KimlienHg

(không máy tính)
SMO2004 tớ thấy là khó nhất trong mấy năm gần đây.
Các đề Junior section nói chung lớp 9+không phụ thuộc máy tính là làm ngon, chỉ khổ là hết lớp 8 đã phải thi.
Post tạm mấy bài cuối, dài quá trời.

28. In the diagram below (mọi người tự vẽ nhé, hì), O is the centre of the circle through A, B, C and D. If $ \widehat{AOB} = 90^o$ and the arcs AC, CD and DB have the same length, find the ratio:
(area of circle) : (area of shaded part)
(theo hình vẽ thì C,D giữa A,B; C giữa A, D. H,K là hình chiếu C,D trên OB. Shaded region là miền giới hạn bởi CH, HK,KD và cung CD nhỏ)

29. It is known that $ n = 10^{100} -1 $ has 100 digit 9. How many digit 9 are there in $ \ n^3$?

30. Let a and b be positive integers such that $ \dfrac{2}{3} < \dfrac{a}{b} < \dfrac{5}{7} $
Find the value of $ a +b$ when b has the minimum value.

31. Matt, who is often late for appointments, walks up a moving escalator in the MRT station one step at a time. When he moves quickly at the rate of 2 steps per sec, her reaches the top after taking 24 steps. When he is tired, he climbs at the rate of 1 step per second and reaches the top after taking 15 steps. How long would Matt take to reach the top if, on a rare day, he just stood on the escalator?

32. (Given a triangle ABC).
In the diagram, AB = 2BC and P lies within the triangle ABC such that $ \widehat{APB} = \widehat{BPC} = \widehat{CPA}$. Given that $ \widehat{ABC} = 60^o$ and the area of the triangle ABC = 70 cm^2 , find the area of triangle APC (in cm^2).

33. A set S of positive integers less than or equal to 2004 satisfies the condition that no number is S is double in value of another number in S. What is the largest possible number of integers in S?

34. Find the number of digits in N where N is the product of all the positive divisors of 100 000 000.
(Note: For any positive integer A we regard A itself as one of its divisors)

35. A positive integer d is said to be Strictly Descending if, in its decimal representation: $ d = d_m d_{m-1} ...d_2 d_1 $ (gạch đầu), we have
$ \9 \geq d_m > ... >d_2 > d_1 \geq 0$
For instance, 541 and 93210 are strictly descending integers. Find the number of strictly descending integers which are less than $\{10}^5$.
(Note: We regard all single_digit positive integers are strictly descending)