De thi Toan HOMC nam 2013

<span class='text_page_counter'>(1)</span>Hanoi Mathematical Society Hanoi Opens Mathematics Competition 2013 Junior Section. Sunday, March 24, 2013. Important: Answer all 15 questions. Enter yor answers on the answer sheet provided. For the multiple choice questions, enter only the letters ( A,B,C,D or E) corresponding to the correct answers in the answer sheet. No calculators are allowed.. Multiple Choice Questions : Q1 : Write 2013 as a sum of m prime numbers. The smallest value of m is: (A) : 2 (B) : 3 (C) : 4 (D) : 1 (E) : None of the above. 2 Q2 : How many natural numbers n are there so that n + 2014 is a perfect square. (A) : 1 (B) : 2 (C) : 3 (D) : 4 (E) : None of the above. Q3 : The largest integer not exceeding [(n + 1)  ] - [n  ], where n is a natural number,  = 2013 2014 , is : (A) : 1 (B) : 2 (C) : 3 (D) : 4 (E) : None of the above. 20 Q4 : Let A be an even number but not divisible by 10. The last two digits of A are :. (A) : 46 (B) : 56 (C) : 66 (D) : 76 (E) : None of the above. Q5 : The number of integer solutions x of the equation below: (12 x  1)(6 x  1)(4 x  1)(3 x  1) 330 is : (A) : 0 (B) : 1 (C) : 2 (D) : 3 (E) : None of the above.. Short Questions 2. Q6 : Let ABC be a triangle with area 1 ( cm ). Points D,E and F lie on the sides AB,BC and CA, respectively. Prove that : 2. Min{Area of ADF, area of BED, area of CEF}  1/4 ( cm ). Q7 : Let ABC be a triangle with  A = 90,  B = 60 and BC = 1cm. Draw outside of ABC three regular triangles ABD, ACE and BCF. Determine the area of DEF. Q8 : Let ABCDE be a convex pentagon. Gives that SABC = SBCD = SCDE = SDEA = SEAB = 2 ( cm2 )..

<span class='text_page_counter'>(2)</span> Find the area of the pentagon. Q9 : Solve the following system in positive numbers.  x  y 1  1 2   xy x 2  y 2 10  Q10 : Consider the set of all rectangles with a given perimeter p. Find the largest value of S M = 2S  p  2 Where S is denoted the area of the rectangle. Q11 : The positive numbers a,b,c,d,e are such that the following identify hold for all number x ( x  a )( x  b)( x  c)  x 3  3dx 2  3x  e3 . Find the smallest value of d. 2 Q12 : If f ( x) ax  bx  c satisfies the condition. | f ( x )|  1,  x ä   1,1 Prove that the equation f ( x)=2 x 2 − 1 has two real roots. Q13 : Solve the system of equations. 1 1 1  x  y 6    3  2 5  x y 6 Q14 : Solve the system of equations.  x3  y x 2  1  3 2 2 y  z 2 y  1 3z 3  x 3z 2  1  * Q15 : Denote by Q and N the set of all rational and positive integer numbers, respectively. ax  b Q * Suppose that x for every x  N .. ax  b Ax  B  Cx for all x  N* . Prove that there exist integers A, B , C such that x. GV Nguyễn Minh Sang THCS Lâm Thao –Phú Thọ ( Chưa đánh máy được lời giải tôi sẽ đưa lên sau).

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