Hanoi Mathematical Society
Hanoi Open Mathematical Olympiad 2012
Junior Section
Sunday, 11 April 2011Important:
Answer all 15 questions.
Enter your answers on the answer sheet provided.
For the multiple choise questions, enter only the lettlers (A, B, C, D or E) corresponding to the correct answers in the answer sheet,
No caculators are allowed.
Q1. Assum that $a-b=-(a-b).$ Then:
$$(A) \; a=b; \qquad (B) \; a<b; \qquad (C) \; a>b \qquad (D) \; \text{ It is impossible to compare those of a and b.}$$
Q2. Let be given a parallegogram $ABCD$ with the area of $12 \ \text{cm}^2$. The line through $A$ and the midpoint $M$ of $BC$ mects $BD$ at $N.$ Compute the area of the quadrilateral $MNDC.$
$$(A) \; 4 \text{cm}^2; \qquad (B) \; 5 \text{cm}^2; \qquad (C ) \; 6 \text{cm}^2; \qquad (D) \; 7 \text{cm}^2; \qquad (E) \; \text{None of the above.}$$
Q3. For any possitive integer $a$, let $\left[ a\right]$ denote the smallest prime factor of $a.$ Which of the following numbers is equal to $\left[ 35 \right]$ ?
$$(A) \; \left[10 \right]; \qquad (B) \; \left[ 15 \right]; \qquad (C ) \; \left[45 \right]; \qquad (D) \; \left[ 55 \right]; \qquad (E) \; \left[75 \right].$$
Q4. A man travels from town $A$ to town $E$ through $B,C$ and $D$ with uniform speeds 3km/h, 2km/h, 6km/h and 3km/h on the horizontal, up slope, down slope and horizontal road, respectively. If the road between town $A$ and town $E$ can be classified as horizontal, up slope, down slope and horizontal and total length of each typr of road is the same, what is the average speed of his journey?
$$(A) \; 2 \text{km/h} \qquad (B) \; 2,5 \text{km/h} ; \qquad (C ) \; 3 \text{km/h} ; \qquad (D) \; 3,5 \text{km/h} ; \qquad (E) \; 4 \text{km/h}.$$
Q5. How many different 4-digit even integers can be form from the elements of the set $\{ 1,2,3,4,5 \}.$
$$(A) \; 4; \qquad (B) \; 5; \qquad (C ) \; 8; \qquad (D) \; 9; \qquad (E) \; \text{None of the above.}$$
Q6. At $3:00$ AM, the temperature was $13^o$ below zero. By none it has risen to $32^o$. What is the average hourly increase in temperature ?
Q7. Find all integers $n$ such that $60+2n-n^2$ is a perfect square.
Q8. Given a triangle $ABC$ and $2$ point $K \in AB, \; N \in BC$ such that $BK=2AK, \; CN=2BN$ and $Q$ is the common point of $AN$ and $CK$. Compute $\dfrac{ S_{ \triangle ABC}}{S_{\triangle BCQ}}.$
Q9. Evaluate the integer part of the number
$$H= \sqrt{1+2011^2+ \frac{2011^2}{2012^2}}+ \frac{2011}{2012}.$$
Q10. Solve the following equation $$\frac{1}{(x+29)^2}+ \frac{1}{(1+30)^2}= \frac{13}{36}.$$
Q11. Let be given a sequense $a_1=5, \; a_2=8$ and $a_{n+1}=a_n+3a_{n-1}, \qquad n=1,2,3,...$ Calculate the greatest common divisor of $a_{2011}$ and $a_{2012}$.
Q12. Find all positive integers $P$ such that the sum and product of all its divisors are $2P$ and $P^2$, respectively.
Q13. Determine the greatest value of the sum $M=11xy+3x+2012yz$, where $x,y,z$ are non negative integers satisfying condition $x+y+z=1000.$
Q14. Let be given a trinagle $ABC$ with $\angle A=90^o$ and the bisectrices of angles $B$ and $C$ meet at $I$. Suppose that $IH$ is perpendicular to $BC$ ($H$ belongs to $BC$). If $HB=5 \text{cm}, \; HC=8 \text{cm}$, compute the area of $\triangle ABC$.
Q15. Determine the greatest value of the sum $M=xy+yz+xyz$, where $x,y,z$ are real numbers satisfying the following condition $x^2+2y^2+5z^2=22.$
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Bài viết đã được chỉnh sửa nội dung bởi Phạm Quang Toàn: 10-06-2012 - 10:03