these grades? (Each correct answer is worth 4 points) and how old is the captain? Explain the reasoning that led to the result. Definition: A prime number is a strictly positive inte-ger that has exactly two divisors : 1 and itself. The first prime numbers are: 2 ; 3 ; 5 ; 7 . . . E.5935 An association wants to create a logo. This logo has been designed from the following construction : ABC is a right-angled triangle in A , we pose : AC = x ; AB = y ; BC = z , we have drawn the semicircles of diameters [ AB ] , [ AC ] , [ BC ] and the square AEFI such that E ∈ [ AB ) , F ∈ [ BC ) and I ∈ [ AC ) . 1 For this question we consider the following figure : a Calculate the length of the side of the square AEFI as a function of x and y . b What can be said about the point I if the triangle ABC is isosceles? (justify) c We assume y =4 . Can the area of the square AEFI be equal to 9 ? (justify) 2 Let K be the foot of the height from A of the triangle ABC . We pose AK = h . We therefore have the following figure : a Justify that : ( x + y ) 2 = z 2 +2 · z · h . b Similarly express ( x − y ) 2 as a function of z and h . Show that h is less than half of z . c Is it possible that : z =10 and h =4.8 ? If yes, determine the values of x and y . 3 Compare the area of the triangle ABC to the area of the shaded surface. E.8132 For all natural numbers m and n , we call the triangle of m by n , and we note m Δ n , the number defined by the following rules, which we admit are possible : 0Δ n = n + 1 n Δ0 = n − 1 Δ1 as soon as n =0 ; n +1 Δ m +1 = n Δ ( n +1)Δ m Warning, m Δ n is not necessarily equal to n Δ m . Some results 1 a Show that : 1Δ0=2 et 1Δ1=3 b Calculate 1Δ2 c More generally, determine, for any natural number n , the value of 1Δ n . We can pose u n =1Δ n and check that the sequence u n is arithmetic. 2 a Calculate 2Δ0 , 2Δ1 and 2Δ2 . b Justify, that for any natural number n : 2Δ n =2 n +3 3 a Calculate 3Δ0 , 3Δ1 and 3Δ2 . b Demonstrate that, for any natural number n , 3Δ n is equal to 2 n +3 − 3 . We can pose v n =3Δ n and show that, for any n greater than or equal to 1 : v n = 2 · v n − 1 + 3 . Illustration from 3Δ n One artist illustrated the values 3Δ0 and 3Δ1 in this way: 4 Draw on the copy a third figure that would logically com-plete this sequence of drawings and illustrate the value of 3Δ2 . 5 Suppose the side of a square in figure 1 measures 1 cm . a Determine the respective areas of figures 1 and 2 . b What would be the area of the figure illustrating 3Δ n ? Any overlaps will be ignored. https://chingmath.fr chapExoCorrec/5935 sacados/5935 ABCIEF ABCK chapExoCorrec/8132 sacados/8132 Figure 2Figure 1
10. Unclassified financial years E.7317 Thermal exchanges In architecture, the ratio of the outer surface - including the base in contact with the ground - of a building is called its compactness factor, measured in m 2 , to its volume, measured in m 3 . The compactness factor c = S v , expressed in m − 1 , gives a first rough assessment of the thermal performance of a hous-ing construction. 1 Compactness calculations for some usual volumes, drawn below. a Determine the compactness factor of the cube with side a . b Determine that of a half-sphere of radius r . Recall that the volume of a sphere of radius r is 4 3 · ı · r 3 and that its surface area is 4 · ı · r 2 . c Determine that of a regular square-based pyramid of side a , and vertical height a . d How do you think the compactness factor relates to a building’s thermal performance? 2 We propose to study the compactness factor of a right block of volume 1 whose dimensions are x , y and z . a Verify that for all numbers a , b and c : a 3 + b 3 + c 3 − 3 · a · b · c = 1 2 · a + b + c a − b 2 + b − c 2 + c − a 2 b Deduce that for all positive real numbers a , b and c : a 3 + b 3 + c 3 3 · a · b · c c Deduce that for all positive real numbers A , B and C whose product is equal to 1 : A + B + C 3 d Show that the compactness factor of this paving stone is : c = 2 · 1 x + 1 y + 1 z E.7318 Liber abaci 4 000 years ago, the ancient Egyptians used a very surpris-ing arithmetic property in calculus : any rational number p q strictly positive is written as a sum of unit fractions, i.e. in-verses of positive integers, all different from each other. Since then, such a decomposition has been called an ˇEgyptian writ-ingı. Thus, la somme 1 6 + 1 17 + 1 102 est-elle une ˇEgyptian writ-ingı of the quotient 4 17 , while the sums 1 17 + 1 17 + 1 17 + 1 17 and 1 17 + 3 17 are not. Many questions about these writings remain, to this day, open. 1 Why are the last two decompositions given in the pream-ble not ˇ Egyptian writing ı? Propose an Egyptian writ-ing of 2 3 involving two unit fractions, then another of 2 3 involving three. 2 An algorithm Let p and q be two integers such that 0 <p<q . The quotient p q is therefore an element of 0 ; 1 . k ← 1 p 1 ← p q 1 ← q . As long as p k =0 Determine the smallest positive integer n k such as: 1 n k p k q k . Ainsi : 1 n k p k q k < 1 n k − 1 p k+1 ← p k · n k − q k q k+1 ← q k · n k Ainsi : p k +1 q k +1 = p k q k − 1 n k Increment k i.e. increase the value of the counter k by one unit. End of As long as a Here, we run the algorithm on the quotient p q = 4 17 . At the start of the first loop round : k =1 ; p 1 =4 ; q 1 =17 . We then determine n 1 =5 . Then p 2 =3 , q 2 =85 and k is 2 before entering the second loop. Continue until complete stop. Que vaut 1 n 1 + 1 n 2 + 1 n 3 + 1 n 4 ? Are the four unit fractions distinct? b It is assumed that the algorithm ends at the end of the N ème loop. Justify that it yields an ˇ Egyptian writing ı of the quotient p q . c Justify clearly that the algorithm cannot be unlimited. This algorithm can therefore give a ˇ writing égyptienne ı of any rational number element of 0 ; 1 . It belongs to a class of algorithms known as ˇ gloutons ı and is attributed to Leonardo of Pisa, author of the Liber abaci (1202) . The adjective ˇgloutonı applies to algorithms making, at each step, an optimal choice. Global optimality is not necessar- https://chingmath.fr chapExoCorrec/7317 sacados/7317 chapExoCorrec/7318 sacados/7318
