2011 AITMO 隊際賽試題

(1)

注意：

允許學生個人、非營利性的圖書館或公立學校合理使用

本基金會網站所提供之各項試題及其解答。可直接下載

而不須申請。

重版、系統地複製或大量重製這些資料的任何部分，必

須獲得財團法人臺北市九章數學教育基金會的授權許

可。

申請此項授權請電郵

ccmp@seed.net.tw

Notice:

Individual students, nonprofit libraries, or schools are

permitted to make fair use of the papers and its

solutions. Republication, systematic copying, or

multiple reproduction of any part of this material is

permitted only under license from the Chiuchang

Mathematics Foundation.

Requests for such permission should be made by

e-mailing Mr. Wen-Hsien SUN

ccmp@seed.net.tw

(2)

Team Contest

Time limit: 60 minutes 2011/11/15

Team Name：

Problem 1 Problem 2 Problem 3 Problem 4 Problem 5 Total Score

Problem 6 Problem 7 Problem 8 Problem 9 Problem 10

Instructions:

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

Remember to write down your team name in the space indicated on the first page.

There are 10 problems in the Team Contest, arranged in increasing order of difficulty, each problem is worth 40 points and the total is 400 points. Each question is printed on a separate sheet of paper. Complete solutions of problem 1, 2, 3, 4, 5, 8 and 9 are required for full credits. Partial credits may be awarded. Only Arabic Numerical answer or drawing in Problem number 6, 7 and 10 are needed.

Diagrams are NOT drawn to scale. They are intended only as aids.

The four team members are allowed 10 minutes to discuss and distribute the first 8 problems among themselves. Each student must solve at least one problem by themselves. Each will then have 35 minutes to write the

solutions of their allotted problem independently with no further discussion or exchange of problems. The four team members are allowed 15 minutes to solve the last 2 problems together.

No calculator or calculating device or electronic devices are allowed. Answer the problems with pencil, blue or black ball pen.

All papers shall be collected at the end of this test.

Malpi International School

(3)

TEAM CONTEST

Team：

Score：

1. Find all possible three-digit numbers abc such that when the tens digit of the given three-digit number was deleted, a two-digit number ac will be formed so that abc= ×9 ac+ ×4 c. For example, 155= × + ×9 15 4 5.

Panauti, Kavre, Nepal.

(4)

TEAM CONTEST

Team：

Score：

2. Find all positive real solutions of the simultaneous equations:

2 3 2 3 2 3 3 (1) 3 (2) 3 (3) x y z y z x z x y + + = − − − + + = − − − + + = − − −

(5)

TEAM CONTEST

Team：

Score：

3. Prove that a2 + + +b2 c2 d2 +ab+ac+ad +bc+bd +cd ≥20, where a, b, c and

d are positive real numbers such that abcd = 4.

(6)

TEAM CONTEST

Team：

Score：

4. Consider a 10 × 15 chessboard. We wish to travel the shortest distance along the grid lines from the bottom left corner to the top right corner. What is the ratio of the number of paths which start with a move to the right to the number of paths which start a move up?

(7)

TEAM CONTEST

Team：

Score：

5. The figure below shows a triangle ABC. D is the midpoint of BC and E lies on

AC such that AE : EC = 2 : 3. If F is a point of AB such that the area of triangle DEF is twice the area of triangle BDF, find the ratio of AF : FB.

A F E D C B Malpi International School

(8)

TEAM CONTEST

Team：

Score：

6. The 25 squares of a 5×5 chessboard are labeled as shown in the diagram below. The chessboard is cut up into a unit square and eight copies the shape shown in the diagram below on the right. These copies may be rotated. Of the 25 squares, identify by their labels those which may be the unit square.

ANSWER:

A B C D E F G H I J K M N O P Q R S T U V W X Y Z

(9)

TEAM CONTEST

Team：

Score：

7. Find 38 consecutive positive integers, such that the sum of the digits of each of them is not divisible by 11.

(10)

TEAM CONTEST

Team：

Score：

8. Circle with center I is inscribed in triangle ABC and touches the sides AC and BC in points M and N. The line MN intersect the line AB at P, as B is between A and

P. If BP = CM , find∠AIC, in degree.

ANSWER:

°

A N M I C B P

(11)

TEAM CONTEST

Team：

Score：

9. Nikolai and Peter share 21 peanuts as follows:

Peter will divide the whole piles into two, with at least two peanuts in each, after which Nikolai will subdivide each pile into two, with at least one peanut in each. Nikolai agrees in advance to take the middle two piles.

No matter how Peter divided, what is the maximum number of peanuts that Nikolai is guaranteed to get? What is Nikolai’s strategy?

(12)

TEAM CONTEST

Team：

Score：

10. In a soccer tournament, there are 10 participating team. Each team plays with each of the others once. A win used to be worth 2 points, but is now worth 3 points. A draw is worth 1 point, and a loss worth no points. It happens that there is a winning team in the tournament under the new system, and yet finishes last under the old system. Reconstruct the possible result of each game in the below table. Using W represent team A wins team B, using 0 represent team A loss to team B and 1 represent team A and team B are tied.

Team (B) Team (A) 1 2 3 4 5 6 7 8 9 10 The total points under older system The total points under new system 1 2 3 4 5 6 7 8 9 10