Kangourou sans Frontières

Transcription

Kangourou sans Frontières
Selected problems for KSF contest 2014
KSF 2014 – selected problems PreEcolier
3 points
# 1 (2451). Ladybird
will sit on a flower that has five petals and three leaves. On
which of the following flowers will ladybird sit?
(A)
(B)
(D)
(E)
(C)
# 2 (2854). In what order do you meet the shapes starting from the arrow?
1
(A) N, , •
(B) N, •, (C) •, N, (D) , N, •
(E) , •, N
# 3 (2564). How many more grey squares than white ones can you see?
(A) 6
(B) 7
(C) 8
(D) 9
(E) 10
# 4 (3312). Put the animials in line from the smallest to the largest. What animal is in the middle?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
# 5 (2563).
Ann has twelve of these tiles
one line with the design. Ann starts at the left side. How does the line end?
(A)
(B)
(C)
(D)
2
. She makes
(E)
# 6 (3277). Which is the shadow of the girl?
(A)
(B)
(D)
(E)
(C)
# 7 (2856). A square was composed of 25 small squares, but some of these small squares are lost.
How many are lost?
(A) 6
(B) 7
(C) 8
(D) 10
# 8 (3279). How many ducks balance the crocodile?
3
(E) 12
(A)
(B)
(D)
(E)
(C)
4 points
# 9 (3526). When the ant
goes from home
following these arrows: → 3, ↑ 3, → 3, ↑ 1, it
comes to the ladybird
Which animal would it come to, if it goes from home following these arrows: → 2, ↓ 2, → 3, ↑ 3,
→ 2, ↑ 2?
(A)
(B)
(C)
(D)
(E)
(D) 4
(E) 5
# 10 (3607). The kangaroo is inside how many circles?
(A) 1
(B) 2
(C) 3
# 11 (2865). A square was cut into 4 parts as shown in the picture. Which of the following shapes
4
cannot be made with these 4 parts?
(A)
(B)
(D)
(E)
(C)
# 12 (2569). Which form fits exactly the one given above?
(A)
(B)
(D)
(E)
(C)
# 13 (2867). Walking from K to O along the lines pick up the letters KANGAROO in the correct
order. What is the lenght of the shortest walk in meters?
1m
1m
(A) 16 m
(B) 17 m
(C) 18 m
(D) 19 m
(E) 20 m
# 14 (3433). How many numbers are greater than 10 and less than or equal to 31 which can be
written with digits 1, 2 or 3 only? You can repeat digits.
(A) 2
(B) 4
(D) 7
(C) 6
5
(E) 8
# 15 (2570). Seven sticks lie on top of each other.
2 is at the bottom. Stick 6 is at the top. Which stick is in the middle?
(A) 1
(B) 3
(C) 4
(D) 5
Stick
(E) 7
# 16 (3311). How many frogs did the three pelicans catch?
(A) 1
(B) 2
(C) 4
(D) 9
(E) 12
5 points
# 17 (3307). The chess board is damaged. How many black squares on the right side of the line
are missing?
(A) 11
(B) 12
(C) 13
(D) 14
(E) 15
# 18 (3398). Rabbit Venya eats cabbages and carrots. Each day he eats either 10 carrots, or 2
6
cabbages. Last week Venya ate 6 cabbages. How many carrots did he eat?
(A) 20
(B) 30
(C) 34
(D) 40
(E) 50
# 19 (2455). What should you put in the square to get a correct diagram?
(A) −38
(C) −45
(B) : 8
(D) ·6
(E) : 6
# 20 (2684). Put the digits 2, 3, 4 and 5 in the squares and calculate the sum to get the largest
+
value. What is that value?
(A) 68
(B) 77
(C) 86
(D) 95
(E) 97
# 21 (2870). The central cell of the square was removed. We cut it into equal pieces. Which piece
is not possible to get?
(A)
(B)
(C)
(D)
(E)
# 22 (2539). To get the product of 2 x 3 x 15, Bill has to press the keys of his calculator seven
times:
Bill wants to multiply all numbers from 3 to 21, using his
calculator. At least, how many times will he press the keys of his calculator?
(A) 19
(B) 31
(D) 50
(C) 37
(E) 60
# 23 (2879). Fedya has 4 red cubes, 3 blue cubes, 2 green cubes and 1 yellow cube. He builds a
tower (see the picture) in such a way that no two adjacent cubes have the same colour. What is the
?
colour of the middle cube?
(A) red
(B) blue
(C) green
7
(D) yellow
(E) impossible to determine
# 24 (2571). Cogwheel A turns round completely once. At which place is x now?
(A) a
(B) b
(C) c
(D) d
8
(E) e
KSF 2014 – selected problems Ecolier
3 points
# 1 (3062). Which drawing is the central part of the picture with the star?
(A)
(B)
(C)
(D)
(E)
# 2 (3340). Jacky wants to insert the digit 3 somewhere in the number 2014. Where should she
insert the digit 3 if she wants her five-digit number to be as small as possible?
(A) in front of 2014
(B) between the 2 and the 0
(C) between the 0 and the 1
(D) between the 1 and the 4
(E) behind 2014
# 3 (3284). Which houses are made using exactly the same pieces of triangular or rectangular
shape?
New diagram required.
(A) 1, 4
(B) 3, 4
(C) 1, 4, 5
(D) 3, 4, 5
(E) 1, 2, 4, 5
# 4 (3048). When Koko the Koala does not sleep, he eats 50 grams of leaves per hour. Yesterday,
he slept 20 hours. How many grams of leaves did he eat yesterday?
(A) 0
(B) 50
(C) 100
(D) 200
(E) 400
# 5 (2578). Maria subtracts and gets as results the numbers from zero to five. She connects the
dots, starting at the dot with the result 0 and ending at the dot with the result 5. Which figure does
she get?
