Team Contest
Instructions
Team members may discuss among themselves silently.
With the permission of invigilators, team members may go to the washroom but they should not communicate with others there.
Answers and all working must be written on these answer sheets. The use of calculator is not allowed.
Each team is required to answer six questions in three hours. One question will be issued every 20 minutes.
Each question carries 20 marks.
Team Code:______________________
Question 1
A piece of square paper ABCE is folded and opened flat several times as follows:
Step 1
The square ABCD is folded along a line EF, where E lies on BC and F lies on AD, so that AB coincides with CD. The pieces of paper is then opened flat and the points E and F are marked.
Step 2
The line segment EA is marked.
Step 3
The square is folded along another line EG, where G lies on the side AB, so that EB coincides with EH, where H is a point on AE. The piece of paper is opened flag again.
Step 4
The square is folded again so that AB coincides with AE. Denote by X the point on AB which coincides with H.
Prove that AB BX× =AX2.
Question 2
The figure shows a 5×5 square, which is partitioned into 25 unit squares. Four non-negative integers have been entered into four of these unit squares as shown. Fill in the remaining unit squares with one positive integer each so that the sums of the integers in each column and in each row are all equal.
82
79
103
0
Question 3
Consider the sequence
2
1 1001, 2 1004, 3 1009, , n 1000 ,
a = a = a = ⋯ a = +n ⋯
where n is a positive integer. Denote by dn the highest common factors of an and an+1. Find the
Question 4
After the release of examination results, it was found that the five classes had unequal average scores and that only two teachers each guessed the ranks of exactly two classes correctly. The other three teachers guessed wrongly all the ranks of the five classes. Find the ranks of the five classes.
Question 5
Find the positive integer solutions of the equation
1 1 1
Each team is given 50 squares of side 4 cm and 50 equilateral triangles of side 4 cm. Use these squares and triangles to construct convex polyhedrons (not necessarily regular).
(1) Stick your squares and triangles on the cardboard;
(2) When you stick the figures together, you must stick only the sides of the squares and the triangles;
(3) Make as many different types of convex polyhedrons as possible (polyhedrons with the same numbers of sides, faces and vertices are counted as the same tyoe);
(4) Stick firmly the polyhedrons you make and submit them for marking;
(5) Number your polyhedrons in order and write your team code on each; fill in the checklist to list out the numbers of squares and triangles, the numbers of sides, faces and vertices of the polyhedrons. Submit the completed checklist together with your polyhedrons.
