Problems and Solutions AIMO Trial Grade 5 2019

Asia International Mathematical Olympiad Union (AIMO Union) is a collaborative international organization among mathematical research organisations and individuals.

It is the mission of AIMO Union to provide teenagers with an internaional platform for the purposes of learning Mathematics, cultural exchange, intriguing their mathmatical insights, improving their qualities of scientific thinking minds and hence enhancing the friendship and cooperation of teenagers from different countires/regions.

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Berikut ini Soal dan Solusi AIMO trial grade 5, semoga bermanfaat.

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1. The greatest prime number less than 100 is 97, and the next is 89. If all the prime numbers less than 100 are multiplied together, is the product an odd number or an even number?

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2. Dylan is facing southeast direction originally. After he turns himself towards right by an angle equals to 2019 right angles, which direction does he facing now?

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3. There are 12 rows of seats in a lecture theatre. There are 16 seats in the first row. The number of seats in every row is 2 greater than that of the row in front of it. How many seats are there in the lecture theatre in total?

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4. Find the value of the following expression.

\(2222222×5555555 ÷1234567654321\)

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5. In a Mathematics examination, the average marks of boys and girls are 34 and 59 respectively. There are 12 boys and 18 girls. Find the average mark of these 30 students.

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6. There is a cuboid. The length is longer than the width by 1 centimetre, the width is longer than the height by 1 centimetre. The volume of that cuboid is 2730 cubic centimetres, what is the total surface area of that cuboid in square centimetres?

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7. Find the remainder when \(2019^{324}\) is divided by 15.

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8. How many four-digit numbers without repeating digits can be formed by using 2, 0, 1 and 9?

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9. Find the value of the following expression.

\(100^2 – 99^2 + 98^2 – 97^2 + 96^2 – 95^2+…+ 2^2 -1^2\)

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10. Find the value of the following expression.

\(20192019×324324 – 2021019×3240324\)

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11. The speeds of Amy and Bella were 72 metres per minute and 54 metres per minute respectively. They walk towards each other from places A and B respectively. They met at a position 45 metres apart from the mid-point of A and B. How many metres was the distance between A and B?

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12. Chris is doing a multiplication of two numbers. He mistakenly writes the last two digits of the multiplicand as 98 instead of 89, and he gets 98154 as the result. If the correct product is 97047, what is the sum of the multiplicand and the multiplier?

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13. In the regular hexagon shown in Figure 13, there is an equilateral triangle where its three vertices are all lying at the mid-points of the sides of hexagon. If the area of the equilateral triangle is 2019 square centimetres, what is the area of the hexagon in square centimetres?

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14. Cut a rectangular cardboard with length 1968 centimetres and width 2256 centimetres into many identical
small squares, which have side length of integer in centimetres. There should not be any waste of cardboard and we make the area of each square to be the largest. How many squares can we get?

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15. How many triangles are there in Figure below?

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16. There are 7 contestants in a GO chess competition. Every two contestants have one match, and the winner
will score 1 mark while the loser will score none. No draw is possible. Finally the score of 7 contestants are \(A, A, B, C, D, D\) and \(E\) respectively, where \(A > B > C > D > E\). Find the seven-digit number \(\overline{AABCDDE}\) . (If \(P = 3, Q = 5\) and \(R = 8\) , then \(\overline{PQR} = 358\))

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17. Given that the value of \((1×2×3 + 2×3×4 + 3×4×5 + …+ 2017×2018×2019)×4\) can be expressed as a product of 4 consecutive natural numbers. Find the greatest one among those 4 consecutive natural numbers.

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18. A, B, C and D are four identical houses. First Margaret works 8 hours a day and David works 4 hours a day, they need 165 days to renovate house A. Then Margaret works 4 hours a day and David works 8 hours a day this time, they need 195 days to renovate house B. Now houses C and D are renovated by Margaret and David separately, and both workers work 8 hours a day, what is the difference between the numbers of days Margaret and David required to renovate the houses?

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19. Cut a square paper with side length 20 centimetres into squares. The side length of the first square cut is 1 centimetre. Then the side length of every square is 1 centimetre longer than the previous one. At most how many squares can be cut?

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20. If all digits of a number with non-zero units digit are reversed, the new number generated is called
“reverse” to the original one. For example the reverse number of 137 is 731.
If the Greatest Common Divisor of two natural numbers is 1, then they are called “co-prime”. For
example 20 and 21 are co-prime.
How many pairs of two-digit numbers are “reverse” and “co-prime” with each other?
(’13 and 31’ and ’31 and 13’ are counted as 1 pair)

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