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<b>Individual Contest</b>

Time limit: 120 minutes

2009/11/30

English Version

2009 Asia
Inter-Cities
Teenagers
Mathematic

s Olympiad
2009 Asia

Inter-Cities
Teenagers
Mathematic
s Olympiad

<b>Instructions:</b>



Do not turn to the first page until you are told to do so.



Remember to write down your team name, your name and

Contestant number in the spaces indicated on the first page.



The Individual Contest is composed of two sections with a total

of 120 points.



Section A consists of 12 questions in which blanks are to be

filled in and only ARABIC NUMERAL answers are required.

For problems involving more than one answer, points are given

only when ALL answers are correct. Each question is worth 5

points. There is no penalty for a wrong answer.



Section B consists of 3 problems of a computational nature, and

the solutions should include detailed explanations. Each problem

is worth 20 points, and partial credit may be awarded.



You have a total of 120 minutes to complete the competition.



No calculator, calculating device, watches or electronic devices

are allowed.

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<b>Individual Contest</b>

<i><b>Time limit: 120 minutes</b></i>

<i><b>2009/11/30</b></i>

<i>Name:</i>

<i>No.</i>

<i>Team:</i>

<i>Score:</i>

<b>Section A.</b>

<i>In this section, there are 12 questions. Fill in the correct answer on the space</i>
<i>provided at the end of each question. Each correct answer is worth 5 points.</i>

1. Arrange the numbers 2847, 3539, 5363, 7308 and 11242 from the largest to the
smallest.

<i><b>Answer </b></i><b>:</b> ＞ ＞ ＞ ＞

2. <i>ABCDEFGH</i> is an octagon in which all eight angles are equal. If <i>AB</i> = 7, <i>BC</i> = 4,

<i>CD</i> = 2, <i>DE</i> = 5, <i>EF</i> = 6 and <i>FG</i> = 2, determine the sum of the lengths of <i>GH</i> and

<i>HA</i>.

<i><b>Answer </b></i><b>:</b>

3. How many four-digit multiples of 9 are there if each of the digits are odd and
distinct?

<i><b>Answer </b></i><b>:</b>

4. A circle is tangent to a line at <i>A</i>. From a point <i>P</i> on the circle, a line is drawn
such that <i>PN</i> is perpendicular to <i>AN</i>. If <i>PN</i> = 9 and <i>AN</i> = 15, determine the radius
of the circle.

<i>H</i>

<i>G</i> <i>F</i> <i>E</i>

<i>D</i>
<i>C</i>
<i>B</i>
<i>A</i>

<i>N</i>
<i>A</i>

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<i><b>Answer </b></i><b>:</b>

5. From the first 30 positive integers, what is the maximum number of integers that
can be chosen such that the product is a perfect square?.

<i><b>Answer </b></i><b>:</b>

6. Ace, Bea and Cec are each given a positive integer. They do not know the

numbers given to the others, but are told that the sum of the three numbers is 15.
Ace announces that he can deduce that the other two have diferent numbers,

while Bea independently announces that she can deduce that no two of the three
numbers are the same. Hearing both announcement, Cec announces that he
knows all three numbers. What are they?

<i><b>Answer </b></i><b>:</b> <i><b>A</b></i><b>= , </b><i><b>B</b></i><b>= , </b><i><b>C</b></i><b>= </b>

7. On the blackboard is a 3×3 magic square. The sum of the three numbers in each
row, each column and each diagonal is the same. As shown in the diagram
below, all but three of the numbers are erased. What is the number represented
by <i>x</i> in the cell at the upper left corner?

<i><b>Answer </b></i><b>:</b>

8. <i>ABCD</i> is a square of side length 2009. <i>M</i> and <i>N</i> are points on the extension of the
diagonal <i>AC</i> such that ∠<i>MBN</i>=135°. Determine the minimum length of <i>MN</i>.

<i><b>Answer </b></i><b>:</b>

9. Let <i>x</i> and <i>y</i> be positive integers such that <i>x y</i> <i>y x</i>  7<i>x</i>  7<i>y</i>  7<i>xy</i> 7.

Determine <i>x</i>+<i>y</i>.

<i>N</i>

<i>M</i>
<i>D</i>
<i>A</i>

<i>B</i>
<i>C</i>

<i>x</i> 21 94

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<i><b>Answer </b></i><b>:</b>

10. There is a certain integer such that when we get its cube and its square, then each
of the digits of the cube or square surprisingly contain only the numerals
1,2,3,4,5,6,7 and 8 exactly once in them. Determine this integer.

<i><b>Answer </b></i><b>:</b>

11. We can express 2009 as the sum of four different numbers each of which consists
of at least two digits and all the digits are identical, 2009=1111+777+88+33.
What is the minimum number of addends needed to express 9002 in the same
manner?

<i><b>Answer </b></i><b>:</b>

12. A farmer has ten baskets of eggs containing 12, 13, 14, 16, 18, 19, 22, 24, 29 and
34 eggs respectively. Some baskets have chicken eggs while other baskets have
duck eggs. He sells one basket and finds that the number of remaining chicken
eggs is three times the number of the remaining duck eggs. How many eggs were
in the basket he sold?

<i><b>Answer </b></i><b>:</b>

<b>Section B.</b>

<i>Answer the following 3 questions, show your detailed solution on the space</i>
<i>provided after each question. Each question is worth 20 points.</i>

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2. You are transporting mangoes by aircraft from Manila to Singapore. There are 12
planes available with the following weight capacities: 2, 2, 3, 3, 4, 7, 8, 8, 10, 10,
11 and 13 tons. Since no two planes may be assigned to the same route, then you
may direct each plane to one of the following 12 routes:

Bangkok–Singapore Hong Kong–Kuala Lumpur

Hong Kong–Singapore Jakarta–Singapore

Kuala Lumpur–Bangkok Kuala Lumpur–Singapore

Manila–Hong Kong Manila–Jakarta

Manila–Kuala Lumpur Manila–Taipei

Taipei–Bangkok Taipei–Hong Kong

What is the maximum number of tons of mangoes you can ship from Manila to
Singapore?

3. <i>A</i>, <i>B</i>, <i>C </i>and <i>D</i> are four consecutive points on a circle, such that <i>AB</i> = 1, <i>BC</i> = 2,

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