2011 Kangaroo Math Student Test

Q1. In the picture on the right a number should be written next to each point. The sum of the numbers on the corners of each side of the hexagon should be equal. Two numbers have already been inserted. Which number should be in the place marked ‘x’?

3 points

Q2. Three racers take part in a Formula-1-Race: Michael, Fernando and Sebastian. From the start Michael is in the lead in front of Fernando who is in front of Sebastian. In the course of the race Michael and Fernando overtake each other 9 times, Fernando and Sebastian 10 times and Michael and Sebastian 11 times. In which order do those three end the race?

3 points

Q4. Jan cannot draw very accurately but nevertheless he tried to produce a roadmap of his village. The relative position of the houses and the street crossings are all correct but three of th e roads are actually straight and only Qurwik street is not. Who lives in Qurwik street?

3 points

Q6. Given are a regular hexagon with si de-length 1, six squares and six equilateral triangles as shown on the right. How big is the perimeter of this tessellation? √ ଷ

3 points

Q7. A rectangular piece of paper is wrappe d around a cylinder. Then an angled straight cut is made through the points X and Y of the cylinder as shown on the left. The lower part of the piece of paper is then unrolled. Which of the following pictures could show the result?

3 points

Q9. Andrew wrote down all odd numbers from 1 to 2011 on a board. Bob then deleted all multiples of three. How many numbers remained on the board?

3 points

Q10. Max and Hugo roll a number of dice in order to decide who has to be the first one to jump into the cold lake. If there is no six, then Max has to jump. If there is one six, then Hugo has to jump and if there are several sixes neither will have to jump in. How many dice do they have to use so that the probability of either of them having to jump in is equal?

3 points

Q11. A rectan g g le is split i n n to three sm a a ller rectang l l es. One of w w hich has th e e measurem e e nts 7 by 11. Another one has t t he measure m m ents 4 by 8 . Determine the measure m m ents of the third rectan g g le so that it s area is as large a a s possible.

4 points

Q12. Michael wants to wr i i te whole nu m m bers into t h h e empty fie l l ds of the 3× 3 table on t h h e right so that the s s um of the n u u mbers in e a a ch 2×2 squa r r e equals 10 . . Four numb e e rs have alr e e ady been written d d own. Whic h h of the follo w w ing values could be th e e sum of the r r emaining fi v v e numbers ? ?

4 points

Q13. 48 child r r en are goin g g on a ski tri p p . Six of whi ch go with e e xactly one s i i bling, nine g g o with exa c c tly two siblings a a nd four wit h h three sibli n n gs. The re m m aining child r r en go with o o ut siblings. How many f f amilies are g g oing on th e e trip?

4 points

Q14. How ma n n y graphs o f f the functio n n s ݕ=ݔ , ݕ =െ ݔ , ݕ = = + √ ݔ , ݕ= െ √ ݔ , ݕ=+ √ െ െ ݔ , ݕ=െ √ √ െݔ , ݕ=+ ඥ ඥ | ݔ | , ݕ=െ ඥ ඥ | ݔ | are incl u u ded in the s s ketch on the right ? ?

4 points

Q15. The rear window wi p p er of a car i s s made in a w w ay so that t h h e rod r and the wiper bl a a de w are eq u u ally long a n n d are conne cted at an a n n gle α . The w w iper rotates a r r ound the ce n n tre of rotati on O and w i i pes over the area shown on the right. Ca lculate the a n n gle β betw e e en the right edge of the c c leaned area and the tangent o o f the curve d d upper edge . . ଷగିఈ ఈ ଷగ గ ఈ

4 points

Q16. We have three horiz o o ntal lines a n n d three para l l lel, sloped l ines. Both o f f the circles s h h own touch f f our of the li n n es. X, Y a n n d Z are the a a reas of the g g rey regions. D D is the are a a of the paral lelogram P Q Q RS. At leas t t how many o o f the areas X, Y, Z and D d d oes one ha v v e to know i n n order to be able to dete r r mine the area o o f the parall e e logram T?

