xi. pangea matematika verseny 2019 5. évfolyam · in 1 minute and 24 seconds. how long would it...

XI. PANGEA Matematika Verseny 2019. Döntő 5. évfolyam

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1. What number should be written at the place of the question

mark?

A) 7 B) 9 C) 10 D) 11 E) 12

2. A car racer completed the first lap at a Formula One Grand Prix

in 1 minute and 24 seconds. How long would it take him to

cover sixty-two and a half laps if he could complete each lap in

this same time?

A) 1 hour 24 minutes 48 seconds

B) 1 hour 25 minutes 30 seconds

C) 1 hour 26 minutes 30 seconds

D) 1 hour 26 minutes 48 seconds

E) 1 hour 27 minutes 30 seconds

294 392 539

49

6 8 ?



3. Uncle Valentine counted his farm animals, and then he said:

”There are three kinds of animals on my farm: horses, cows and

pigs. All of the animals are horses, except two; all of them are

cows, except two; and all of them are pigs, except two.”

How many horses does Uncle Valentine have?

A) 7 B) 6 C) 4 D) 3 E) 1

4. On an old ruler with a centimeter scale most of the numbers

are not visible any more. Only the numbers 0, 3 and 7 and their

marks are visible. We would like to make a 2 cm long line

segment. What is the minimum number of steps required?

A) 6 B) 5 C) 4 D) 3 E) 2



5. It started to rain in Kőszeg at 6 p.m. Will the sun be out 102

hours later in Budapest, about 200 kilometres away?

A) No, it won’t.

B) The sun might be out by that time.

C) It will probably be out, because the rain might stop in 102

hours.

D) It will probably be raining, as weather fronts usually move

from West to East.

E) We can’t know for sure.

6. How many of the following statements are false?

– If rectangles are added to a square in a way that one of their

sides touches the square, a new square cannot be created.

– There is no rectangle that is also a square.

– All squares can be divided into rectangles.

– All rectangles can be divided into squares in only one way.

A) 0 B) 1 C) 2 D) 3 E) 4



7. There are two bricks in one of the pans of a two–armed balance

scale. The other pan keeps balance with this, holding a quarter

of a brick and a 7 kg weight. How many kilograms does one

brick weigh?

A) 2 kg

B) 2 and a half kg

C) 3 kg

D) 3 and a half kg

E) 4 kg

8. The floor of a toilet meaures 90×90 cm. We would like to cover

the floor with 20×20 cm tiles that are placed next to each other

without gaps. Tiles can be cut but only if there isn’t enough

room left for a full tile.

What is the minimum number of tiles to cover the floor? How

much waste will we have in this case?

A) 21 tiles are needed and ¾ of the last will be the waste.

B) 16 tiles are needed and there will be no waste.

C) 25 tiles are needed and 8 half-tiles and a ¾ tile will be the

waste.

D) 20 tiles are needed and there will be no waste.

E) 20 tiles are needed and ¼ of the last will be the waste.



9. Gábor has built a big cube out of smaller

cubes such that there are four small cubes

along any edge of the big cube. Looking from

the top and from the four sides, Gábor saw a

total of 80 little squares. Then he pulled out

two cubes from the middle of the top face,

next to each other. How many little squares

could he see now?

A) 88 B) 86 C) 84 D) 82 E) 80

10. We need a jug of milk and 2/3 cups of sugar to make a cake. If

we want to make a bigger cake, we need four and a half jugs of

milk. How much sugar do we need, assuming that we do not

change the ratio of the ingredients?

A) 4 and a half cups

B) 3 and a half cups

C) 3 cups

D) 2 and a half cups

E) 2 cups



11. Norbi invited his friend, Pete, to come over to listen to music

in the evening. They agreed on the time of the visit, too. When

Pete was leaving home he turned back and saw their wall clock

in the mirror. It looked as if the arms of the clock had shown

6:27 pm. He rang the bell at Norbi’s front door 27 minutes later.

Norbi opened the door and said:

”If you had arrived 5 minutes earlier, then you would have come

exactly one third of an hour earlier than we have agreed on.”

What time did they agree to meet?

A) half past six

B) a quarter past six

C) six o’clock

D) a quarter to six

E) half past five



12. Peter and Paul have their birthdays on the same day of the

year but Paul is older. Paul says to Peter on his birthday:

”Now I am twice as old as you were when I was celebrating my

11th birthday. Together we are 39 years old now.”

How old is Peter now?

A) 10 B) 13 C) 16 D) 19 E) 20

13. A pendulum clock strikes six times at six o’clock in the

morning which takes 3 ¾ seconds. How long do the strikes take

at 11 o’clock in the morning? The duration of the strike itself is

considered to be zero.

A) 6 ¼ seconds

B) 6 7/8 seconds

C) 7 ¼ seconds

D) 7 ½ seconds

E) 8 ¼ seconds



14. Find all four-digit positive integers in which the sum of the

first two digits is 13, the sum of the third and fourth digits is 11,

and the number formed by the two middle digits is divisible by 9.

How many such four-digit numbers are there? Any digit can be

used more than once.

A) 2 B) 3 C) 4 D) 5 E) 6

15. Find the pattern and give the value of the question mark.

A) 3 B) 20 C) 21 D) 32 E) 33

7 4

8 5

3

9 6

8 7

15

8 9

4 7

?

4 6

2 3

0



Questions 16-20. must be solved on the "Solutions" sheet. (Show

your work, giving the final solution only is not enough.)

Where there is a diagram included, work within the diagram.

16. The Leaning Tower of Pisa was completed at around the end

of the 1300s, and it was already leaning at that time. In the

roughly 600 years that followed, the tower continued leaning.

Although detailed information is not available, let us assume

that the topmost point of the tower moved another 35

centimetres by 1490, then in each 100 years after that it moved

5 cm more than in the previous 100 years. In 1990 they started

to pull the tower back with steel cables and counterweights.

Consequently, by 2001 the leaning was reduced by 45

centimetres and by 2018 the topmost point moved backwards

yet another 3 centimetres. Now it is 387 centimetres away from

the vertical position.

Consider the point that is one third of the way up along the

tower from the ground. How far away was this point from the

vertical position at the time the building of the tower was

completed?

17. An airplane flies around the Earth moving in one direction,

covering 40 000 km. When it starts off the Sun is exactly above

the plane. How many kilometers and in which direction does the

plane have to travel in one hour if the Sun must be above the

plane all the time? (Round your answer to the nearest

kilometre.)



18. Emperor Augustus was born on 23 September, 63 BC. He died

on 19 August, 14 AD. How many years did he live if we count full

years only?

19. Two friends meet again after 5 years and ask about each

other.

”Imagine,” says one of them, ”I have 3 daughters born since we

last met. The product of their ages is 36.”

Determine the sum of the ages of the three daughters.

20. Zoltán went to play roulette on his birthday. The cost of one

playing chip was 100 Ft. He bought one chip. He chose the

simplest strategy: he always placed his chips on red. If the ball

did not land on red, he lost his chips, but he bought new chips

for twice as much money as he had lost and placed all of them

on red again. He continued until red finally came out for the first

time. Then he got all of his chips back and won the same number

of extra chips. At this point he stopped playing. Did Zoltán win or

lose in the end? Explain your answer in details.

xi. pangea matematika verseny 2019 5. évfolyam · in 1 minute and 24 seconds. how long would it...

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