Đề thi Olympic Hà Nội mở rộng 2010

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JUNIOR SECTION
​

Sunday, 28 March 2010

PART I:
​

Question 1: Compare the numbers:

and

A.

B.

C.

Explain.

Question 2: The number of integer n from the set

such that

is divisible by 7, is
A. 0
B. 1
C. 2
D. 3
E. Non of the above
Explain.

Question 3: The last 5 digits of the number

are
A. 65625
B. 45625
C. 25625
D. 15625
E. Non of the above.
Explain

Question 4: How many real numbers

such that the corresponding number

is an integer.
A. 0
B. 1
C. 8
D. 9
E. Non of the above
Explain.

Question 5: Each box in 2 . 2 table can be colored black or white. How many different colorings of the table are there?
A. 4
B. 8
C. 16
D. 32
E. Non of the above
Explain

Question 6: The greatest integer less than

are
A. 721
B. 722
C. 723
D. 724
E. Non of the above.
Explain

__________________________________________________________

PART II:
​

Question 7: Determine all positive integer a such that the equation

has two prime roots, i.e. both roots are prime numbers.

Question 8: If n and

are both perfect squares, find n.

Question 9: Let be given a triangle ABC and points A, M, N belong to BC, AB, AC, respestively. Suppose that MD is parallel to AC and ND is parallel to AB. If

compute

Question 10: Find the maximum value of

 

[duynhan1]

]

Question 10: Find the maximum value of

[TEX] \frac32 - M = \frac12 ( \frac{y}{2x+y} + \frac{z}{2y+z} + \frac{x}{2z+x} )[/TEX]

[TEX]\Leftrightarrow 3- 2M = \frac{y^2}{2xy+y^2} + \frac{z^2}{2yz+z^2} + \frac{x^2}{2zx+x^2} \geq 1 [/TEX]

[TEX]\Leftrightarrow M \leq 1[/TEX]

 

[ms.sun]

JUNIOR SECTION​

Sunday, 28 March 2010

PART I:​

Question 1: Compare the numbers:

and

A.

B.

C.

Explain.

chém câu dễ nhất
đặt [TEX]333.......333 (2010 c/số 3) =a [/TEX]
[TEX]444......4444 (2010 c/số 4) =b[/TEX]
[TEX] \Rightarrow P=2ab ; Q= b(2a+1)=2ab+b [/TEX]
[TEX] \Rightarrow P <Q [/TEX]

 

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