4 points

Q17. In the (x , , y)-plane th e e co-ordinate axes are po s s itioned as u s s ual. Point A A (1, -10) wh i i ch 2 is on the parabola y = = ax + bx + c was mark e e d. Afterwa r r ds the co-or d d inate axis a nd the majo r r ity of the p a a rabola were deleted. W h h ich of the f o o llowing stat ements coul d d be false? 2

4 points

Q18. The side s s AB, BC, C C D, DE, EF a a nd FA of a h h exagon all t t ouch the sa m m e circle. T h h e measure m m ents of the sides AB, B C, CD, DE a a nd EF are i n n this order 4 4 , 5, 6, 7 and 8. How lon g g is side FA?

4 points

Q19. Which i s s the smalles t t possible p o o sitive, whol e e number va l l ue of the ex p p ression K ⋅ A ⋅ N N ⋅ G ⋅ A ⋅ R ⋅ O ⋅ O if di f f ferent letter s s stand for d i i fferent digit s s not equal t o o 0 and the s s ame G G ⋅ A ⋅ M ⋅ E letters st a a nd for the s a a me digits?

4 points

Q20. The brothers Gerhard and Günther pass on informatio n about the members of their chess club. Gerhard says: “All members of our club are male with five exceptions.” Günther says: “In each group of six members there are at least four female memb ers.” How many members does the chess club have?

4 points

Q21. In a drum there are a number of balls. A different positive whole number is written on each ball. On 30 of the balls numbers that are divisible by 6 are written, on 20 balls numbers that are divisible by 7 are written and on 10 balls numbers that are divisible by 42 are written. What is the minimum number of balls in the drum?

5 points

Q22. Given are the two arithmetic sequences 5, 20, 35, … and 35, 61, 87,… . How many different arithmetic sequences of positive whole numbers do both sequences have as subsequences?

5 points

Q23. The function sequence ݂ ଵ ሺ ݔ ሻ ,݂ ଶ ሺ ݔ ሻ ,⋯ , fulfills the conditions ݂ ଵ ሺ ݔ ሻ =ݔ and ݂ ௡ାଵ ሺ ݔ ሻ = ଵି௙ ೙ ሺ ೣ ሻ Determine the value of ݂ ଶ଴ଵଵ ሺ 2011 ሻ .

5 points

Q24. In a box there are red and green balls. If two balls ar e taken out of the box at random, the probability of them both being the same colour is ½. Which of the following could be the total number of balls in the box?

5 points

Q25. An airline does not charge for luggage if it is belo w a certain weight. For each additional kg of weight there is a charge. Mr. and Mrs. Raiss had 60 kg of luggage and paid 3 €. Mr. Wander also had 60 kg of luggage but had to pay 10.50 €. How many kg of l uggage per passenger were transported for free?

5 points

Q27. An archer tries his art on the target shown below on the right. With each of his three arrows he always hits the target. How many different sco res could he total with three arrows?

5 points

Q28. Let a, b and c be positive whole numbers for which the following holds true a = 2b = 3c . What is the minimum number of factors of abc if 1 and abc are counted as well?

5 points

Q29. Twenty different positive whole numbers are written into a 4 × 5 table. Two numbers in cells that have one common sideline, always have a common factor greater than 1. Determine the smallest possible value of n, if n is to be the biggest number in the table.

5 points

Q30. A 3 × 3 × 3 die is assembled out of 27 identical small dice. One plane, perpendicular to one of the space diagonals of the die goes through the midpoint of the die. How many of the smaller dice are cut by this plane?

5 points

2011 Student (Grade 11 - 12)

Q1. In the picture on the right a number should be written next to each point. The sum of the numbers on the corners of each side of the hexagon should be equal. Two numbers have already been inserted. Which number should be in the place marked ‘x’?

