25 MARTIE 2017

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CLASA a IV-a Here are some suggestions to help you do your best:

Read carefully each question and think about the answer before choosing your response.

RULES

Part I has four multiple choice exercises.

Part II must be solved into English

Part III must be translated into English, and then solved in English as well.

PART I. 1. Victor and his father go to the theatre. Their seat numbers are 77 and 78.

The sign below shows where the seats are.

Which way should they go? A B C D E

2. A rectangle is partly hidden behind a curtain. What is the shape of hidden part?

A A circle

B A square C A hexagon D A rectangle E A triangle

Seats 1 to 25

Seats 26-50

Seats 51-75

Seats 76-100

Seats 101-125

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3. Which of the following is true about this picture?

4. The sum of digits of the number 2017 is 2+0+1+7=10. What is the next number greater than 2017 with the same sum of its digits?

A 2008 B 2026 C 2035 D 2107 E 2133

PART II. Hedgehog Pogonici complained to his friends: “If I had picked up twice as many apples as I really did, I would have 24 apples more than I have now.” How many apples did Pogonici pick up? PART III.

La concursul , participă 80 de elevi. Ei au avut de rezolvat 4 probleme. Se știe că prima problemă a fost rezolvată de 67 de elevi, a doua problemă a fost rezolvată de 61 de elevi, a treia problemă nu a fost rezolvată de 20 de elevi, iar a patra problemă a fost rezolvată de 58 de elevi. Arătați că sunt cel puțin 6 elevi care au rezolvat toate problemele.

A There are as many circles as squares. B There are fewer circles than triangles C There are twice as many circles as triangles D There are two triangles more than circles E There are more squares than triangles

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CLASA a V-a Here are some suggestions to help you do your best:

Read carefully each question and think about the answer before choosing your response.

RULES

Part I has four multiple choice exercises.

Part II must be solved into English

Part III must be translated into English, and then solved in English as well.

PART I. 1. Leo cuts pizza into quarters. Then he cuts every quarter into thirds. What part of the whole pizza is one of the pieces?

A a third B a quarter C a seventh D an eighth E a twelfth

2. How many natural numbers are there between 3,17 and 20,16 ?

A 15 B 16 C 17 D 18 E 19

3. Adrian is writing down all the numbers with the following properties: - the first digit is 1; - each of the following digits is at least as big as the one before it; -the sum of the digits is 5. How many numbers will he write?

A 4 B 5 C 6 D 7 E 8

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4. Lisa has 90 marbles. Anna has 10 more marbles than Lisa but 50 less marbles than Octavian. How many marbles do they have together?

A 100 B 150 C 240 D 250 E 340

PART II. The number 137 can be divided by a non-zero natural number and thus we get the quotient which is half the divisor, and the rest which is a one-digit number. Find the divisor, the quotient and the rest.(G.M.12/2016) PART III. Într-un grup sunt 17 prieteni. Numele și prenumele lor încep cu literele M, E, T sau S. Demonstrați că există cel puțin doi prieteni ale căror inițiale ale numelui și prenumelui coincide.

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CLASA a VI-a Here are some suggestions to help you do your best:

Read carefully each question and think about the answer before choosing your response.

RULES

Part I has four multiple choice exercises.

Part II must be solved into English

Part III must be translated into English, and then solved in English as well.

PART I. 1. Four points A, B, C, D are given on a line, in some order, so that AB=1, BC=2, CD=3, DA=4. Which two points are the farthest from each other?

A A and D B B and D C B and C D A and C E impossible to determine

2. The number 2581953764 is written on a strip of paper. Ana cuts the strip twice and gets three numbers. Then she adds these 3 numbers. What is the smallest possible sum she can get?

A 2675 B 2975 C 2978 D 4217 E 4298

3. The total number of participants in a math club is between 50 and 100. The club teacher wanted to divide them in teams. He tried to group the students in 5, or 6, or 12 per team, and noticed that there were always 3 students left. How many students are there in the math club?

A 51 B 61 C 63 D 75 E Not enough information

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4. Three buses provide small tours of the city during the day. They all depart at 8 o’clock each morning from the rail station. The tours last 20 minutes, 30 minutes, and 24 minutes, respectively. When will it be the first time when the three bus drivers meet again at the railway station?

A At 9:30 am B At 10:00 am C At 12:00 noon D At 10:30 am E At 8:00 am next day

PART II. Each of the 9 line segments in the figure is to be coloured either blue, green or red. The three sides of each triangle should have three different colours. Three of the line segments have already been coloured, as shown. What colour can the line segment marked with 𝑥 have? PART III. Fie numerele naturale a=2n+1, b=5n+1, c=n, d=n+1, unde n este număr natural. a) Determinați valorile lui n pentru care a și b au divizori comuni diferiți de 1. b) Arătați că [a,c] +[a,d] este pătrat perfect. Am notat cu [x,y] cel mai mic multiplu comun al numerelor x și y. (G.M.11/2016)

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CLASA a VII-a Here are some suggestions to help you do your best:

Read carefully each question and think about the answer before choosing your response.

RULES

Part I has four multiple choice exercises.

Part II must be solved into English

Part III must be translated into English, and then solved in English as well.

