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ccmp@seed.net.tw
Individual Contest
Time limit: 120 minutes 2013/12/28
Name:
No.
Team:
--- Jury use only ---
Section A Score
1 2 3 4 5 6 7 8 9 10 11 12 Total Score
Section B Score
1 2 3
Instructions:
z Do not turn to the first page until you are told to do so.
z Remember to write down your team name, your name and contestant
number in the spaces indicated on the first page.
z The Individual Contest is composed of two sections with a total of 120
points.
z 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.
z 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.
z Diagrams are NOT drawn to scale. They are intended only as aids.
z You have a total of 120 minutes to complete the competition.
z No calculator, calculating device, watches or electronic devices are allowed.
z Answers must be in pencil or in blue or black ball point pen.
2013 Asia Inter-Cities Teenagers Mathematics Olympiad Page 1
Individual Contest
Time limit: 120 minutes 2013/12/28
Name: No.
Team:
Score:
Section A.
In this section, there are 12 questions. Fill in the correct answer on the space provided at the end of each question. Each correct answer is worth 5 points. Be sure to read carefully exactly what the question is asking.
1. In the following addition, the same letter stands for the same digit and different letters stand for different digits. What is the maximum value of the five-digit number AITMO? A I T M O A I T M O A I T M O + A I T M O T A L E N T Answer :
2. The diagram below show a city with 11 streets numbered 0 to 10 from West to East, and 11 avenues numbered 0 to 10 from South to North. Eleven students live in junctions marked by black dots. They decide to meet one another at one of the junctions which minimizes the total walking distances from their homes along the grid lines to that junction. What is the product of its street number and avenue number?
Answer :
3. The first digit from the left of a five-digit number is 6, the third from the left is 7 and the fourth 2. When this number is divided by 99, the remainder is 0. What is the quotient? Answer : 10 0 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 0
4. Determine the order pair of real numbers (x, y) that satisfy the equation
2 2
(4x +12x+13)(4y −20y+33) 32= .
Answer :
5. Each of A, B, C and D either always tells the truth or always lies. They are asked how many of them are truth-tellers. They make the following responses.
A: The number is 0, 1 or 3. B: The number is 1, 2 or 3.
C: Excluding me, the number is 0, 1 or 3. D: Excluding me, the number is 1, 2 or 3. What is the correct number?
Answer :
6. Consider the nine numbers 1!, 2!, 3!, 4!, 5!, 6!, 7!, 8! and 9!. What is the maximum number of them we can choose so that the product of the chosen numbers is the square of an integer?
Answer :
7. In the figure below, PT touches a circle with centre O at R. Diameter SQ when
produced meets PT at P. If ∠SPR = x° and ∠QRP = y°, what is the value of
x+2y?
Answer :
8. What is the maximum value of n such that the number n! can be represented as a product of (n−3) consecutive positive integers?
Answer :
9. What is the largest positive integer less than ( 6+ 5)6?
Answer :
10. The Fibonacci sequence starts with 1, 1, 2, 3, 5, 8, 13, ... .(Start from the 3rd term, each term is the sum of the two previous terms). Let Fn be the nth term of this
sequence. S is defined as 1 1 2 3 5 2 4 8 16 32 2 n n F S = + + + + + +" +" . Calculate the value of S.
Answer :
11. There are 4 sixteen-year-old girls, 5 fifteen-year-old girls and 7 fourteen-year-old girls. In how many ways can they be formed into 8 pairs so that the 2 girls in each pair are of different ages?
Answer : ways x T S R Q P y O
2013 Asia Inter-Cities Teenagers Mathematics Olympiad Page 3 12. O is the centre of a circle with radius 10 cm. A line, through a point A at the
distance 20 cm from O, intersects the circle at points В and С, with B between A and C. If the distance from point O to BC is 6 cm, find the length of the segment
АВ, in cm.
Answer : cm
Section B.
Answer the following 3 questions. Show your detailed solution on the space provided after each question. Each question is worth 20 points.
1. The area of square ABCD is 60 cm2.The point P lies on the side BC , the point Q lies on the side CD and BP：CP = CQ：DQ = 1：3. If BQ and AP intersect at point
M, and BQ and AC intersect at point K, find the area of triangle AMK, in cm2.
Answer : cm2 A O C B A K Q P D C B M
2. One morning, the King sent the Queen 33 messages and the Queen sent the King 61 messages. Each was carried by a messenger. The messages left the respective Palaces one after another, all travelling at the same constant speed on the same road which connected the two Palaces. All meeting between two messengers occurred in the afternoon. Whenever two messages met, they exchanged the messages they were carrying and headed the other way. How many exchanges took place during that day?
2013 Asia Inter-Cities Teenagers Mathematics Olympiad Page 5 3. Determine all possible positive integers k such that all the zeroes(or roots) of the
equation in terms of x which is x3 +(k +12)x2 +(43−k x) −56 0= are integers.