2-2
6-5
8-6
11-8
13-9
17-12
(A)
(B)
(C)
(D)
(E)
# 6 (3493). Adam built fewer sandcastles than Martin but more than Susan. Lucy built more
sandcastles than Adam and more than Martin. Dana built more sandcastles than Martin but fewer
1
than Lucy. Who of them built the most sandcastles?
(A) Martin
(B) Adam
(C) Susan
(D) Dana
(E) Lucy
# 7 (2574). Monica writes numbers in the diagram so that each number is the product of the two
numbers below it. Which number should she write in the grey cell?
(A) 0
(B) 1
(C) 2
(D) 4
(E) 8
# 8 (2580). Ann has four pieces as shown.
With these pieces
she can completely cover the shape. Where should she put the piece
?
(A)
(B)
(D)
(E)
(C)
4 points
# 9 (3189). Mr. Brown has painted flowers on the store window (see picture). How does thess
flowers look like from the other side of the window?
2
(A)
(B)
(C)
(D)
(E)
# 10 (3115). Some candies were in a bowl. Sally took half of the candies. Then Tom took half of
the candies left in the bowl. After that Clara took half of the remaining candies. In the end there
were 6 candies in the bowl. How many candies were in the bowl at the beginning?
(A) 12
(B) 18
(C) 20
(D) 24
(E) 48
# 11 (3157). Which tile must be added in the picture so that the white area is as big as the black
?
area?
(A)
(B)
(D)
(E)
(C)
# 12 (3004). Paula shoots arrows at the following target. When she misses, she obtains zero points.
Paula shoots two arrows and adds the number of points. Which of the following sums cannot be her
30
50
70
score?
(A) 60
(B) 70
(C) 80
(D) 90
(E) 100
# 13 (2529). Mary had equal number of grey, black, and striped tokens. She used some of these
3
tokens to make a pile. You can see all used tokens in the figure. She still has five tokens which are
not on the pile. How many black tokens did she have at the beginning?
(A) 5
(B) 6
(C) 7
(D) 15
(E) 18
# 14 (3399). Rabbit Borya likes cabbages and carrots very much. In a day he eats either 9 carrots,
or 2 cabbages, or 1 cabbage and 4 carrots. During one week Borya has eaten 30 carrots. How many
cabbages has he eaten during this week?
(A) 6
(B) 7
(C) 8
(D) 9
(E) 10
# 15 (3498). The solid in the picture was made by sticking eight equal cubes together. How does
this solid look from above?
(A)
(B)
(C)
(D)
# 16 (2582).
are there in this picture?
(A) 180
(B) 181
(E)
How many dots
(C) 182
(D) 183
5 points
4
(E) 265
# 17 (3360). On Kangaroo planet each kangyear has 20 kangmonths and each kangmonth has 6
kangweeks. How many kangweeks are there in one quarter of a kangyear?
(A) 9
(B) 30
(C) 60
(D) 90
(E) 120
# 18 (2586).
Seven children are standing in a circle. No two boys are standing
next to each other. No three girls are standing next to each other. Which of these is true for the
number of girls standing in the circle?
(A) only 3 is possible
(B) 3 and 4 are possible
(D) 4 and 5 are possible
(E) only 5 is possible
(C) only 4 is possible
# 19 (3652). Eve arranged cards in a line as it is shown in the figure below. At each move Eve is
allowed to interchange any two cards. What is the smallest number of moves Eve needs to get the
word KANGAROO?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6
# 20 (2575). A sequence of triangles of diamonds is made. The first three stages are shown. In
each stage a line of diamonds is added. In the bottom line the outer diamonds are white. All other
diamonds in the triangle are black. How many black diamonds has the figure in stage 6?
1
(A) 19
2
(B) 21
3
(C) 26
(D) 28
(E) 34
# 21 (3316). Kangaroo Hamish bought toys and gave the shop-assistant 150 Kangcoins. He received
20 Kangcoins back. Then he changed his mind and exchanged one of the toys for another. He got
back an additional 5 Kangcoins. What toys did Hamish leave the store with?
5
(A) the carriage and the plane
(B) the carriage and the bus
(C) the carriage and the tram
(D) the motorcycle and the tram
(E) the bus, the motorcycle and the tram
# 22 (2754). Write each of the numbers 0, 1, 2, 3, 4, 5, 6 in the squares to make the addition
+
correct. Which digit will be in the grey square?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6
# 23 (3108). What is the largest number of small squares which can be shaded so that no square
made of four shaded small squares appears on the figure?
(A) 18
(B) 19
(C) 20
(D) 21
(E) 22
# 24 (3402). Nick has written each of the numbers from 1 to 9 in the cells of the 3 × 3 table. Only
four of these numbers can be seen in the figure. Nick has noticed that for the number 5 the sum of the
numbers in the neighbouring cells equals 13 (neighbouring cells are cells sharing sides). He noticed
the same applies for the number 6. Which number has Nick written in the shaded cell?
(A) 5
(B) 6
(C) 7
(D) 8
6
(E) 9
KSF 2014 – selected problems Benjamin
3 points
# 1 (3005). Arno spelled the word KANGAROO with cards showing one letter at a time. Unfortunately some cards were tipped. Tipping back twice he can correct the letter K and tipping once he
can correct the A - see the figures. How many times does he need to tilt for all of the letters to be
correct?
KA
K
K
A
K
(A) 4
R
(B) 5
(C) 6
A
N
A
G
(D) 7
(E) 8
# 2 (3554). A cake weights 900 g. Paul cuts it in 4 pieces. The biggest piece is as heavy as the 3
others weight altogether. What’s the weight of the biggest piece?
(A) 250 g
(B) 300 g
(C) 400 g
(D) 450 g
(E) 600 g
# 3 (2592). Two great rings, one grey, one white, are linked in each other. Peter, in front of the
rings, sees the rings as in the picture.
(A)
(B)
(D)
(E)
Paul is behind the rings. What does he see?
