Kaohsiung First Invitational World Youth
Mathematics Intercity Competition
Team Contest
Team: __________________________
Time allowed: 3 hours
There are five questions. Each question carries 24 marks.
Rules: (i) Team members are allowed to discuss among themselves silently. (ii) If you wish to go to the toilet, you must obtain the permission of an
invigilator. Do not talk to anyone else there.
(iii) Show your steps and answers on these answer sheets. (iv) Please do not take away any question paper or materials.
Question 1
(a) Find the prime factors of the number 98 + 76 + 54 +32 + 1. (b) Find any two prime factors of the number 230 + 330.
Question 2
Write the numbers 1, 3, 5, ..., 59 on the blank cards provided, each number appearing on one card only.
(a) Arrange the cards numbered 1, 3, 5, 7, 9, 11, 13, 15, 17, 19 in a pile so that the arrangement satisfies the following requirement:
If the topmost card is moved to the bottom of pile, then the card numbered 1 will be topmost. Put this card numbered 1 on one side. Move the three topmost cards to the bottom of the pile. Then the card numbered 3 will be topmost. Put this card
numbered 3 on one side. Repeat this procedure to move the topmost 2k+1 (2≦k≦8) cards to the bottom of the pile in order so that the topmost card will be the one numbered 2k+1 after the move. The last one is the card 19.
Write down the original arrangement of these cards.
(b) Repeat the procedure in (a) for the cards numbered 1, 3, 5, ..., 59 in a pile so that the arrangement satisfies the following requirement:
numbered 3 on one side. Repeat this procedure to move the topmost 2k+1 (2≦k≦ 28) cards to the bottom of the pile in order so that the topmost card will be the one numbered 2k+1 after the move. The last card is the one 59.
Write down the original arrangement of these cards.
Question 3
(a) Find a group of distinct positive integers, including the number 5, such that the sum of their reciprocals is 1.
(b) Find a group of distinct positive integers, including the number 1999, such that the sum of their reciprocals is 1.
Question 4
(a) Describe, with justification, how to dissect a square into exactly 1999 smaller non-overlapping squares, which need not be identical with each other;
(b) Cut the logo (Figure 1) on a separate sheet provided in a minimal number of pieces that fill the grid (checkboard with central 2 ?2 hole) shown in Figure 2 below. Solution must indicate the number of parts clearly. You may flip over some pieces if necessary.
Question 5
A 5×5 square is divided into 25 unit squares. One of the numbers 1, 2, 3, 4, 5 is inserted into each of the unit squares in such a way that each row, each column and each of the two diagonals contains each of the five numbers once and only once. The sum of the numbers in the four squares immediately below the diagonal from top left to bottom right (shown as shaded part in the figure below) is called the score.
What is the highest possible score? Justify your claim and construct the corresponding 5×5 square. You may be awarded partial marks if the score for your 5×5 square constructed is not the highest possible one.
