CHU VAN AN SECONDARY SCHOOL TEST NUMBER 02 GIFTED STUDENTS INVESTIGATION TEST SUBJECT MATH IN ENGLISH GRADE 8 Duration 120 minutes PART I MULTIPLE – CHOICE (100 marks) Question 1 Caculate 2 275 45 240[.]
CHU VAN AN SECONDARY SCHOOL GIFTED STUDENTS INVESTIGATION TEST - SUBJECT: MATH IN ENGLISH - GRADE TEST NUMBER 02 Duration: 120 minutes PART I: MULTIPLE – CHOICE (100 marks) 752 452 Question 1: Caculate ? 240 A 10 ; B 12 ; Question 2: When the number N does it end in ? A 3; B 4; C 15 ; D 18 ; 11.22.33 99 is writen as a dicimal number, how many zeros C 5; D 6; Question 3: CE and BD are angle bisectors of BFC 110o , find the measure of BAC A 30o ; Question 4: If A ; B 35o ; y x z B y z ; E None of the above ABC which intersect at point F If C 40o ; x E 30 D 45o ; E 70o x x for three positive numbers x, y and z , all different, then y y C ; D ; Question 5: The polynomials x3 x2 ax and x x2 remainder when divided by x Find the value of a ? A 10; B 11; C 12; D 13; ? E 2a x 16 leaves the same E None of the above Question 6: There are twenty people in a room, with a men and b women Each pair of men shakes hands, and each pair of women shakes hands, but there are no handshakes between a man and a woman The total number of handshakes is 106 Determine the value of a.b A 72; B 75; C 80; D 84; E None of the above Question 7: Rectangles FENR and HGMS are inscribed in triangle ABC as shown The area of triangle ABC is 60 and its base, BC , has length 10 If EN GM then the sum of the areas of the two rectangles is: A E G N B M A 30; B 32, 5; C 36; F H R S C D 36, 5; 2n2 13n 20 an integer ? n2 n C ; D ; E 37, Question 8: For how many integers n is A ; B ; E Question 9: For how many n in 1; 2; 3; ;100 is the tens digit of n2 odd ? A 10 ; B 20 ; C 30 ; D 40 ; E 50 Question 10: In the figure, ABCD is an isosceles trapezoid with side lengths AD BC 5, AB and DC 10 The point C is on DF and B is the midpoint of hypotenuse DE in the right triangle DEF Then CF ? E A B D A 3, 25 ; B 3, ; C C 3, 75 ; F E 4, 25 D ; PART II: COMPOSE (200 marks) Problem Prove that if n is a perfect cube then n2 3n Problem Let a, b, c and d be real numbers such that a2 Determine the value of ab cd cannot be a perfect cube b2 c2 d2 and ac bd Problem Let CH be the altitude of triangle ABC with ACB 900 The bisector of BAC intersects CH , CB at P, M respectively The bisector of ABC intersects CH , CA at Q, N respectively Prove that the line passing through the midpoints of PM and QN is parallel to line AB Problem Let a, b and c be positive real numbers such that abc of the expression P a2 ab a b2 bc b c2 ca c -THE END Find the minimum value CHU VAN AN SECONDARY SCHOOL GIFTED STUDENTS INVESTIGATION TEST - SUBJECT: MATH IN ENGLISH - GRADE TEST NUMBER 02 Duration: 120 minutes PART I: MULTIPLE – CHOICE (100 marks) ANSWERS AND MARKS Question 10 Answer C C C E A D E C B D 10,0 10,0 10,0 10,0 10,0 10,0 10,0 10,0 10,0 10,0 Marks PART II: COMPOSE (200 marks) Problem Answer Marks Suppose by way of contradiction that n2 (50 marks) Hence n n2 3n Note that n n2 and since n Since a2 (50 marks) b2 is a cube 20 is a cube 3n 3n n3 n2 3n 1 20 is not a cube, we obtain a contradiction 10 n , a and b are not both From ac We may assume that a Substituting into c It follows that a 2 d d Hence ab (50 marks) cd ab d a2 b2 d a2 a2 bd a 20 bd a c b2 d a2 , we have bd d2 b a2 d2 a2 15 15 C N N E Q F P A X H Y B Let E , F be the midpoints of QN , PM respectively Let X , Y be the intersection of CE , CF with AB respectively Now CMP So CM 90 CAM CP then CF 90o AF BAM APH CPM 10 15 Since AF bisects CAY Hence CAF YAF A.S A so CF FY 15 Similarly CE EX By the midpoint theorem, we have EF parallel to line XY , which is the same as line AB (50 marks) We have P a2 ab a b2 bc b Using AM-GM inequality, we get P P P P 2a ab a 2a ab a 1 2ab 2abc abc ab a abca abc 2ab ab a ab a ab a ab a MinP c2 ca c 2a 2b ab a bc b when a b ab 2c ca c 10 15 20 15 c ... DE in the right triangle DEF Then CF ? E A B D A 3, 25 ; B 3, ; C C 3, 75 ; F E 4, 25 D ; PART II: COMPOSE (200 marks) Problem Prove that if n is a perfect cube then n2 3n Problem Let a, b, c... Question 10 Answer C C C E A D E C B D 10,0 10,0 10,0 10,0 10,0 10,0 10,0 10,0 10,0 10,0 Marks PART II: COMPOSE (200 marks) Problem Answer Marks Suppose by way of contradiction that n2 (50 marks)... and since n Since a2 (50 marks) b2 is a cube 20 is a cube 3n 3n n3 n2 3n 1 20 is not a cube, we obtain a contradiction 10 n , a and b are not both From ac We may assume that a Substituting into