(C)
# 4 (3319). In the following addition, some of the digits have been replaced by stars.
1∗2
+1 ∗ 3
1∗4
−−−−−−−
= 309
What is the sum of the missing digits?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 10
# 5 (3042). What is the difference between the smallest 5-digit number and the largest 4-digit
number?
(A) 1
(B) 10
(C) 1111
(D) 9000
(E) 9900
# 6 (3385). A square of perimeter 48 cm is cut into 2 pieces to make a recangle (see picture). What
is the perimeter of the rectangle?If we cut a rectangle in half and place one piece above the other we
create a square whose area is 144 cm2 . What is the perimeter of the rectangle?
1
(A) 24 cm
(B) 30 cm
(C) 48 cm
(D) 60 cm
(E) 72 cm
# 7 (3006). Katrin has 38 matches. She builds a triangle and a square, using all the matches. Each
side of the triangle consists of 6 matches. How many matches are in each side of the square?
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8
# 8 (3160). The pearl necklace in the picture contains dark grey pearls and shiny white pearls.
Arno wants to have 5 of the dark grey pearls. He can only take pearls from either end of the
necklace, and so he has to take some of the white pearls also. What is the smallest number of white
pearls Arno has to take?
(A) 2
(B) 3
(C) 4
(D) 5
(E) 6
# 9 (2496). Harry participated in a broom flight contest which consisted of 5 laps. The times
when Harry passed the starting point are shown in the picture. Which lap took the shortest time?
(A) the first
(B) the second
(D) the fourth
(E) the fifth
(C) the third
# 10 (3205). Ben’s digital watch is not working properly. The three horizontal lines in the rightmost
digit on the watch are not displayed. Ben is looking at his watch and the time has just changed from
the one shown on the left to the one shown on the right. What time is it now?
(A) 12:40
(B) 12:42
(C) 12:44
(D) 12:47
(E) 12:49
4 points
# 11 (3161). Which tile must be added to the picture so that the white area is as large as the black
area?
?
(A)
(B)
(C)
(D)
2
(E) It is impossible.
# 12 (3394). Henry and John started walking from the same point. Henry went 1 km north, 2 km
west, 4 km south and finally 1 km west. John went 1 km east, 4 km south and 4 km west. Which of
the following must be the final part of his walk in order to reach the same point as Henry?
(A) He has already reached the same point.
(B) 1 km north.
(C) 1 km north-west.
(D) More than 1 km north-west.
(E) 1 km west.
# 13 (3114). At the summer camp, 7 pupils eat ice cream every day, 9 pupils eat ice cream every
second day and the rest of the pupils don’t eat ice cream at all. Yesterday, 13 pupils had ice cream.
How many pupils will eat ice cream today?
(A) 7
(B) 8
(C) 9
(D) 10
(E) it cannot be determined
# 14 (2399). Kangaroos A, B, C, D and E are sitting in that order, clockwise, around a circular
table. Exactly when the bell rings, each kangaroo but one exchanges its position with a neighbour.
The resulting positions, clockwise and starting with A, are A, E, B, D, C. Which kangaroo did not
move?
(A) A
(B) B
(C) C
(D) D
(E) E
# 15 (3057). A square can be formed using four of these five pieces. Which one will not be used?
A
(A) A
B
(B) B
C
D
(C) C
E
(D) D
(E) E
# 16 (2492). A natural number has three digits. When we multiply the digits we get 135. What
result do we get if we add the digits?
(A) 14
(B) 15
(C) 16
(D) 17
(E) 18
# 17 (3357). In a restaurant there are 16 tables, each having either 3, 4 or 6 chairs. Together, the
tables having 3 or 4 chairs can accommodate 36 people. Knowing that the restaurant can accommodate
72 people, how many tables are there with 3 chairs?
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8
# 18 (2759). The points A, B, C, D, E, F are on a straight line in that order. We know that
AF = 35, AC = 12, BD = 11, CE = 12 and DF = 16. What is the distance BE?
(A) 13
(B) 14
(C) 15
(D) 16
(E) 17
# 19 (3595). Parisa set her stones in groups on the desk. After she arranged the stones in groups
of 3, she found that there were 2 stones left. Then she arranged the stones in groups of 5, and again
3
there were 2 stones left. At least how many more stones does she need so that there won’t be any left
when she arranges them in groups of 3 and in groups of 5?
(A) 3
(B) 1
(C) 4
(D) 10
(E) 13
# 20 (3113). The faces of a cube are numbered 1, 2, 3, 4, 5, and 6. The faces 1 and 6 have a
common edge. The same is true for faces 1 and 5, faces 1 and 2, faces 6 and 5, faces 6 and 4, and
faces 6 and 2. Which number is on the face opposite the one with number 4?
(A) 1
(B) 2
(C) 3
(D) 5
(E) it cannot be determined
5 points
# 21 (2594). The 3 × 3 × 3 cube in the picture is made of 27 small cubes.
How many small cubes do you have to take away to see the following result when looking from the
right, from above, and from the front?
(A) 4
(B) 5
(C) 6
(D) 7
(E) 9
# 22 (3110). There are 5 songs: song A lasts 3 min, song B 2 min 30 s, song C 2 min, song D 1 min
30 s, and song E 4 min. These 5 songs are playing in the order A, B, C, D, E in a loop without any
breaks. Song C was playing when Andy left home. He returned home exactly one hour later. Which
song was playing when Andy got home?
(A) A
(B) B
(C) C
(D) D
(E) E
# 23 (3403). Dan entered the numbers 1 to 9 in the cells of a 3x3 table. He began by placing the
numbers 1, 2, 3 and 4 as shown in the picture.
It happened that for the number 5, the sum of the numbers in the adjacent cells (having a common
side) is equal to 9. What is the sum of the numbers adjacent to the number 6?