1234512345 12A12345B1234C1234D ABCDEGrst ily achieved, as evidenced by the two decompositions of 4 17 encountered in this problem. E.7319 There are n counters vertically. They are black on one side, white on the other, and are numbered from 1 to n . At the start of the game, each pawn randomly presents its black or white face. At each move - which we call a operation throughout the sequel - one of the pawns and all its neighbors on top are turned over. The drawing opposite shows an example of the change made to an initial configuration by an operation with the third to-ken. The aim of the game is to find a sequence of operations such that all pawns show their white face. 1 Does the order of two operations matter? 2 What is the combined effect of two identical operations? 3 Indicate the numbers of the pawns to be turned over to see only white faces, in the situations shown below. 4 The following algorithm for a configuration of n squares is given : For k ranging from n to 1 in steps of − 1 If token k is black, perform an operation with this token End Pour a Explain why this algorithm whitens the column in a minimum of operations. How many operations does it implement at most? b Give an example of a configuration of n boxes requir-ing n operations. E.6753 The ˇ K ı brand padlock man-ufacturer wants to print a logo for its company. This logo takes the form of a stylized capital letter K , in-scribed in a square ABCD , of side one unit of length, and meeting the following conditions C 1 and C 2 : Condition C 1 : the letter K must consist of three lines : one of the lines is the segment [ AD ] ; a second line has as ends the point A and a point E of the segment [ DC ] ; the third line has as its end point the point B and a point G located on the second line. Condition C 2 : the area of each of the three surfaces bounded by the three lines drawn in the square must be between 0.3 and 0.4 , the unit of area being that of the square. These areas are noted r , s , t in the figures below. A design studio offers the design shown opposite. To carry out the following study, we place ourselves in the orthonormal reference frame A ; −−→ AB ; −−→ AD . The three lines are segments and the three areas are equal: r = s = t = 1 3 Determine the coordinates of the oints E and G . https://chingmath.fr chapExoCorrec/7319 sacados/7319 1234512345 12A12345B1234C1234D chapExoCorrec/6753 sacados/6753 ABCDEGrst
E.9548 In this problem, we consider only non-zero natural numbers. For each of these integers, we number the digits of its decimal writing from left to right. The first digit on the left cannot be 0 . For example, for the number 3021 , the digit 3 receives the number 1 , the digit 0 the number 2 , the digit 2 the number 3 and the digit 1 the number 4 . We call ˇ large even ı any number in which each digit in even position, if any, is at least auassi greater than its directly neighboring digits (if any) . We call ˇ large odd ı any number in which each digit in odd position is at least as large as its directly neighboring digits (if it has any) . For example: number 3021 is a large odd, but not a large even ; the numbers 3 , 2 , 7 and 777 are both large even and large odd ; the number 2019 is neither a large even nor a large odd. 1 Is the number 384 957 a large even? A large odd? 2 Determine the numbers that are both large even and large odd. 3 Among numbers written with two digits, are there more large even or large odd? 4 a Can the number 3021 be written as the sum of two large odd numbers with the same number of digits? b Can the number 3021 be written as the sum of two large even numbers with the same number of digits? 5 Prove that any integer can be written as the sum of two large odd (nothing is imposed here as to the number of digits of these two large odd) . 6 Show that any large odd number strictly less than 100 can be written as the sum of two large even numbers (with no constraints on the number of digits of these large even numbers) . 7 Determine the smallest odd great greater than or equal to 2 that cannot be written as the sum of two great peers (without constraint as to the number of digits of these great peers) . 8 Complete the pseudocode below (or be inspired by it) to write an algorithm (to be transcribed on your copy) , which, starting with an array ˇ T ı representing a number ˇ N ı (for example 384957) from ˇ nb ı digits (here 6 ) . nb = 6 T=[3, 8, 4, 9, 5, 7] result = 1 i = 1 while (result == 1) and (i <=nb): ... i = i+2 print(r) https://chingmath.fr sacados/9548