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Vòng 1 - Vòng sơ loại
(ĐỀ THI CHÍNH THỨC)
VÒNG 1-CUỘC THI TOÁN TIẾNG ANH
*Tất cả các câu hỏi trong vòng 1 các bạn chỉ ghi đáp án vào ô giải. Đáp án phải thật chuẩn xác, không chấp nhận bất cứ lỗi nào.
*Đáp án phải được giải bằng tiếng Anh (nếu có)
Đối với những bạn không qua được vòng 1: + 3GP
Đối với những bạn qua vòng 1: + 5GP
* Có 30 câu trắc nghiệm trong vòng 1 (Ước tính làm khoảng 45-60 phút)
* Các câu hỏi sẽ được xếp độ khó theo thang chuẩn 1-7, trong đó:
+ Các câu có độ khó (1)(2)(3)(4): Không có kí hiệu gì
3) Tất cả các câu trả lời đều được đặt ở dạng số thập phân
[VD] CHẤP NHẬN: 0,(6) KHÔNG CHẤP NHẬN: 2/3, 4/6, 8/12, ...
Nếu yêu cầu là phân số thì phải là phân số tối giản
CHẤP NHẬN: 2/3 KHÔNG CHẤP NHẬN: 4/6, 8/12
4) Tỉ số ở dạng phân số tối giản
CHẤP NHẬN: 7/2 KHÔNG CHẤP NHẬN:
5) Chỉ trả lời đáp án không, nếu cho lời giải vào thì trừ 25% số điểm câu đó.
From question 1 to question 10: 2 marks/question
QUESTION 1: Consider the pattern below:
222 = 121 × (1 + 2 + 1)
3332 = 12,321 × (1 + 2 + 3 + 2 + 1)
44442 = 1,234,321 × (1 + 2 + 3 + 4 + 3 + 2 + 1)
For what positive value of n will n2
= 12,345,654,321 × (1 + 2 + 3 + 4 + 5 + 6 + 5 + 4 + 3 + 2 + 1)?
QUESTION 2: In a golf long-drive competition, Jason Zuback hits his first five drives 394, 401, 387, 414 and 421 yards, respectively. How long must he hit his sixth drive to ensure that the mean of his six drives is at least 400 yards?
QUESTION 3: For what fraction of the day is the hour hand or minute hand (or both the hour and minute hands) of an analog clock in the upper half of the clock? Express your answer as a common fraction.
QUESTION 4: A fair coin is flipped, and a standard die is rolled. What is the probability that the coin lands heads up and the die shows a prime number? Express your answer as a common fraction.
QUESTION 5: How many pairs of numbers (a, b) satisfy rules I and II shown here?
I. a = 0 or b = −1 or b = 1
II. a = −1 or a = 1 or b = 0
QUESTION 6: A standard 52-card deck of playing cards includes four aces. What is the probability that two cards selected randomly, without replacement, will both be aces? Express your answer as a common fraction.
QUESTION 7: A bag contains five chips numbered 2 through 6. Danya draws chips from the bag one at a time and sets them aside. After each draw, she totals the numbers on all the chips she has already drawn. What is the probability that at any point in this process her total will equal 10? Express your answer as a decimal to the nearest tenth.
QUESTION 8: Let @x@ be defined for all positive integer values of x as the product of all of the factors of 2x. For example, @7@ = 14 × 7 × 2 × 1 = 196. What is the value of @3@?
QUESTION 9: A triangle has three vertices given by coordinates (2, 2),
(2, −6) and (−5, −9). What is the area of the triangle?
From question 10 to question 17: 3 marks/question
QUESTION 10: Three days ago, there were p cupcakes on the counter. Two days ago, exactly 20% of the cupcakes were eaten. Today, there are 30% fewer cupcakes than yesterday and half as many as there were three days ago. If a whole number of cupcakes were eaten every day, what is the least possible value of p?
QUESTION 11: What numeral in base 8 is equivalent to 3325 (denoting 332 base 5)?
QUESTION 12: Two standard six-sided dice are rolled. One of the dice represents the numerator and the other represents the denominator of a fraction. The fraction is simplified, if possible. How many distinct fractions less than one can be generated by this method?
QUESTION 13: What is the value of (1.4×10−7)(2.4×108)1.2×109 when written in simplest form? Express your answer in scientific notation to two significant digits.
QUESTION 14: The formula P = F/A indicates the relationship between pressure (P), force (F) and area (A). In newtons, what is the maximum force that could be applied to a square area with side length 4 meters so that the pressure does not exceed 25 newtons per square meter?