(A) 14
(B) 15
(C) 17
(D) 28
(E) 29
# 24 (3415). Trees grow on only side of Park Avenue. There are 60 trees in total. Every second
tree is a maple, and every third tree is either a linden or a maple. The remaining trees are birches.
How many birches are there?
(A) 10
(B) 15
(C) 20
(D) 24
4
(E) 30
# 25 (3489). A thin colourful ribbon is stuck on a transparent plastic cube (see the picture).
Which of the following pictures doesn’t show the cube from any perspective?
(A)
(B)
(C)
(D)
(E)
# 26 (3096). The king and his messengers are travelling from the castle to the summer palace at a
speed of 5 km/h. Every hour, the king sends a messenger back to the castle, who travels at a speed of
10 km/h. What is the time interval between any two consecutive messengers arriving at the castle?
(A) 30 min
(B) 60 min
(C) 75 min
(D) 90 min
(E) 120 min
# 27 (3594). There were 3 one-digit numbers on the blackboard. Ali added them up, and got
15. Then he erased one of the numbers and wrote the number 3 in its place. Then Reza multiplied
the three numbers on the blackboard and got 36. What are the possibilities for the number that Ali
erased?
(A) either 6 or 7
(B) either 7 or 8
(C) only 6
(D) only 7
(E) only 8
# 28 (3400). Rabbit Vasya loves cabbages and carrots. In a day, he eats either 9 carrots, or 2
cabbages, or 1 cabbage and 4 carrots. But some days he only eats grass. Over the last 10 days, Vasya
ate a total of 30 carrots and 9 cabbages. On how many of these 10 days did he eat only grass?
(A) 0
(B) 1
(C) 2
(D) 3
(E) 4
# 29 (3635). In Fabuland, every sunny day is immediately preceded by two consecutive rainy days.
Also, five days after any rainy day, it is another rainy day. It is sunny today. For how many days at
most can we predict the weather with certainty?
(A) 1 day
(B) 2 days
(C) 4 days
(D) We cannot predict even one day ahead
(E) We can predict the weather every day from here on
# 30 (2602). Granny has 10 grandchildren. Alice is the eldest. One day, Granny notices that
her grandchildren all have different ages. If the sum of her grandchildrens’ ages is 180, what is the
youngest Alice could be?
(A) 19
(B) 20
(C) 21
(D) 22
5
(E) 23
KSF 2014 – selected problems Cadet
3 points
# 1 (2778). Each year, the date of the Kangaroo competition is the third Thursday of March. What
is the latest possible date of the competition in any year?
(A) 14th March
(B) 15th March
(C) 20th March
(D) 21st March
(E) 22nd March
# 2 (2729). How many quadrilaterals of any size are shown in the figure?
New Diagram required
(A) 0
(B) 1
(C) 2
(D) 4
(E) 5
# 3 (3555). What is the result of: 2014 · 2014 : 2014 − 2014?
(A) 0
(B) 1
(C) 2013
(D) 2014
(E) 4028
# 4 (3379). The area of rectangle ABCD is 10. Points M and N are midpoints of the sides AD
and BC. What is the area of quadrilateral M BN D?
(A) 0,5
(B) 5
(C) 2,5
(D) 7.5
(E) 10
# 5 (2620). The product of two numbers is 36 and their sum is 37. What is their difference?
(A) 1
(B) 4
(C) 10
(D) 26
(E) 35
# 6 (2816). Wanda has several square pieces of paper of area 4. She cuts them into squares and
right-angled triangles in the manner shown in the first diagram. She takes some of the pieces and
makes the bird shown in the second diagram. What is the area of the bird?
New diagram required (without the eye)
(A) 3
(B) 4
(C) 9/2
(D) 5
1
(E) 6
# 7 (3469). A bucket was half full. A cleaner added 2 litres to the bucket. The bucket was then
three-quarters full. What is the capacity of the bucket?
(A) 10 l
(B) 8 l
(C) 6 l
(D) 4 l
(E) 2 l
# 8 (3012). Georg built the shape shown using seven unit cubes. How many such cubes does he
have to add to make a cube with edges of length 3?
(A) 12
(B) 14
(C) 16
(D) 18
(E) 20
# 9 (3128). Which of the following calculations gives the largest result?
(A) 44 × 777
(B) 55 × 666
(D) 88 × 333
(E) 99 × 222
(C) 77 × 444
# 10 (3165). The necklace in the picture contains grey beads and white beads.
Arno takes one bead after another from the necklace. He always takes a bead from one of the ends.
He stops as soon as he has taken the fifth grey bead. What is the largest number of white beads that
Arno can take?
(A) 4
(B) 5
(C) 6
(D) 7
(E) 8
4 points
# 11 (2413). Jack has a piano lesson twice a week and Hannah has a piano lesson every other week.
In a given term, Jack has 15 more lessons than Hannah. How many weeks long is their term?
(A) 30
(B) 25
(C) 20
(D) 15
(E) 10
# 12 (3216). In the diagram, the area of each circle is 1cm2 . The area common to two overlapping
circles is 81 cm2 . What is the area of the region covered by the five circles?
(A) 4cm2
(B) 92 cm2
(C)
35
2
8 cm
2
(D)
39
2
8 cm
(E)
19
2
4 cm
# 13 (3296). This year a grandmother, her daughter and her granddaughter noticed that the sum
of their ages is 100 years. Each of their ages is a power of 2. How old is the granddaughter?
(A) 1
(B) 2
(C) 4
(D) 8
(E) 16
# 14 (2553). Five equal rectangles are placed inside a square with side 24 cm, as shown in the
diagram. What is the area of one rectangle?
(A) 12 cm2
(B) 16 cm2
(C) 18 cm2
(D) 24 cm2
(E) 32 cm2
# 15 (3391). The heart and the arrow are in the positions shown in the figure. At the same time
the heart and the arrow start moving. The arrow moves three places clockwise and the heart moves
four places anticlockwise and then stop. They continue the same routine over and over again. After
how many routines will the heart and the arrow land in the same triangular region for the first time?
