MathDB

2012 Hanoi Open Mathematics Competitions

Part of Hanoi Open Mathematics Competition

Subcontests

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Q4 - Hanoi Open Mathematical Olympiad 2012 - Junior Section

Q4. A man travels from town AA to town EE through B,CB,C and DD 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 AA and town EE 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)  2km/h(B)  2,5km/h;(C)  3km/h;(D)  3,5km/h;(E)  4km/h.(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}.

x=\frac12$ ($\sqrt[3]{2+\sqrt3}+\sqrt[3]{2-\sqrt3} (HOMC 2012 Q4)

What is the largest integer less than or equal to 4x3−3x4x^3 - 3x, where x=2+33+2−332x=\frac{\sqrt[3]{2+\sqrt3}+\sqrt[3]{2-\sqrt3}}{2} ?
(A) 11, (B) 22, (C) 33, (D) 44, (E) None of the above.
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Q3- Hanoi Open Mathematical Olympiad 2012 - Junior Section

Q3. For any possitive integer aa, let [a]\left[ a\right] denote the smallest prime factor of a.a. Which of the following numbers is equal to [35]\left[ 35 \right] ? (A)  [10];(B)  [15];(C)  [45];(D)  [55];(E)  [75].(A) \; \left[10 \right]; \qquad (B) \; \left[ 15 \right]; \qquad (C ) \; \left[45 \right]; \qquad (D) \; \left[ 55 \right]; \qquad (E) \; \left[75 \right].

computational inside a trapezoid (HOMC 2012 Q3)

Let be given a trapezoidal ABCDABCD with the based edges BC=3BC = 3 cm, DA=6DA = 6 cm (AD//BCAD // BC). Then the length of the line EFEF (E∈AB,F∈CDE \in AB , F \in CD and EF//ADEF // AD) through the common point MM of ACAC and BDBD is
(A) 3,53,5 cm (B): 44 cm (C) 4,54,5 cm (D) 55 cm (E) None of the above
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Q2 - Hanoi Open Mathematical Olympiad 2012 - Junior Section

Q2. Let be given a parallegogram ABCDABCD with the area of 12 cm212 \ \text{cm}^2. The line through AA and the midpoint MM of BCBC mects BDBD at N.N. Compute the area of the quadrilateral MNDC.MNDC. (A)  4cm2;(B)  5cm2;(C)  6cm2;(D)  7cm2;(E)  None of the above.(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.}

compare P =2^a,Q =3, T=2^b, where a=\sqrt2 , 1+1/\sqrt2 (HOMC 2012 Q2)

Compare the numbers P=2a,Q=3,T=2bP = 2^a,Q = 3, T = 2^b, where a=2,b=1+12a=\sqrt2 , b=1+\frac{1}{\sqrt2}
(A) P<Q<TP < Q < T, (B) T<P<QT < P < Q, (C) P<T<QP < T < Q, (D) T<Q<PT < Q < P, (E) Q<P<T Q < P < T
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