Cuộc thi Toán Tiếng Anh do Quoc Tran Anh Le tổ chức

Vòng 2 - Vòng sơ khảo

Trong vòng 2, mình sẽ lấy 10 người xuất sắc nhất vào vòng 3. Trong đó:

@ Đạt hơn 80 điểm: +3 điểm vào vòng 3.

@ 3 bạn cao điểm nhất: +2 điểm vào vòng 3.

@ 3 bạn cao điểm tiếp theo: +1 điểm vào vòng 3.

@ 4 bạn còn lại: Không cộng điểm vào vòng 3.

From question 1 to question 5: 4 marks/question

QUESTION 1: There is a \(\dfrac{2}{3}\) chance of rain for each of three days. If the weather on each day is independent of the weather on the other two days, what is the probability that it will rain on none of the three days? Express your answer as a common fraction.

QUESTION 2: A drawer contains five brown socks, five black socks and five gray socks. Randomly selecting socks from this drawer, what is the minimum number of socks that must be selected to guarantee at least two matching pairs of socks? A matching pair is two socks of the same color.

QUESTION 3: When the circuit containing blinking lights A and B is turned on, lights A and B blink together. Then A blinks once every 5 seconds and B blinks once every 11 seconds. Lindsey looks at the two lights just in time to see A blink alone. What is the percent probability that the next light to blink will be A blinking alone? 

QUESTION 4: A family farm is equally divided among three heirs: Jim, Jan and John. John’s share of the farm is then equally divided among his three heirs: Peter, Paul and Patricia. Paul decides to sell his share of the farm, and then later the family decides to sell the remainder of the farm all at once. What portion of the proceeds from the most recent sale should Jim receive? Express your answer as a common fraction. 

QUESTION 5: The product of the first three terms of an arithmetic sequence of integers is a prime number. What is the sum of the three numbers?

From question 6 to question 15: 5 marks/question 

QUESTION 6: If RATS × 4 = STAR, and each letter represents a different digit from 0 to 9, inclusive, what is the value of S + T + A + R?

QUESTION 7: An Euler Airline flight is getting ready to take off. Gary McDonald, the pilot, starts from rest at the edge of the runway. He needs to accelerate to a speed of 300 km/h in 30 seconds. Acceleration is defined as the change in speed per unit time. What is Gary’s average acceleration, in meters per second per second, which is equivalent to meters per second squared, during takeoff? Express your answer as a decimal to the nearest tenth.

QUESTION 8: The graph of a linear equation contains the points (1, a), (2, b) and (4, 18). What is the value of \(\dfrac{3}{2}b-a\)?

QUESTION 9: In the four by four grid shown, move from the 1 in the lower left corner to the 7 in the upper right corner. On each move, go up, down, right or left, but do not touch any cell more than once. Add the numbers as you go. What is the maximum possible value that can be obtained, including the 1 and the 7? 

4 5 6 7
3 4 5 6
2 3 4 5
1 2 3 4

QUESTION 10: In the land of Binaria, the currency consists of coins worth 1¢, 2¢, 4¢, 8¢, 16¢, 32¢ and 64¢. Bina has two of each coin. How many combinations of her coins have a combined value of 50¢?

QUESTION 11 BIT HARD: A dartboard consists of three concentric circles with radii 10, 5 and 1, respectively, measured in inches. The area between the largest and middle circles is colored green, the area between the middle and smallest circles is colored yellow, and the area within the smallest circle, the bull’s-eye, is colored red. If a thrown dart is guaranteed to hit the board, but its position on the board is uniformly random, what is the probability that it lands in the yellow portion of the board? Express your answer as a common fraction. 

QUESTION 12 BIT HARD: In equilateral triangle ABC, M is the midpoint of side AB. If CMN is also an equilateral triangle, what fraction of the area of triangle \(\Delta\)ABC lies inside of \(\Delta\)CMN? Express your answer as a common fraction.

QUESTION 13 BIT HARD: A circle passes through the origin and (8, 0). It has a radius of 5, and its center is in the first quadrant. What are the coordinates of its center? Express your answer as an ordered pair.

QUESTION 14 BIT HARD: The quadratic equation x\(^2\) − 7x + 5 = 0 has two real roots, m and n. What is the value of \(\dfrac{1}{m}\) + \(\dfrac{1}{n}\) ? Express your answer as a common fraction.

QUESTION 15 BIT HARD: Alex Zhu bikes between home and school every day. He uses the same route to go to and from school, but it takes him 20 minutes to bike to school and only 15 minutes to bike back. If his average biking pace for the whole round-trip is 7 minutes per mile, how many miles long is the trip from home to school? Express your answer as a decimal to the nearest tenth.

From question 16 to question 20: 6 marks/question

QUESTION 16 BIT HARD: In a regular octagon, the diagonals have three possible lengths —“short,” “medium,” and “long.” What is the ratio of the length of the medium diagonal to the long diagonal? Express your answer as a decimal to the nearest thousandth.

QUESTION 17 BIT HARD: Kendra starts at a positive integer k and counts up by 4s until she hits exactly 200. Mason starts at a positive integer m and counts up by 6s until he hits exactly 200. If it takes Kendra half as many steps to reach 200 as it takes Mason, what is the greatest possible value of k − m?

QUESTION 18 BIT HARD: When three squares are arranged as shown, seven unique regions are formed. What is the maximum number of regions that can be formed by three congruent, overlapping squares?

QUESTION 19 BIT HARD: A wheel that makes 10 revolutions per minute takes 18 seconds to travel 15 feet. In feet, what is the diameter of the wheel? Express your answer as a decimal to the nearest tenth.

QUESTION 20 HARD: If the median of the ordered set {0, \(\dfrac{2}{5}x\), x, 11.5x, 5, 9} is 2, what is the mean? Express your answer as a decimal to the nearest hundredth.

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