(A) 7
(B) 8
(C) 9
(D) 10
(E) It will never happen
# 16 (3560). The diagram shows the triangle ABC in which BH is a perpendicular height and AD
is the angle bisector at A. The obtuse angle between BH and AD is four times the angle DAB (see
the diagram). What is the angle CAB?
(A) 30◦
(B) 45◦
(C) 60◦
(D) 75◦
(E) 90◦
# 17 (3146). Six boys share a flat with two bathrooms which they use every morning beginning at
7:00 o’clock. There is never more than one person in either bathroom at any one time They spend
8, 10, 12, 17, 21 and 22 minutes at a stretch in the bathroom respectively. What is the earliest time
that they can finish using the bathrooms?
(A) 7:45
(B) 7:46
(C) 7:47
(D) 7:48
(E) 7:50
# 18 (3550). A rectangle has sides of length 6 cm and 11 cm. One long side is selected. The
bisectors of the angles at either end of that side are drawn. These bisectors divide the other long side
3
into three parts. What are the lengths of these parts?
(A) 1 cm, 9 cm, 1 cm
(B) 2 cm, 7 cm, 2 cm
(D) 4 cm, 3 cm, 4 cm
(E) 5 cm, 1 cm, 5 cm
(C) 3 cm, 5 cm, 3 cm
# 19 (3414). Captain Sparrow and his pirate crew dug up several gold coins. They divide the
coins amongst themselves so that each person gets the same number of coins. If there were four fewer
pirates, then each person would get 10 more coins. However, if there were 50 fewer coins, then each
person would get 5 fewer coins. How many coins dig they dig up?
(A) 80
(B) 100
(C) 120
(D) 150
(E) 250
# 20 (2985). The average of two positive numbers is 30% less than one of them. By what percentage
is the average greater than the other number?
(A) 75%
(B) 70%
(C) 30%
(D) 25%
(E) 20%
5 points
# 21 (3404). Andy enters all the digits from 1 to 9 in the cells of a 3x3 table, so that each cell
contains one digit. He has already entered 1, 2, 3 and 4, as shown. Two numbers are considered to
be ’neighbours’ if their cells share an edge. After entering all the numbers he notices that the sum of
the neighbours of 9 is 15. What is the sum of the neighbours of 8?
(A) 12
(B) 18
(C) 20
(D) 26
(E) 27
# 22 (3150). An antique scale is not working properly. If something is lighter than 1000 g, the scale
shows the correct weight. However, if something is heavier than or equal to 1000 g, the scale can show
any number above 1000 g. We have 5 weights A g, B g, C g, D g, E g each under 1000 g. When
they are weighed in pairs, the scale shows the following: B + D = 1200, C + E = 2100, B + E = 800,
B + C = 900, A + E = 700. Which of the weights is the heaviest?
(A) A
(B) B
(C) C
(D) D
(E) E
# 23 (2509). Quadrilateral ABCD has right angles only at vertices A and D. The numbers show
the areas of two of the triangles. What is the area of ABCD?
(A) 60
(B) 45
(C) 40
(D) 35
(E) 30
# 24 (2987). Liz and Mary compete in solving problems. Each of them is given the same list of
100 problems. For any problem, the first of them to solve it gets 4 points, while the second to solve
it gets 1 point. Liz solved 60 problems, and Mary also solved 60 problems. Together, they got 312
4
points. How many problems were solved by both of them?
(A) 53
(B) 54
(C) 55
(D) 56
(E) 57
# 25 (3731). David rides his bicycle from Edinburgh to his croft. He was going to arrive at 15:00,
but he spent 2/3 of the planned time covering 3/4 of the distance. After that, he rode more slowly
and arrived exactly on time. What is the ratio of the speed for the first part of the journey to the
speed for the second part?
(A) 5 : 4
(B) 4 : 3
(C) 3 : 2
(D) 2 : 1
(E) 3 : 1
# 26 (3149). We have four identical cubes (see picture). They are arranged so that a big black
circle appears on one face, as shown in the second picture. What can be seen on the opposite face?
(A)
(B)
(C)
(D)
(E)
# 27 (2766). A group of 25 people consists of knights, serfs and damsels. Each knight always tells
the truth, each serf always lies, and each damsel alternates between telling the truth and lying. When
each of them was asked: ”Are you a knight?”, 17 of them said ”Yes”. When each of them was then
asked: ”Are you a damsel?”, 12 of them said ”Yes”. When each of them was then asked: ”Are you a
serf?”, 8 of them said ”Yes”. How many knights are in the group?
(A) 4
(B) 5
(C) 9
(D) 13
(E) 17
# 28 (3662). Several different positive integers are written on the board. Exactly two of them are
divisible by 2 and exactly 13 of them are divisible by 13. Let M be the greatest of these numbers.
What is the smallest possible value of M ?
(A) 169
(B) 260
(C) 273
(D) 299
(E) 325
# 29 (3148). On a pond there are 16 water lily leaves in a 4 by 4 pattern as shown. A frog sits on a
leaf in one of the corners. It then jumps from one leaf to another either horizontally or vertically. The
frog always jumps over at least one leaf and never lands on the same leaf twice. What is the greatest
number of leaves (including the one it sits on) that the frog can reach?
(A) 16
(B) 15
(C) 14
(D) 13
(E) 12
# 30 (3732). A 5 × 5 square is made from 1 × 1 tiles, all with the same pattern, as shown. Any two
adjacent tiles have the same colour along the shared edge. The perimeter of the large square consists
5
of black and white segments of length 1. What is the smallest possible number of such unit black
segments?
(A) 4
(B) 5
(C) 6
(E) 8
6
(D) 7
KSF 2014 – selected problems Junior
3 points
# 1 (2777). The date of the Kangaroo competition is the third Thursday in March in each year.
What is the first possible date of the competition?
