HANS WOYDA MATHEMATICS QUIZ COMPETITION 1999/2000



HANS WOYDA MATHEMATICS QUIZ COMPETITION 2010/2011

KNOCK-OUT 2

SECTION 1 - STARTERS (INDIVIDUAL)

Marks: 2 marks to either or both competitors for the correct answer

Time: 30 seconds.

Year

7-9 1) A digital photograph measures 5 cm × 7 cm. It is made up of square pixels of side 0.1 mm. Find how many pixels there are in the photograph.

10-11 2) The diagram shows the rough positions of a coastguard O and two ships A and B. The coastguard can see ship A on a bearing of 059º, and ship B at a range of 10 km. Given that A and B are 8 km apart, and A is 6km away from O, find the bearing of B from A.

12 3) Find how many of the digits of the value of 99972 are 0.

13 4) Given that [pic] , evaluate cos2θ – 4sin2θ .

7-9 5) Each child in a class of 25 children sends every other child a birthday card. Each card costs £2. Find the cost of the cards sent during the year.

10-11 6) Given that x > 0, simplify [pic].

12 7) A ladder AB with midpoint M initially rests upright against a vertical wall. The end B slides along the horizontal floor, with end A sliding down the wall as shown, until the ladder rests on the floor. Describe the locus of M.

13 8) Given that [pic] , b = [pic] , c = [pic] , d = [pic] , write the letters a, b, c, d in order of increasing magnitude.

HANS WOYDA MATHEMATICS QUIZ COMPETITION 2010/2011

KNOCK-OUT 2

SECTION 2 - GEOMETRY AND TRIGONOMETRY (PAIRS)

Marks: 2 marks to either or both pairs for the correct answer

Time: 90 seconds.

Year

7-11 1) The diagram shows an equilateral triangle and a square. Find the size of angle x.

12-13 2) The diagram shows a regular icosagon (20 sided polygon). Find the size of angle ABC.

7-11 3) AD is a diameter of a circle with centre O. Points B and C lie on that circle, as shown. Angle BCA = 35º. Find the size of angle BAD

12-13 4) CE touches a circle at D, as shown. AB = AD. Angle CDA = θ º. Find an expression for angle CEA in terms of θ.

HANS WOYDA MATHEMATICS QUIZ COMPETITION 2010/2011

KNOCK-OUT 2

SECTION 3 - MENTAL ARITHMETIC AND PROBABILITY (INDIVIDUAL)

Marks: 2 or 1 to opponent

Time: 60 seconds

Questions 1 and 2 are to be done mentally

Year

7-9 1) Find the sum of the integers from 10 to 20 (inclusive).

2) Find how many zeros there are at the end of the product of all the integers from 10 to 20 (inclusive)

Pencil and paper may be used in the remaining questions

10-11 3) The probability that I stop at the first set of traffic lights is [pic] , and the probability that I stop at the second set is [pic] . The two sets of lights operate independently. Find the probability that I stop at least once in passing the two sets of lights.

4) Find the probability of rolling two even numbers when I roll two normal unbiased dice.

12 5) Three normal unbiased dice are rolled. Find the probability that they all show the same number.

6) Three normal unbiased dice are rolled. Find the probability that they all show different numbers.

13 7) The letters in my Scrabble® rack are A, B, C, C, C, D, D. Find in how many different ways I can arrange these letters.

8) Find how many different three-letter “words” (they do not have to make sense) I can make with these letters: A, B, C, D, E, F, F.

HANS WOYDA MATHEMATICS QUIZ COMPETITION 2010/2011

KNOCK-OUT 2

SECTION 4 - TEAM QUESTION

Time: 5 minutes.

Imagine you have these ten dominos:

[pic]

You have to arrange six of them in this pattern:

As usual with dominos, squares that touch must contain equal numbers of dots.

For example:

which you may write

like this to save time:

Find as many such arrangements as you can in which the square marked * contains no dots. There is a bonus mark if you state how many solutions there are, even if you do not show them all on the answer sheet.

Marks: Give 1 point for each correct answer and -1 point for each wrong answer; no penalty for omissions, giving a total of T points. Each team scores T/7 marks, rounded to the nearest integer, with minimum score zero, plus 1 bonus mark if the number of solutions is stated.

HANS WOYDA MATHEMATICS QUIZ COMPETITION 2010/2011

KNOCK-OUT 2

SECTION 4 - TEAM QUESTION

Answer sheet

(provide several copies)

HANS WOYDA MATHEMATICS QUIZ COMPETITION 2010/2011

KNOCK-OUT 2

SECTION 4 - TEAM QUESTION

Solutions

There are two types of solution, in each of which a, b, c is a permutation of 1, 2, 3

and x is either 1 or 2 or 3.