QUESTION 15: Sara told Jo, “If you give me three of your marbles, I will have twice as many as you.” Jo responded, “If you give me just two of your marbles, I will have twice as many as you.” How many marbles does Sara have?
QUESTION 16: If the digits 7, 8, 2, 3 and 0 are used, each exactly once, to form a three-digit positive integer and a two-digit positive integer that differ by exactly 288, what is the sum of the three-digit integer and the two-digit integer?
QUESTION 17: Let #x represent the greatest even integer less than x. If 20 < x < 30, what is the maximum possible value of #(5x) − #(4x)?
From question 18 to question 25: 4 marks/question
QUESTION 18: How many ordered triples of integers (m, n, p) exist such that mn = p, np = m and mp = n?
QUESTION 19: What is the sum of the integers strictly between 1 and 100 that are multiples of neither 2 nor 3?
QUESTION 20: Two different integers are randomly selected from the set of positive integers less than 10. What is the probability that their product is a perfect square? Express your answer as a common fraction.
QUESTION 21: Nine people are forming three teams of three people each for a game of XFlag. Each team has one player who is the captain. The nine participants are Alana, Benny, Chico, Danzig, Elias, Frederico, Gina, Hsin-Hsin and Illiana. They are very particular about which players can be on a team together. Frederico must be with Hsin-Hsin or Illiana. Elias, Frederico and Gina must be on different teams. Hsin-Hsin and Illiana must be on different teams. Chico and Danzig must be on the same team; neither is a captain. Danzig cannot be on a team with Gina as captain. Frederico cannot be on a team with Alana as captain. Hsin-Hsin cannot be on a team with Elias as captain. Alana and Benny are captains. Who is the third captain?
QUESTION 22: A couple getting married today can be expected to have 0, 1, 2, 3, 4 or 5 children with probabilities of 20%, 20%, 30%, 20%, 8% and 2%, respectively. What is the mean number of children a couple getting married today can be expected to have? Express your answer to the nearest whole number.
QUESTION 23: Six numbers 1,2,3,4,5,6 are all used to make a 1-digit, 2-digit and 3-digit number. The sum of the 1-digit and 2-digit number is 47, and the sum of the 2-digit and 3-digit number is 358. What is the sum of these three numbers.
QUESTION 24: Arnold, Benji and Celal found an old scale. When Arnold and Benji stepped on the scale, it showed a weight of 158 pounds. When Benji and Celal stepped on the scale, it showed a weight of 176 pounds. When all three of them stepped on the scale, it accurately showed a weight of 250 pounds but then promptly broke under the strain. However, they already had enough information to determine each of their weights. How much does Benji weigh?
QUESTION 25: Using the number keypad shown, it is possible to convert any three-digit number to various three-letter strings by choosing one letter for each number. For example, 223 can be used to make BAD, ACE, CCF, and 24 others, some of which are real words and some of which are not. Using only the buttons 2 through 9, what is the smallest three-digit number such that none of its possible three-letter strings are real words? (Exclude proper nouns and abbreviations.)
From question 26 to question 30: 5 marks/question.
QUESTION 26: How many rectangles of any size are in the figure shown?
QUESTION 27: James and John take turns spinning the pointer of a fair spinner that is divided into three congruent sectors, with 1 win-sector and 2 pass-sector. The first player whose spin lands on the WIN sector is the winner of the game. If James goes first, what is the probability that he wins the game? Express your answer as a common fraction.
QUESTION 28: A set S contains some, but not all, of the positive integers from 3 to 7. Some statements describing S are given below. The statement numbered n is true if the number n is in S and false if n is not in S. What is the product of the numbers that are in S?
3. The sum of the numbers in S is odd.
4. The sum of the numbers in S is less than 15.
5. S contains exactly one composite number.
6. S contains exactly one prime number.
7. The product of the numbers in S is odd.
QUESTION 29: How many different combinations of pennies, nickels, dimes and quarters are possible in the cup holder of Terry’s car if he counts 15 coins total?
QUESTION 30: For the integer 263, neither pair of consecutive digits (the pair 2 and 6 and the pair 6 and 3) is relatively prime. But the nonconsecutive digits 2 and 3 are relatively prime. What is the greatest positive integer n that satisfies the following three conditions?
(1) None of the digits is zero.
(2) No pair of consecutive digits is relatively prime.
(3) All non-consecutive digits are relatively prime.