Q3. If 2 = 15 and 15 = 32 then xy equals

Q4. Jan cannot draw very accurately but nevertheless he tried to produce a roadmap of his village. The relative position of the houses and the street crossings are all correct but three of th e roads are actually straight and only Qurwik street is not. Who lives in Qurwik street?

Q5. All four-digit numbers whose digit sum is 4 are wr itten down in descending order. In which position is the number 2011?

Q6. Given are a regular hexagon with si de-length 1, six squares and six equilateral triangles as shown on the right. How big is the perimeter of this tessellation? √ ଷ

Q7. A rectangular piece of paper is wrappe d around a cylinder. Then an angled straight cut is made through the points X and Y of the cylinder as shown on the left. The lower part of the piece of paper is then unrolled. Which of the following pictures could show the result?

Q8. Determine the area of the quadrilateral PQRS pictured on the right, where PS = RS, ∠ PSR = ∠ PQR = 90°, ST ∞ PQ, and ST = 5.

Q9. Andrew wrote down all odd numbers from 1 to 2011 on a board. Bob then deleted all multiples of three. How many numbers remained on the board?

Q11. A rectan g g le is split i n n to three sm a a ller rectang l l es. One of w w hich has th e e measurem e e nts 7 by 11. Another one has t t he measure m m ents 4 by 8 . Determine the measure m m ents of the third rectan g g le so that it s area is as large a a s possible.

Q13. 48 child r r en are goin g g on a ski tri p p . Six of whi ch go with e e xactly one s i i bling, nine g g o with exa c c tly two siblings a a nd four wit h h three sibli n n gs. The re m m aining child r r en go with o o ut siblings. How many f f amilies are g g oing on th e e trip?

Q14. How ma n n y graphs o f f the functio n n s ݕ=ݔ , ݕ =െ ݔ , ݕ = = + √ ݔ , ݕ= െ √ ݔ , ݕ=+ √ െ െ ݔ , ݕ=െ √ √ െݔ , ݕ=+ ඥ ඥ | ݔ | , ݕ=െ ඥ ඥ | ݔ | are incl u u ded in the s s ketch on the right ? ?

Q18. The side s s AB, BC, C C D, DE, EF a a nd FA of a h h exagon all t t ouch the sa m m e circle. T h h e measure m m ents of the sides AB, B C, CD, DE a a nd EF are i n n this order 4 4 , 5, 6, 7 and 8. How lon g g is side FA?

Q22. Given are the two arithmetic sequences 5, 20, 35, … and 35, 61, 87,… . How many different arithmetic sequences of positive whole numbers do both sequences have as subsequences?

Q23. The function sequence ݂ ଵ ሺ ݔ ሻ ,݂ ଶ ሺ ݔ ሻ ,⋯ , fulfills the conditions ݂ ଵ ሺ ݔ ሻ =ݔ and ݂ ௡ାଵ ሺ ݔ ሻ = ଵି௙ ೙ ሺ ೣ ሻ Determine the value of ݂ ଶ଴ଵଵ ሺ 2011 ሻ .

Q24. In a box there are red and green balls. If two balls ar e taken out of the box at random, the probability of them both being the same colour is ½. Which of the following could be the total number of balls in the box?

Q26. Determine the sum of all positive whole numbers x less than 100 so that x² - 81 is a multiple of 100.

Q27. An archer tries his art on the target shown below on the right. With each of his three arrows he always hits the target. How many different sco res could he total with three arrows?

Q28. Let a, b and c be positive whole numbers for which the following holds true a = 2b = 3c . What is the minimum number of factors of abc if 1 and abc are counted as well?

Q29. Twenty different positive whole numbers are written into a 4 × 5 table. Two numbers in cells that have one common sideline, always have a common factor greater than 1. Determine the smallest possible value of n, if n is to be the biggest number in the table.

Q30. A 3 × 3 × 3 die is assembled out of 27 identical small dice. One plane, perpendicular to one of the space diagonals of the die goes through the midpoint of the die. How many of the smaller dice are cut by this plane?