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MATHEMATICS
EXPLORATION PROBLEMS
Name : Index Number :
Country :
15th International Mathematics and Science Olympiad Zhejiang Province, China
1 October 2018
Instructions:
1. Write your name, country and index number on every page of the AnswerSheet.
2. Write your answers only in the Answer Sheet.
3. Answer all questions in Arabic Numerals.
4. There are 6 questions in this paper.
5. Each question is worth 6 marks and partial credit may be awarded.
6. You have 120 minutes to complete this paper.
7. You are provided with some manipulatives for exploration on some questions.
International Mathematics and Science Olympiad 2018
EXPLORATION PROBLEMS
(1) Use the digits from 1 to 7 to make the equation below correct. Each letter represents a different one-digit number.
a ＋ b c ÷ d ＝ e f
List down all possible equations if d > . 3
(2) There are 27 unit cubes, each of size 1 1 1× × , are glued together to form a larger 3 3 3× × cube. Each face on each of the 1 1 1× × unit cubes contains a positive integer such that:
(i) Two opposite faces of any unit cube must always contain an even number and an odd number. Moreover, the even number must be twice the odd number.
(ii)When any two unit cubes are placed face-to-face, the two touching faces must always contain an even number and an odd number. Moreover, the odd number must be 1 more than the even number.
(a) Place the numbers in the faces of A', B' and C' which are opposite to the faces of A, B, and C, respectively. Find the total sum of the numbers on the faces of A', B' and C'. (2 Marks)
(b) Find the total sum of the numbers on the faces of all the unit cubes. (4 Marks) 1 62 7 14 5 9 5 3 1 78 1 30 1 7 5 5 1 62 1 9 1 30 3 46 14 7 3 Face C Face B Face A
(3) There are 5 points on the plane, such that no three points lie on one line. Every pair of them is connected by a segment. It is known that it is possible to color five of these segments in red and the rest in blue, so that all segments of each color form a simple pentagon. Draw one example.
The figures below are some examples of a simple pentagon:
The following figures below are NOT simple pentagons (because of self intersections) :
(4) In a 4 4× table, we put a coin on each cell. The coins on some cells are tails up (T), while all the other coins are heads up (H). In each move, you are allowed to change all entries in any row, column or diagonal. Diagonals may be of length 1, 2, 3 or 4. Make the entries in the tables below all H using the required number of moves. Show your steps.
(a) in 4 moves (b) in 6 moves (c) in 6 moves
H T H H H T H H H H H H
H H H H H H H H H T H H
T H H H H H H H H H H H
H H H H H H T H H H H H
(1 Mark) (2 Marks) (3 Marks)
(5) (a) In a sequence of positive integers, each term after the first term is the sum of its preceding term and the largest digit of that term. What is the largest possible number of successive odd terms in such a sequence? (3 Marks) (b) In the following sequence {00, 01, 02, 03, …,39}, the terms were
rearranged so that each term after the first is obtained from the preceding one by increasing or decreasing one of its digits by 1. For example, 29 can be followed by 19, 39 or 28, but not by 30 or 20. What is the largest
number of terms that can remain in their original places? (3 Marks)
(6) (a) A chess piece can start anywhere on a 7 7× chessboard. It can jump over 4 or 5 vacant squares either vertically or horizontally, but it cannot visit the same square twice. At most how many squares can it visit? (1 Mark)
(b) A chess piece can start anywhere on a 9 9× chessboard. It can jump over 5 or 6 vacant squares either vertically or horizontally, but it cannot visit the same square twice. At most how many squares can it visit? (2 Marks) (c) A chess piece can start anywhere on a 15 15× chessboard. It can jump
over 8 or 9 vacant squares either vertically or horizontally, but it cannot visit the same square twice. At most how many squares can it visit? (3 Marks)