(A) 14
(B) 15
(C) 20
(D) 21
(E) 22
# 2 (2407). The MSC Fabiola holds a record as being the largest container ship to enter San
Francisco Bay. It carries 12500 containers which if placed end to end would stretch about 75 km.
Roughly, what is the length of one container?
(A) 6 m
(B) 16 m
(C) 60 m
(D) 160 m
(E) 600 m
# 3 (2472). If a, b and c denote the lengths of the lines in the picture, then which of the following
is correct?
(A) a < b < c
(B) a < c < b
(C) b < a < c
# 4 (3196). Which number is in the middle of
(A)
11
15
(B)
7
8
(C)
(D) b < c < a
(E) c < b < a
2
4
and ?
3
5
3
4
(D)
6
15
(E)
5
8
# 5 (2785). In the number 2014 the last digit is bigger then the sum of the other three digits. How
many years ago did this last occur?
(A) 1
(B) 3
(C) 5
(D) 7
(E) 11
# 6 (2642).
The length of the edges of the big regular hexagon is two
times the length of the edges of the small regular hexagon. The small hexagon has an area of 4 cm2 .
What is the area of the big hexagon?
(A) 16 cm2
(B) 14 cm2
(C) 12 cm2
(D) 10 cm2
(E) 8 cm2
# 7 (2842). What is the negation of the following statement ”Everybody solved more than 20
problems”?
(A) Nobody solved more than 20 problems.
(B) Somebody solved less than 21 problems.
(C) Everybody solved less than 21 problems.
1
(D) Somebody solved exactly 20 problems.
(E) Somebody solved more than 20 problems.
# 8 (3152). In a coordinate system Tom drew a square. One of its diagonals lies on the x-axis. The
coordinates of the two vertices on the x-axis are (−1, 0) and (5, 0). Which of the following are the
coordinates of another vertex of this square?
(A) (2, 0)
(B) (2, 3)
(C) (2, −6)
(D) (3, 5)
(E) (3, −1)
# 9 (3386). In a certain village, the ratio between adult men and adult women is 2 : 3 and the ratio
between adult women and children is 8 : 1. What is the ratio between adults (men and women) and
children?
(A) 5 : 1
(B) 10 : 3
(C) 13 : 1
(D) 12 : 1
(E) 40 : 3
# 10 (2643).
The big wheel of this bicycle has perimeter 4.2
metres. The small wheel has perimeter 0.9 metres. At a certain moment, the valve of both wheels are
at their lowest point. The bicycle rolls to the left. After how many metres will both valves first be at
their lowest point together again?
(A) 4.2
(B) 6.3
(C) 12.6
(D) 25.2
(E) 37.8
4 points
# 11 (3298). A grandmother, her daughter and her granddaughter can this year say that the sum
of their ages is 100. In which year was the granddaughter born if each age is a power of 2?
(A) 1998
(B) 2006
(C) 2010
(D) 2012
(E) 2013
# 12 (3154). Paul put some rectangular paintings on the wall. For each picture he put one nail into
the wall 2.5 m above the floor and attached a 2 m long string at the two upper corners. Which of the
following pictures is closest to the floor (format: width in cm × height in cm)?
(A) 60 × 40
(B) 120 × 50
(C) 120 × 90
(D) 160 × 60
(E) 160 × 100
# 13 (3153). Six girls share a flat with two bathrooms which they use every morning beginning at
7:00 o’clock. They use the bathroom one at a time, and sit down to eat breakfast together as soon as
the last girl has finished. They spend 9, 11, 13, 18, 22 and 23 minutes in the bathroom respectively.
Being well organized, what is the earliest they can have breakfast together?
(A) 7:48
(B) 7:49
(C) 7:50
2
(D) 7:51
(E) 8:03
# 14 (2549). In the following figure there is a regular octagon. The shaded area measures 3 cm2 .
Find the area of the octagon in cm2 .
√
(A) 8 + 4 2
(B) 9
√
(C) 8 2
(D) 12
(E) 14
# 15 (3456). A new kind of crocodile has been discovered in Africa. The length of his tail is a
third of his entire length. His head is 93 cm long and its length is a quarter of the crocodile‘s length
without his tail. How long is this crocodile in cm?
(A) 558
(B) 496
(C) 490
(D) 372
(E) 186
# 16 (3461). In the picture there is a special dice. Numbers on the opposite faces always make the
same sum. The numbers that we cannot see in the picture are all prime numbers. Which number is
opposite of 14?
(A) 11
(B) 13
(C) 17
(D) 19
(E) 23
# 17 (2635). Ann has walked 8 km with a velocity of 4 km/h. Now she will run some time with a
velocity of 8 km/h. How long does she have to run in order to have an overall average velocity of 5
km/h?
(A) 15 min
(B) 20 min
(C) 30 min
(D) 35 min
(E) 40 min
# 18 (3100). A chess player played 40 matches and scored 25 points (a win counts as one point, a
draw counts as half a point, and a loss counts as zero points). How many more matches did he win
than lose?
(A) 5
(B) 7
(C) 10
(D) 12
(E) 15
# 19 (3459). Triplets Jane, Danielle and Hannah wanted to buy identical hats. However, Jane
lacked a third of their price, Danielle a quarter and Hanna a fifth. When the hats became 9,40 EUR
cheaper, the sisters joined their savings and each of them bought a hat. Not a cent was left. What
was the price of a hat before the price reduction?
(A) 12 EUR
(B) 16 EUR
(C) 28 EUR
# 20 (2394). Let p, q, r be positive integers and p +
pqr?
(A) 6
(B) 10
(C) 18
(D) 36 EUR
1
q+ r1
=
25
19 .
Which of the following is equal to
(D) 36
5 points
3
(E) 112 EUR
(E) 42
# 21 (3709). In the equation, N × U × (M + B + E + R) = 33, each letter stands for a different
digit (0, 1, 2, ..., 9). How many different ways are there to choose the values of the letters?
