5 5 5 5 5 5 VIETNAM MATHEMATICAL SOCIETY Mathematical Young Talent Search 1. As shown in the diagram, the flowers are arranged following a pattern: the first row has 1 flower, the second row has 3 flowers, the third row has 5 flowers, and the forth row has 7 flowers. How many flowers are there in the 25th row? V V V V V V V V V V V V V V V V V V V V V V V V V . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. After three tests, the average of a Andy’s marks is 13. After the fourth test, the average of the four marks is now 14. What was the the boy’s fourth test mark? 3. The figure shown has area 336 cm2 , consisting of 21 small identical squares. An ant is travelling along a path indicated by the red arrows, which has length ` cm, find the value of `. 4. At a conference, 1 4 of the attendants occupied 2 3 of the chairs. The rest of the people decided to stand. If there were three chairs unoccupied in the conference, how many people are there in the conference? 5. Vinh put all the toy fish and birds in the following patterns that has six rows and 22 columns such that the number of birds is 1 3 that of fish. How many fish are there altogether? g g g g g f · · · f g f f f g g f g · · · f g g g g g g g g · · · f f f . . . . . . . . . . . . . . . . . . · · · . . . . . . . . . g g g f g g · · · f g g MYTS Mathematical Young Talent Search 25 5 5 5 5 5 5 6. The vaccination rate among intended children in a particular city is 94%. If 24 children (within that age limit) were not vaccinated, how many children are there in the city that should have been vaccinated? 7. Seven circular cards are placed around a circle on the table as shown. Which card has the maximum area in contact with the table? 46 2 5 1 7 3 8. On the date 15/03/2016, Vũ discovered that the sum of the first four digits is equal to the sum of the last four digits. What is the last date of the year has the same property? 9. The consecutive numbers 1316, 1317, . . . , 1382, 1383 are written on a sheet of paper. Binh decides to circle all the multiples of 5. How many numbers will be circled? 10. A rectangular sheet of paper is divded into nine rectangular pieces with straight parallel lines. If all the nine pieces are colored, what is the minimum number of colors required such that no two adjacent pieces have the same colors? 11. Even whole numbers 2, 4, 6, 8, . . . 2016 are filled in the table (with five columns) such that one cell is skipped after every group of three numbers. Which column contains the number 2016? A B C D E 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 . . . . . . . . . . . . . . . MYTS Mathematical Young Talent Search 26 5 5 5 5 5 5 12. An ant is travelling along the surface of a wooden cube along the path indicated by te arrows from A to B. If the total surface area of the cube is 96 cm2 , find the length of the path. A B 13. There are 24 doves in the figure, each dove is at one cell. At least how many doves need to leave from where they are now so that the columns, and the rows of the table have the same number of doves? f f f f f f f f f f f f f f f f f f f f f f f f 14. Nine identical cubes are attached to each other. The surfaces of the resulting block are painted. What is the total surface area in cm2 if each side of the cube is 2 cm? 15. Each of the small equilateral triangle in the triangular grid has area 3 cm2 . Find the area of the triangle that is shaded. MYTS Mathematical Young Talent Search 27 5 5 5 5 5 5 16. Each of the figures below is built up by identical matchsticks, all following the same pattern. At least how many matchsticks are required to build the 25th figure? Figure 1 Figure 2 Figure 3 17. A boy, Dung, puts all his marbles in seven boxes placed in a row, with at least two marbles per box. He remarks that the product of the number of marbles in three consecutive boxes is always 60. Given that the second box contains 4 marbles, what is the maximum number of marbles owned by the boy? 18. ABCD is a rectangle and E, F are two points on the side AB in the order A, E, F, B such that AE = F B = 5 cm. It is known that AD = 4 cm and that the area of the triangle DEF is the third of the area of the rectangle. What is the length EF? A B D C E F 19. An olympic games flag consists of five identical intersecting circles. If we can rearrange these circles, what is the maximum number of intersections they have? 20. An urn contains marbles of three colors: blue, red, and yellow. The total number of marbles in the urn is 25. If we take 21 marbles (without looking), we can assure that we get marbles of three different colors. How many yellow marbles are there in the urn? 21. Cut the figure (along the lines only) into two congruent parts. Then calcuate the sum of the numbers in each part. What is the value of the larger sum? 1 12 2 3 4 5 6 7 8 9 10 11 MYTS Mathematical Young Talent Search 28 5 5 5 5 5 5 22. Today is Sunday. Exactly in 259 days’s time the indenpendence day of the South Africa republic (independence from the UK in 1931) is celebrated on the 11 of December, 2016. What day of the week is the day? Figure below is the extract of March and April 2016. March 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 April 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 23. In a chess tournament, one competitor plays exactly once with the other. If there are a total of 300 games, how many competitors are there in the tournment? 24. A rectangular block K is made up of 2 × 3 × 4 identical unit cube. The total surface area of block K is 52 square unit. If we remove one cube from the block, what is the minimum value of the total surface area of the resulting block? 25. Four children Khuê, Linh, Mạnh, and Nga, competed in a mathematical contest. When asked about their results, they give the following response Khuê: I got the highest score in the group. Linh: I got the lowest score in the group. Mạnh: I am not the one who got the lowest scoret. Nga: I am neither the person who got top score nor the one who got the lowest score. Just one of the persons is telling lies, who is the person that got the highest score? MYTS Mathematical Young Talent Search 29 5 5 5 5 5 5 26. Van prepared a triangular piece of paper and then she cut two identical square pieces of paper from the original piece, resulting in a polygon with perimeter 110 cm. If the perimeter of the triangle is 86 cm and its area is 282 cm2 , find the area of the resulting polygon? 27. A whole number N is called generous if it is divisible by all of its digits and also divisible by the sum of its digits. For instance, 12 is a generous number while 102 is not. Find the least generous number that is divisible by 17? 28. One of the four girls broke the flower vase in the room by accident. When asked who was the person that broke the vase, the gave the following response. Lan : It is not me! Vân : It is Yen who broke it! Hằng : No, it is Van who did it! Yến : Van is the liar. Only one of the four is telling the truth. Who broke the vase? 29. There are two pipes of which one supplies hot water and the other supplies cold water. The pipe with hot water can fill a tank in 23 minutes, while the one with cold water can fill the same tank in 17 minutes. If you open the hot pipe first, how soon do you need to open the cold pipe so that you have one and a half more hot water than cold water when the tank is full? 30. In the figure, ABCD is a trapezium with AB k CD. The two diagonals meet at K. Let L be the point on line segment BD such that BL = LD. The areas of triangles ABK, DAL are 18 cm2 , and 21 cm2 ; and the area of shaded triangle ALC is x cm2 . Find the value of x. A B D C 18 21 L K MYTS Mathematical Young Talent Search 30

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