So there are 36 solutions altogether.

HANS WOYDA MATHEMATICS QUIZ COMPETITION 2010/2011

KNOCK-OUT 2

SECTION 5 - CALCULATORS (INDIVIDUAL)

Marks: 2 to either or both competitors for the correct answer

Time: 90 seconds

You are reminded that the written questions are to be given simultaneously to the respective pupils at the beginning of this section.

Year

7-9 1) Given that x = 2.345, evaluate [pic] , giving your answer correct to 4 significant figures

10-11 2) Given that [pic] and [pic] , evaluate [pic] , giving your answer correct to 3 significant figures.

12 3) Starting with x1 = 2, continue the iteration [pic] until successive terms agree correct to 3 significant figures, and give this value.

13 4) The equation [pic] (where θ is in radians) has one solution with 0 < θ < π/2. Find this solution, giving your answer correct to 3 significant figures.

HANS WOYDA MATHEMATICS QUIZ COMPETITION 2010/2011

KNOCK-OUT 2

SECTION 6 - ALGEBRA AND CALCULUS (INDIVIDUAL)

Marks: 2 or 1 for opponent

Time: 60 seconds.

Throughout this section x is a positive integer and f(x) = difference between the largest and smallest numbers which can be formed by writing the digits of x in some order.

For example f(341) = 431 – 134 = 297 f(860) = 860 – 68 = 792 (notice that 068 is allowed and means 68) f(39) = 93 – 39 = 54.

f2(x) means f(f(x)), with similar meanings for higher powers.

Year

7-9 1) Find f(684) .

2) Find f(144) .

10-11 3) Find f2(701) .

4) Find f2(232)

12 5) Given that 100 ≤ x ≤ 200, find a value of x for which 0 < f(x) < 100.

6) It can be proved that for 100 ≤ x ≤ 999 the largest possible value of f(x) is 891. Find the largest value of x in this range for which f(x) = 891.

13 7) Given that 100 ≤ x ≤ 999, it can be proved that f5(x) is either 0 or N. Find the value of N.

8) Given that 100 ≤ x ≤ 999, it can be proved that f(x) is either 0 or is divisible by the same two prime factors. Find these two prime factors.

HANS WOYDA MATHEMATICS QUIZ COMPETITION 2010/2011

KNOCK-OUT 2

SECTION 7 - RACE (INDIVIDUAL)

Marks: 2 or 0

Time: 60 seconds.

Year

7-9 1) Write down the number that is five hundred and eighty six less than a million.

10-11 2) Write down the number that is two thousand and ten less than eight hundred thousand.

12 3) Different permutations of the digits 1, 6 and 9 make numbers which are perfect squares. If x = [pic], y = [pic] and z = [pic], find the value of x + y + z.

13 4) Three numbers are in arithmetic progression.

The largest is 347 and their mean is 223.

Find the sum of the three numbers.

7-9 5) I bought seven packets of beef sandwiches and paid with a twenty pound note.

I noticed that the change was exactly the price of each packet of sandwiches. Find how much change I got.

10-11 6) Four circles of radius 10 cm are arranged so that they touch and their centres form a square as shown. Find an exact expression for the shaded area.

12 7) Find which of the following numbers are not divisible by 99:

1980, 9018, 1089, 8019, 8091 .

13 8) Given that [pic] and [pic], express xz in terms of y, giving your answer in the form [pic].

HANS WOYDA MATHEMATICS QUIZ COMPETITION 2010/2011

KNOCK-OUT 2

ANSWERS (allow equivalent answers)

SECTION 1

1. 350 000

2. 329 (º)

3. 3

4. 0

5. (£) 1200

6. 3x – 1

7. Quarter circle (centre O)

8. c, d, b, a

SECTION 2

1. 75 (º)

2. 63 (º)

3. 55 (º)

4. 3θ – 180 (º)

SECTION 3

1. 165

2. 3

3. 4/7

4. 1/4

5. 1/36

6. 5/9

7. 420

8. 135

SECTION 4

Please see question sheet

SECTION 5

1. 1.030 [must be 4 s.f.]

2. -1.481

3. 2.45

4. 1.19

SECTION 6

1. 396

2. 297

3. 594

4. 0

5. 100 or 101 or 110

6. 990

7. 495

8. 3, 11 [both needed]

SECTION 7

1. 999 414

2. 797 990

3. 58

4. 669

5. (£) 2.50

6. 400 - 100π (cm2)

7. 9018, 8091 [both needed]

8. [pic]

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