Đề thi giao lưu học sinh giỏi cấp Huyện năm học 2022-2023 song ngữ Toán 7 (Có đáp án)

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1. PEOPLE’S COMMITTEE OF ENGLISH MATHEMATICS COMPETITION VINH BAO DISTRICT FOR GRADE 7 STUDENTS SCHOOL YEAR: 2022-2023 Time limit: 120 minutes Section A Write the answer to the question in the dot. Question 1. The remainder in the division42021 by the number 7 is ............................................ Question 2. Given A= 3 + 32 + 33 + ............... + 32015 + 32016 . Find the last digit of A. ................................................................................................................................................ Question 3. How many pairs of integers (x; y) satisfy: x + y = 4 - 5xy ................................................................................................................................................ Question 4. Tom has a bag containing 12 blue balls, 6 red balls and 8 yellow balls. After adding 1 some red balls, Tom notices that now there is a chance that a red ball is randomly selected from 3 the bag. How many red balls did Tom add to the bag? ................................................................................................................................................ Question 5. Given the polynomial f(x) = x6 – 2021x5 +2021x4 - 2021x3+ 2021x2 - 2021x + 2021. Find f(2020). ......................................................................................................................................... Question 6. In grade 7 there are 3 classes 7A, 7B, 7C and there are 105 students. Suppose that the 7 3 student of 7A is equal to of the student of 7B, the students of 7B is equal to of students of 7C. 6 4 How many student are there in 7A? ................................................................................................................................................ Question 7. Let ABC be a triangle AC = 3cm, = 135 . Area of triangle ABC is 6cm. calculate the length of side AB? ................................................................................................................................................ Question 8. Let ABC be an isosceles right triangle at A. Draw Cx is the bisector of the exterior angle at the vertex C of triangle ABC. The opposite ray of ray Cx is Cx'. The line xx' intersects AB at D. Calculate angle BDC ................................................................................................................................................ Question 9. How many natural numbers abcd satisfy cd < ab. ................................................................................................................................................ Question 10. A foothbal few right through the glass window of the staff room. One of the four people A, B, C, D breaks the glass.
2. A : D broke it. B: D broke it. C: I didn't do it. D: B is lying. Knowing only one person told the truth, who broke the glass? ................................................................................................................................................ Section B Write your detailed solutions on the exam paper: Question 1. 1.Given three rectangles, know that the area of the first and the second is proportional to 4 and 5, the area of the second and the third are proportional to 7 and 8, the first and the second has the same length and the sum of their widths is 27 cm, the second and the third have the same width, the length of the third is 24 cm. Calculate the area of each of those rectangles. 