PART I. 1. After a party, Livia washed half of the glasses, Alex washed half of the rest and then Sonia washed half of the remaining glasses. There were only three glasses left for David to wash. Half of the guests used one glass each and the other half of the guests used two glasses each. How many people were there at the party?

A 36 B 24 C 18 D 16 E 8

2. The diagram shows four identical rectangles placed inside a square. The perimeter of each rectangle is 16 cm. What is the perimeter of the large square?

A 16 cm B 20 cm C 24 cm D 28 cm E 32 cm

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3. Twelve girls met in a cafeteria. On average, they ate 1,5 cup-cakes. None of them ate more than two cup-cakes and two of them had only mineral water. None of the cup-cakes were shared by the girls. How many girls ate two cup-cakes?

A 8 B 7 C 6 D 5 E 2

4. Two rectangles, ABCD and DBEF are shown in the diagram. The length of AD is 3 cm; the length of AB is 4 cm. What is the area of rectangle DBEF?

A 10 cm2

B 14 cm2 C 16 cm2 D 13 cm2 E 12 cm2

PART II. Let ABCD be a rectangular trapezoid with m(∢𝐴)=m(∢𝐷)=900, 𝐴𝐵 < 𝐶𝐷, și 𝐴𝐶 ⊥ 𝐵𝐷. The points M and N are the symmetrical points of D, respectively C to the intersection point of the diagonals, 𝑀𝑃 ⊥ 𝐶𝐷, 𝑃 ∈ 𝐴𝐶. a)Prove that MADP is a rhombus. b)Prove that 𝐴𝑀 ⊥ 𝑁𝐷 and 𝐵𝑃 ⊥ 𝑃𝐷. (G.M.4/2016) PART III. Într-o sală de spectacole sunt atâtea rânduri câte locuri sunt pe un rând. La niciun spectacol nu rămân bilete nevândute. Directorul teatrului a decis scumpirea cu 50% a biletelor pentru locurile din primele două rânduri. În acest fel, încasările au crescut cu 6,25%. Aflați câte locuri are sala.

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CLASA a VIII-a Here are some suggestions to help you do your best:

Read carefully each question and think about the answer before choosing your response.

RULES

Part I has four multiple choice exercises.

Part II must be solved into English

Part III must be translated into English, and then solved in English as well.

PART I. 1. Teo’s watch is 10 minutes slow, but he believes that it is 5 minutes fast.

Leo’s watch is 5 minutes fast, but he believes that it is 10 minutes slow. Looking at his watch, Teo believes it is now 12:00. What time does Leo think it is now? A 11:30 B 11:45 C 12:00 D 12:30 E 12:45

2. Several distinct positive integers are written on a blackboard. The product

of the smallest two of them is 16. The product of the largest two is 225. What is the sum of all the integers?

A 38 B 42 C 44 D 58 E 243

3. The figure shows a cube with four marked angles. What is the sum of these angles?

A 3150

B 3300

C 3450

D 3600

E 3750

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4. We have boxes numbered as 1, 2, 3, … We put a ball with number 1 into the box number 1. We put the balls numbered 2 and 3 in the box number 2. We put the balls numbered 4, 5, 6 into the box number 3. And so on. What is the box number containing ball 2017?

PART II. Let a, b, c be real numbers, a a negative number and the functions 𝑓, 𝑔: ℝ → ℝ 𝑓(𝑥) = 𝑎𝑥 + 𝑏

𝑔(𝑥) =1 − 𝑎

2𝑥 + 𝑐

Find the real numbers 𝑎, 𝑏, and 𝑐 so that in a system XOY, the graphics of the two functions are perpendicular and the point of their intersection is 𝐴(0,1). PART III.

În cubul ABCDA’B’C’D’ notăm cu M, N, P mijloacele muchiilor AB, B’C’, respectiv DD’. Aflați sinusul unghiului format de dreapta CM și planul (MNP), precum și sinusul unghiului format de planele (MNC), (MNP).

(G.M.12/2016)

A 50 B 60 C 62 D 63 E 64

Suggested answers

Part IlL 4" grade

80 students took part in the 22 METS competition. They had to solve four problems. We know that the first problem was solved by 67 students, the second by 61, the third was not solved by 20 students, whereas the fourth problem was solved by 58 students. Prove that there are at least 6 students who managed to solve all the problems.

Part III, 5' grade

There are seventeen friends in a group. Their surnames and first names begin with the letters M, E, T or S. Prove that there are at least two friends whose surname and first name initials coincide.

Part III, 61 grade

Let a= 2n + 1, b= Sn ± 1, c= n, d'' n + I be some natural numbers, where n is a natural number as well.

a) Find the values of n, for which A and B have common divisors other than L b) Prove that [a,c] + [a,d] is a perfect square.

We consider [x,y] the lowest common multiple of numbers x,y.

Part 111, 7' grade

In a theatre hail there are as many rows as there are seats on a row. Tickets are sold out for each performance. The manager of the theatre has decided to raise the ticket price for the seats in the first two rows by 50%. The earnings have thus increased by 6.25%. Find how many seats there are in the theatre hall.

Part III, 8' grade

In a cube ABCDA B C D we consider M,N,P the mid-points of sides AB, B C respectively DD . Find the sine of the angle formed by side CM and plane MN P and the sine of the angle formed by planes (MNC), (MNP).

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