Hkimo 2020 Session 4 Secondary 1 [PDF]

Zitiervorschau

Secondary 1

1)

Given A and B are 1-digit integers that satisfy the following: I. The 2-digit numbers ̅𝐴𝐵̅̅̅̅̅, ̅𝐵̅̅̅𝐴̅̅, and ̅𝐴𝐴̅̅ are prime. II. A is not equal to B. III. Neither are equal to 3. Find the value of A2 + B2. Answer: 50 The only repeated 2-digit number that is prime is 11, so one of the numbers should be 1. The numbers AB that satisfy the first condition are (11, 17, 13). The second condition cancels out (11), so the remaining numbers are (17, 13). The third condition cancels out (13) so the remaining possible sets for A and B is 1 and 7. Therefore, 12 + 72 = 50.

2)

A set of 4 integers have a total product of 1. How many possible solution sets are there? Answer: 3 Considering that integers can be either positive or negative, - 1x1x1x1=1 - 1 x 1 x (-1) x (-1) = 1 - (-1) x (-1) x (-1) x (-1) = 1 -

3)

According to the pattern shown below, what number should go next? 5, 11, 23, 47, 95… Answer: 191 By careful scrutiny, it can be seen that the pattern follows (xn = 2xn-1 + 1), or in simpler terms, the current term is 1 plus double the previous term. 2(5) + 1 = 11 2(11) + 1 = 23 2(23) + 1 = 47 2(47) + 1 = 95 Following the pattern, the next term should be: 2(95) + 1 = 191

4)

There are 20 different problems in a mathematics examination. The scores of each examinee is tabulated by the following rules: 2 points will be given for a correct answer, 1 point will be deducted from a wrong answer, and 0 marks will be given for a blank answer. Find the minimum number of examinee(s) to ensure that 4 examinees will have the same scores in the exam. Answer: 181 examinees The maximum score is 40 and the minimum score is -20. So there are tentatively 61 scores (40 positive scores, 20 negative scores, and zero). Since the score “39” cannot be obtained in any way, only 60 different scores can be obtained. Now, 60 x 3 +1 = 181 examinees

5)

Andy goes north for 21 meters, goes east for 3 meters, goes north for x meters, and then goes east for 6 meters. At the end of his walk, he is 41 meters away from his original position. How long is x? Answer: 19 meters Recall the Pythagorean theorem:

6)

All 26 letters of the alphabet are individually written in pieces of paper and placed in a box to be picked at random. The first 12 letters picked consisted of 8 consonants and 4 vowels, and surprisingly, the order that the first 12 letters followed the sequence: consonant – consonant – vowel – consonant – consonant – vowel etc., at least how many more letters should be drawn from the box to ensure that another vowel will be drawn? Answer: 14 The letters are drawn at random so the pattern for the first 12 letters does not dictate the order of the next letters. Now, there are 26 different letters. 21 consonants and 5 vowels. After the 12 letters were drawn, only 13 consonants and 1 vowel remain, the worst case is that the vowel could be drawn last. Considering the worst case, all 13 are picked first and then the vowel. 13 + 1 = 14

Algebra

7)

Find the value of

.

Answer: 16

8)

Factorize 12𝑥3 − 16𝑥2 − 3𝑥. Answer: (6𝑥 + 1)(2𝑥 − 3)

9)

How many integral solutions are there for 𝑥 if −105 < 12𝑥 + 4 < 217? Answer: 27 -105 < 12x + 4 < 217 -109 < 12x < 213 -9.083 < x < 17.75 Number of values of x = 9 + 1 + 17 = 27

10)

Find the value of 1𝑥3 + 2𝑥4 + 3𝑥5 + ⋯ + 10𝑥12.

Answer: 495 We can express the terms as an algebraic expression, in the form (𝑛 + 2) from n = 1 to n = 10. Since the terms are being added, we can express it in summation, now in the form:

11)

Find the sum of the squares of 𝑥 and 𝑦 if |3𝑥 − 4𝑦| + |𝑥 + 𝑦 − 7| = 0 Answer: 25 Since x and y are two different variables, two different equations are needed to determine both values. If both absolute terms are equated to 0, two different equations are obtained that satisfy the main equation. Now we have: 3𝑥 − 4𝑦 = 0 𝑎𝑛𝑑 𝑥 + 𝑦 − 7 = 0 By substitution, matrix, or elimination, the values for x and y are 4 and 3 respectively. So, the product is: 𝑥2 + 𝑦2 = 42 + 32 = 16 + 9 = 25