2. Find x know: 3x - 3 + 2x + (- 1)2016 = 3x + 20170 Question 2. Let ABC be a triangle with 3 acute angles (AB < AC). Draw outside triangle ABC the equilateral triangles ABD and ACE. Let I be the intersection of CD and BE; K be the intersection of AB and CD a) Calculate angle DIB b) Let M, N be the midpoints of CD and BE respectively, proving that triangle AMN is equilateral. Question 3. 1. Find all primes p so that : 2 p p2 is also a prime number. 2. Find x, y integer satisfy 3xy – 5 = x2 + 2y. Question 4. 1 1. For two real numbers x, y satisfy x + y = 2. Find the min value of A = y2 + xy + x2 - x 2 2. In a square board consisting of 5 x 5 squares, people write in each square only one of 3 numbers 1; 0; - 1 .Prove that the sums of 5 numbers in each column, row, and diagonal must have at least two equal totals. --------------- The end -----------------
3. PEOPLE’S COMMITTEE OF VINH BAO ENGLISH MATHEMATICS VINH BAO DISTRICT COMPETITION FOR GRADE 7 STUDENTS VINH BAO SECONDARY SCHOOL YEAR: 2022-2023 SCHOOL Section A. 10 marks for each Question 1 2 3 4 5 6 7 8 9 10 Solution 2 0 2 4 1 35 4 22,5o 4905 C Section B. 50 marks for each Question Solutions Marks 1. From Let's call the area of the three rectangles S1;S2;S3 respectively, length and width respectively d1,r1;d2 ,r2 ;d3 ,r3 10 Condition : S1;S2;S3 ; d1,r1;d2 ,r2 ;d3 ,r3 > 0 S1 4 S2 7 we have ; and d1 d2 ;r1 r2 27;r2 r3 ,d3 24 S2 5 S3 8 Since the first and the second has the same length and the sum of their Q.1 widths is 27 cm, 1.1 S 4 r r r r r 27 1 1 1 2 1 2 3 (30 point) S2 5 r2 4 5 9 9 10 Deduce r1 12cm,r2 15cm (accept) Since the second and the third have the same width S2 7 d2 7d3 7.24 d2 21cm S3 8 d3 8 8 2 So S2 d2r2 21.15 315 cm 4 4 S S .315 252 cm2 1 5 2 5 10 8 8 S S .315 360 cm2 3 7 2 7 1b. 3x - 3 + 2x + (- 1)2016 = 3x + 20170(1) ị 3x - 3 + 2x + 1 = 3x + 1 1.2 - 1 - 1 Condition 3x + 1 ³ 0 ị x ³ ị x ³ ị 2x + 1 ³ 0 (30 point) 3 2 But 3x - 3 ³ 0" x ị 3x - 3 + 2x + 1 = 3x - 3 + 2x + 1 10 from(1) ị 3x + 3 + 2x + 1 = 3x + 1
4. ị 3x - 3 = x (x ³ 0) ộ 3 ờ3x - 3 = x ị x = (accept) ờ ị ờ 2 10 ờ 3 ờ3x - 3 = - x ị x = (accept) ở 4 So E Q.2 A D J N K IM B C a) We have AD AB;Dã AC Bã AE; AC AE 10 ADC ABE(s.a.s) ã ã ã ã a ABE ADC, but BKI AKD (two vertical angles) 10 (20 point) Review BIK and DAK deduce Bã IK Dã AK 600 b)Because ADC ABE ( Proved above) 10 b CM EN, ãACM ãAEN (30 point) 10 ACM AEN(c.g.c) AM AN and Cã AM Eã AN 10 Mã AN Cã AE 600.So AMN is an equilateral triangle Q3 1. with p 2 then 2 p p2 4 4 8 isn’t prime number 3.1 p 2 (20 point) With p 3 then 2 p 8 9 17 is prime number. 10 with p 3 then p is prime numberị p oddị 2 p 22k 1  2(mod3) and p2 1(mod3) ị 2 p p2 3 p 2 p 2 but 2 p 3 then 2 p is composite number 10 so p 3 then 2 p p2 is prime number. 2.We have 3xy – 5 = x2 + 2y. 10 3.2 =>3xy – 2y = x2 + 5
5. (30 point) =>y.(3x - 2) = x2 +5 x2 + 5 ị y = 3x - 2 Put yẻ Z => (x2 + 5)M(3x - 2) ị 9.(x2 + 5)M(3x - 2) ị (9x2 + 45)M(3x - 2) 2 10 ị (9x - 4 + 49)M(3x - 2) ị (3x - 2).(3x + 2) + 49M(3x - 2) ị 49M(3x - 2) =>(3x – 2 ) is divisor of 49 (3x - 2)ẻ {- 1;1;7;- 7;49;- 49} ị 3x ẻ {1;3;9;- 5;51;- 47} ị x ẻ {1;3;17}because x ẻ Z 10 With x = 1 => y = 6 With x = 3 => y = 2 With x = 17 => y = 6 So (x;y) = (1; 6); (3;2); (17;6) 1)Because x + y = 2ị y = 2 – x Q.4 1 4.1 A = (2- x)2 + x(2- x) + x2 - x (30 point) 2 1 = .(4- 4x + x2) + 2x - x2 + x2 - x 2 1 1 10 = 2 - 2x + x2 + x = x2 - x + 2 2 2 1 1 = (x2 - 2x + 4) = (x2 - 2x + 1+ 3) 2 2 10 1 ộ 2 ự 1 3 = . ờ(x - 1) + 3ỳ= .(x - 1)2 + 2 ởờ ỷỳ 2 2 1 2 3 Because .(x - 1) ³ 0" x =>A ³ " x 2 2 10 3 So Amin = . The equality holds when (x – 1)2 = 0 =>x=1 2 4.2 (20 point) 2.We have 5 columns, 5 rows and 2 diagonals so there will be 12 sums Each square only accepts one of 3 numbers 1,0 or -1, so each sum can 10 only accept values from - 5 to 5. We have 11 integers from - 5 to 5 which is - 5; - 4 ; .;0;1; .5 10
6. So, according to Dirichle's principle, at least two sums must be equal.