(A) 12
(B) 24
(D) 48
(C) 30
(E) 60
# 22 (3174). On the picture shown Kaan wants to add some line segments such that each of the
seven points has the same number of connections to other points. What is the least number of line
segments Kaan must draw?
(A) 4
(B) 5
(C) 6
(D) 9
(E) 10
# 23 (3173). The picture shows the same cube from two different views. It is built from 27 small
cubes, some of them are black and some are white. What is the largest number of black cubes there
could be?
(A) 5
(B) 7
(C) 8
(D) 9
(E) 10
# 24 (3036). On an island, frogs are always either green or blue. The number of blue frogs increased
by 60% while the number of green frogs decreased by 60%. It turns out that the new ratio of blue
frogs to green frogs is the same as the previous ratio in the opposite order (green frogs to blue frogs).
By what percentage did the overall number of frogs change?
(A) 0%
(B) 20%
(C) 30%
(D) 40%
(E) 50%
# 25 (3233). Tom wrote down several distinct positive integers, not exceeding 100. Their product
was not divisible by 18. At most how many numbers could he have written?
(A) 5
(B) 17
(C) 68
(D) 69
(E) 90
# 26 (2473). Any three vertices of a cube form a triangle. What is the number of all such triangles
whose vertices are not all in the same face of the cube?
(A) 16
(B) 24
(C) 32
(D) 40
(E) 48
# 27 (3617). In the picture, P T is tangent to a circumference C with center O and P B bisects the
angle T P A. Calculate the angle T BP .
4
(A) 30◦
(B) 45◦
(C) 60◦
(D) 75◦
(E) It depends on the position of point P
# 28 (2953). Consider the set of all the 7-digit numbers that can be obtained using, for each number,
all the digits 1, 2, 3,..., 7. List the numbers of the set in increasing order and split the list exactly at
the middle into two parts of the same size. What is the last number of the first half?
(A) 1234567
(B) 3765421
(C) 4123567
(D) 4352617
(E) 4376521
# 29 (3583). Let ABC be a triangle such that AB = 6 cm, AC = 8 cm and BC = 10 cm and M
be the midpoint of BC. AM DE is a square, and M D intersects AC at point F . Find the area of
quadrilateral AF DE in cm2 .
(A)
124
8
(B)
125
8
(C)
126
8
(D)
127
8
(E)
128
8
# 30 (2682). There are 2014 persons in a row. Each of them is either a liar (who always lies) or a
knight (who always tells the truth). Each person says ’There are more liars to my left than knights
to my right’. How many liars are there in the row?
(A) 0
(B) 1
(C) 1007
5
(D) 1008
(E) 2014
KSF 2014 – selected problems Student
3 points
# 1 (3022). If you take a number of 1 × 1 × 1 cubes out of a 5 × 5 × 5 cube, you end up with a solid
figure consisting of columns of the same height, which stand on the same ground plate (see figure).¡br¿
How many small cubes were taken out?
(A) 56
(B) 60
(C) 64
(D) 68
(E) 80
# 2 (3178). Today is Carla’s, Emilie’s and Lilia’s birthday. The sum of their ages is now 44. What
will the sum of their ages be the next time it is a two-digit number with two equal digits?
(A) 55
(B) 66
# 3 (3325). If ab =
(A)
(E)
1
2
(C) 77
(D) 88
(E) 99
what is the value of a−3b ?
1
8
1
6
(C) −8
(B) 8
(D) 6
# 4 (3019). There are 48 balls placed into three baskets of different sizes. The smallest and the
largest basket contain together twice the number of balls that the middle one contains. The smallest
basket contains half the number of balls of the middle one. How many balls are there in the largest
basket?
(A) 16
# 5 (2476).
(A) 22011
(B) 20
22014 −22013
22013 −22012
(C) 24
(D) 30
(E) 32
(C) 22013
(D) 1
(E) 2
=?
(B) 22012
# 6 (2437). Which of these expressions does not contain b + 1 as a factor?
(A) 2b + 2
(B) b2 − 1
(C) b2 + b
(D) −1 − b
(E) b2 + 1
# 7 (3557). How many digits long is the result of the multiplication: (222 )5 · (555 )2 ?
(A) 22
(B) 55
(C) 77
(D) 110
(E) 111
# 8 (3179). Handsome Harry has a secret email account that only four friends know. Today he
received 8 emails in that account. Which of the following is certainly true?
(A) Harry received two emails from each friend.
(B) Harry cannot have received eight emails from one of his friends.
1
(C) Harry received at least one email from each friend.
(D) Harry received at least two emails from one of his friends.
(E) Harry received at least two emails from 2 different friends.
# 9 (3017). Two identical cylinders are cut open along the dotted lines and glued together to form
one bigger cylinder – see figure. What can you say about the volume of the big cylinder compared to
the volume of one small cylinder?
(A) It has twice the volume.
(B) It has 3 times the volume.
(C) It has π times the volume.
(D) It has 4 times the volume.
(E) It has 8 times the volume.
# 10 (2786). In the number 2014 the digits are different and the last digit is greater than the sum
of the other three digits. How many years ago did this occur the last time?
(A) 5
(B) 215
(C) 305
(D) 395
(E) 485
4 points
# 11 (2959). The size of a rectangular box is a × b × c, with a < b < c. If you increase a or b or c
by a given positive number, the volume of the box also increases. In which of the following cases is
the increase of the volume of the box the greatest?
(A) If you increase a.
(B) If you increase b.
(C) If you increase c.
(D) The increase of the volume is the same in A), B), C).
(E) It depends on the values of a, b, c.
# 12 (2479). In a football match, the winner gets 3 points, the loser gets 0 points, while in the case
of a draw, each team gets 1 point. Four teams, A, B, C, D, take part in a football tournament. Each
team plays three games: one against each other team. At the end of the tournament team A has 7
points and teams B and C have 4 points each. How many points does team D have?