12)

Calculate √ 6 + √ 6 + √6 + ⋯ Answer: 3 We can express the above equation as

= x, now, by squaring:

6+𝑥=𝑥 𝑥−6=0 (𝑥 − 3)(𝑥 + 2) = 0 𝑥 = 3, −2 The equation cannot possibly yield a negative value, so we only consider the positive principal root. Therefore, x = 3 𝑥2 −

Number ̅ Theory

̅̅

13)

̅̅

̅ The 10-digit number ̅995348A1B6̅̅ is divisible by 99. Find the (A+B)2. Answer: 81 Since the number is divisible by 99, it must be divisible by both 9 and 11. The sum of all known digits is 45, by the divisibility rules of 9, A + B must be either 0 or 9. By the divisibility rules of 11, 9 + 5 + 4 + 𝐴 + 𝐵̅̅̅ = 9 + 3 + 8 + 1 + 6 ± (0,11𝑧) It can be seen that A+B= 0 does not satisfy this equation but A+B= 9 does. Therefore, A+B = 9 and (A+B)2 = 81

14)

Find the number of all positive factor(s) of 126,293. Answer: 12 Factoring 126, 293: 126,293 = 17 𝑥 17 𝑥 19 𝑥 23 = 172 𝑥 19 𝑥 23 Taking adding one to all the power coefficients and multiplying, (2 + 1) 𝑥 (1 + 1) 𝑥 (1 + 1) = 12

15)

It is given that 𝑎 ⊗ b = (a − b)(a ⋅ a + b ⋅ b + a ⋅ b), Find the value of 5⊗(4⊗(3⊗2)). Answer: 313738910000

Notice that (a − b)(a ⋅ a + b ⋅ b + a ⋅ b) = 𝑎3 − 𝑏3 . So, rewriting will give us: (53 − (43 − (33 − 23))) = 313738910000

16)

Given that 6𝑥2 − 13𝑥 + 3 + 𝑘 is divisible by 𝑥 − 3, find the integer 𝑘.

value of the

Answer: -18 Since 𝑥 − 3 is a factor of 6𝑥2 − 13𝑥 + 3 + 𝑘, we’ll need to determine the other linear factor to find k. 6𝑥2 − 13𝑥 + 3 + 𝑘 = (𝑥 − 3)(𝑎𝑥 + 𝑏) Since the coefficient of the highest power is 6, we can set a to be 6. 6𝑥2 − 13𝑥 + 3 + 𝑘 = (𝑥 − 3)(6𝑥 + 𝑏) Now, from this, we have: −13𝑥 = 6𝑥 ⋅ (−3) + 𝑏 𝑏=5 With b, finding the constant: (−3) ⋅ b = 3 + k −15 = 3 + 𝑘 𝑘 = −18

17) There are 2 consecutive positive numbers. When either of them is divided by 1442, the remainder is 36. Find the sum of the two numbers Answer: 75 Let the two consecutive numbers be 𝑛 and 𝑛 + 1. (𝑛)(𝑛 + 1) + 36 = 1442 𝑛2 + 𝑛 + 36 = 1442 𝑛2 + 𝑛 − 1406 = 0 𝑛 = 37, −38 Since the numbers are positive, 𝑛 = 37 So, the sum is: 𝑛 + (𝑛 + 1) = 37 + 38 = 75

18) Find the remainder when 72345 is divided by 5. Answer: 2 Analyzing the remainders from 71 onwards to determine a pattern: 71 ÷ 5, r = 2

72 ÷ 5, r = 4 73 ÷ 5, r = 3 74 ÷ 5, r = 1 75 ÷ 5, r = 2 … So, for base 7 divided by 5, the remainder depends on the power coefficient following the pattern (2,4,3,1) Using the remainder technique to locate where among the 4 terms of the pattern 2345 is located: 2345 ÷ 4 = 586 R1 Since the remainder is 1, it would fall on the first term of the pattern. Therefore, the last digit is 2. Geometry 19) Points (6, 7) and (14, -5) are the endpoints of the diameter of a circle, what is the center point of the circle? Answer: (10, 1) X coordinate: Y coordinate:

20) What time between 5:45 pm and 6 pm will the minute hand and hour hand of a clock form exactly 125 degrees? Answer: 5:50 pm For this specific problem, the following format can be used: 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑎𝑛𝑔𝑙𝑒 𝑜𝑓 ℎ𝑜𝑢𝑟 ℎ𝑎𝑛𝑑 + ℎ𝑜𝑢𝑟 ℎ𝑎𝑛𝑑 𝑠𝑝𝑒𝑒𝑑 = 𝑚𝑖𝑛𝑢𝑡𝑒 ℎ𝑎𝑛𝑑 𝑠𝑝𝑒𝑒𝑑 − 𝑑𝑒𝑔𝑟𝑒𝑒𝑠 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒 150 + 0.5𝑥 = 6𝑥 − 125 5.5𝑥 = 275 𝑥 = 50 𝑚𝑖𝑛𝑢𝑡𝑒𝑠

21) A total of 860 diagonals can be drawn from a certain regular polygon. How many sides does this polygon have? Answer: 43 sides Recall the formula for the number of diagonals given the number of sides:

Considering the positive root, n = 43

22) Find the area of the triangle bounded by the lines (x + y = 7), (3x + y = 12), and the positive xaxis. 15

Answer: 3.75

or 4

Points of intersection of first line (0,7) and (7,0) Points of intersection of second line (0, 12) and (4,0)

The altitude is the intersection between the two lines, where x = 2.5 The base is the difference between x coordinates of the points of the two lines on the x-axis 7 – 4 =3 Using the triangle area formula (base = 3, height 2.5), the area is

23) A solid cuboid with integral side lengths has a volume of 10013 units. Find the maximum possible surface area of the cuboid (in sq. units). Answer: 2878 units2 The surface area is greatest when the dimensions are closest to each other, in this case, the closest is 17x19x31. Calculating the surface area: SA = 2[lw + wh + lh] SA = 2[17 ⋅ 19 + 19 ⋅ 31 + 31 ⋅ 17] SA = 2878 units2

24) The ratio of the sides of a rectangular prism is 3:5:11. If the longest side and the shortest side differ by 64 units, find the volume of the prism (in cubic units). Answer: 84480 units3 The ratio of the shortest to the longest is 3:11. Gathering the exact lengths: 11x − 3x = 64 8x = 64 x =8 Thus, each proportion mark should be multiplied by 8 to get their exact lengths. 3: 5: 11 ≈ 24:40: 88 Therefore, the volume is: 𝑉 = 𝑙𝑤ℎ = 24 ⋅ 40 ⋅ 88 = 84480 units3

Combinatorics 25) How many ways are there to choose 1 president, 1 vice-president, and a secretary among 6 people? Answer: 120 Unlike the common combinations problems, there is a specific order amongst each set. So, permutation should be used. 6 different people and 3 different positions = 6P3 !

26) How many different 10-letter words can be created by shuffling the word “PHILIPPINE”? Answer: 100,880 There are a total of 10 letters, the letter “I” and letter “P” are repeated thrice. 10!/(3!3!) = 100800

27) How many ways are there to distribute 9 distinct balls into 5 distinct boxes? Answer: 715

Using the general stars and bars

formula

28) A deck of 52 cards are shuffled and arranged randomly. If the first drawn card is a heart, how more many cards must be drawn to ensure that the probability of the next draw is at least 0.5 for a heart? Answer: 27 A deck has 4 suites, each having 13 cards. A heart was drawn so there are only 12 hearts left. The worst case is when all 12 hearts are picked last, and the probability will become 0.5 when there are an equal number of hearts and non-hearts left. So, out of 51 cards, there should be 24 left to ensure the condition on the worst case. The cards drawn should be 51 – 24 = 27 cards.

29) There is a Science books, 3 identical Chinese books, 3 identical Music books, 2 identical English books to be arranged in a shelf. How many different possible arrangement(s) is/are there? Answer: 5040 This is done by calculating the total combinations of the elements and dividing amongst the combinations of each identical element

30) Given (𝑎, 𝑏, 𝑐, 𝑑) is a set of integers bounded by the

d ≥ 1. Find the number of

conditions a ≥ 0, b ≥ -2, c ≥ 0, solution sets of a + b + c + d = 13.

Answer: 455 Summing up the constraints, -2 + 1 = -1, to be added to the right hand sign. The equation hence becomes a + b + c + d = 12. Using the formula n+k-1Ck-1 12+4-1C4-1 = 15C3= 455