Question 1
A pattern of hexagons is made from sticks, as shown below.
If 200 sticks are used, how many hexagons have been formed?
If 200 sticks are used, how many hexagons have been formed?
IMSO 2005 Mathematics – Taiwan Selection – Exploration
Question 2
If you have coins of denominations 1, 2, 5 and 10 dollars, in how many ways can you make up 50 dollars?
Coins of each denomination are available in sufficient quantity. Count combinations of coin numbers, not different orders of the same coins.
Coins of each denomination are available in sufficient quantity. Count combinations of coin numbers, not different orders of the same coins.
Question 3
Place the digits 1 to 7 in the grid so that each digit appears once in every row and once in every column. Each circled number is the product of the four digits surrounding that circle.
The source includes a completed 4-by-4 example explaining the rule and the blank 7-by-7 puzzle:
The source includes a completed 4-by-4 example explaining the rule and the blank 7-by-7 puzzle:
Question 4
The Fibonacci numbers are
$$F_1=1,\quad F_2=1,\quad F_3=2,\quad F_4=3,\quad F_5=5,\quad F_6=8,\quad F_7=13,\ldots$$
The first two are both 1, and each later number is the sum of the two preceding it. What is the last digit of the sum of the first 2005 Fibonacci numbers?
$$F_1=1,\quad F_2=1,\quad F_3=2,\quad F_4=3,\quad F_5=5,\quad F_6=8,\quad F_7=13,\ldots$$
The first two are both 1, and each later number is the sum of the two preceding it. What is the last digit of the sum of the first 2005 Fibonacci numbers?
Question 5
Four darts are thrown at the dartboard shown. The four scores are added, and a miss counts as zero. What is the smallest positive total score that is impossible to obtain?