(A) 0
(B) 1
(C) 2
(D) 3
# 13 (2390). The radii of two concentric circles are in proportion 1 : 3.
2
(E) 4
AC is a diameter of the big circle; BC is a chord of the big circle which is tangent to the smaller;
and the length of AB is 12. Then the radius of the big circle is
(A) 13
(B) 18
(C) 21
(D) 24
(E) 26
# 14 (2391). How many triples (a, b, c) of integers with a > b > c > 1 satisfy
(A) none
(B) 1
(C) 2
(D) 3
1
a
+
1
b
+
1
c
> 1?
(E) infinitely many
# 15 (3545). a, b, c are non zero numbers and n is a positive integer. It is known that the numbers
(−2)2n+3 a2n+2 b2n−1 c3n+2 and (−3)2n+2 a4n+1 b2n+5 c3n−4 have the same sign. Which of the following
is definitely true?
(A) a > 0
(B) b > 0
(C) c > 0
(D) a < 0
(E) b < 0
(D) 10
(E) 12
# 16 (3694). Six weeks is n! seconds. n =?
(A) 6
(B) 7
(C) 8
# 17 (3384). The vertices of a cube are numbered 1 to 8 in such a way that the result of adding
the four numbers of the vertices of a face is the same for all faces. Numbers 1, 4 and 6 are already set
x
............................................................................................
......
... ..
..... .
..... ..
..... ..
..... .....
.....
.
.....
.
.
...
...
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.....
...
.
.
.
.
.
.
.
.
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...
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.....
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...
.
.....
.....
.
.
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....
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.................................................................................................
...
.
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....... .... .... .... .... .... ....... .... .... .... .... ............
.
...
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..
.
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...
.
.
....
.
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.
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.
.
.
..
.... ....
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.................................................................................................
6
on some vertices as shown. What is the value of x? 1
(A) 2
(B) 3
(C) 5
4
(D) 7
(E) 8
# 18 (3156). The label on a package of cream cheese reads: 24 % total fat. The same label also
reads: 64 % fat in dry matter. What is the percentage of water in this cheese?
(A) 88 %
(B) 62.5 %
(C) 49 %
(D) 42 %
(E) 37.5 %
# 19 (2550). Line L passes through the vertex A of a rectangle ABCD. The distance from point
C to L is 2, and the distance from point D to L is 6. If AD is twice AB, find AD.
3
(A) 10
(B) 12
(C) 14
(D) 16
√
(E) 4 3
# 20 (3697). The function f (x) = ax+b satisfies the equalities f (f (f (1))) = 29 and f (f (f (0))) = 2.
What is the value of a?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
5 points
# 21 (3664). There are 10 different positive integers, exactly 5 of them are divisible by 5 and exactly
7 of them are divisible by 7. Let M be the greatest of these 10 numbers. What is the minimum possible
value of M ?
(A) 105
(B) 77
(C) 75
(D) 63
(E) none of these
# 22 (2736). P QRS is a rectangle. T is the midpoint of RS. QT is perpendicular to the diagonal
P R.
T
S
R
P
Q
What is the ratio P Q : QR?
√
(A) 2 : 1
(B) 3 : 1
(C) 3 : 2
(D)
√
2: 1
(E) 5 : 4
# 23 (3059). There are 9 kangaroos called Greatkangs. They are coloured either silver or gold.
When 3 Greatkangs meet by chance, there is a two in three chance that none of them is silver. How
many Greatkangs are gold?
(A) 1
(B) 3
(C) 5
(D) 6
(E) 8
# 24 (2741). A square fits snugly between the horizontal line and two touching circles of radius 1.
What is its side length?
(A)
2
5
(B)
1
4
(C) √12
(D)
4
1
5
(E)
1
2
# 25 (3234). Tom wants to write several distinct positive integers, none of them exceeding 100.
Their product should not be divisible by 54. At most how many integers can he write?
(A) 8
(B) 17
(C) 68
(D) 69
(E) 90
# 26 (2727). Two regular polygons of side length 1 lie on opposite sides of their common side AB.
One of them is a 15-gon ABCD . . . and the other is an n-gon ABZY . . .. What value of n makes the
distance CZ equal to 1?
(A) 10
(B) 12
(C) 15
(D) 16
1
(E) 18
1
# 27 (3236). The equalities k = (2014 + m) n = 1024 n + 1 are given for positive integers k, m, n.
How many different values can the number m take?
(A) None
(B) 1
(C) 2
(D) 3
(E) Infinitely many
# 28 (2739). The diagram shows a polygon whose vertices are the mid-points of the edges of a
cube. An interior angle of the polygon is defined in the normal way: the angle between the two edges
meeting at a vertex.
What is the sum of all the interior angles of the polygon?
(A) 720
(B) 1080
(C) 1200
(D) 1440
(E) 1800
# 29 (2544). The function f : Z → Z satisfies the conditions f (4) = 6 and xf (x) = (x − 3)f (x + 1).
What is the value of f (4)f (7)f (10).....f (2011)f (2014)?
(A) 2013
(B) 2014
(C) 2013 · 2014
(D) 2013!
(E) 2014!
# 30 (2674). In the forests of a magical island three kinds of animals roam: lions, wolves and goats.
Wolves can eat goats, and lions can eat either wolves or goats. However, this being a magical island:
If a wolf eats a goat, it turns into a lion. If a lion eats a goat, it turns into a wolf. If a lion eats a
wolf, it turns into a goat. Originally, there were 17 goats, 55 wolves and 6 lions on the island. What
is the highest possible number of animals remaining on the island after no more eating is possible to
happen?
(A) 1
(B) 6
(D) 23
(C) 17
5
(E) 35

Kangourou sans Frontières

Transcription

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