GV Nguyễn Minh Sang THCS Lâm Thao –Phú Thọ Chưa đánh máy được lời giải tôi sẽ đưa lên sau.[r]
Trang 1Hanoi 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
Q2 : How many natural numbers n are there so that n + 2014 is a perfect square2
(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
Q4 : Let A be an even number but not divisible by 10 The last two digits of A are :20
(A) : 46 (B) : 56 (C) : 66 (D) : 76 (E) : None of the above
Q5 : The number of integer solutions x of the equation below:
(12x1)(6x1)(4x1)(3x 1) 330 is :
(A) : 0 (B) : 1 (C) : 2 (D) : 3 (E) : None of the above
Short Questions
Q6 : Let ABC be a triangle with area 1 (cm ) Points D,E and F lie on the sides AB,BC and CA, 2
respectively Prove that :
Min{Area of ADF, area of BED, area of CEF} 1/4 (cm ).2
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 (
2
cm ).
Trang 2Find the area of the pentagon.
Q9 : Solve the following system in positive numbers
2 2
1
10
x y
Q10 : Consider the set of all rectangles with a given perimeter p Find the largest value of
M = 2 2
S
Sp
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 3dx 3x e
Find the smallest value of d
Q12 : If f x( ) ax 2bx c satisfies the condition
| ( )| 1,f x xä1,1
Prove that the equation f (x)=2 x2− 1 has two real roots.
Q13 : Solve the system of equations
1 1 1
6
3 2 5
6
x y
x y
Q14 : Solve the system of equations
1
Q15 : Denote by Q and N the set of all rational and positive integer numbers, respectively * Suppose that
ax b x
Q for every x N * Prove that there exist integers A, B , C such that
for all x N *
GV Nguyễn Minh Sang THCS Lâm Thao –Phú Thọ ( Chưa đánh máy được lời giải tôi sẽ đưa lên sau)