Download Problem Sheet 2 – Problems based on number Warm-up!

Year 12 Problem-Solving Summer School
Problem Sheet 2 – Problems based on number
Warm-up!
Question 1
Show that, for any natural number n, n(n+1) is even
Solution
Question 2
For what value of n is n! equal to the exact number of seconds in six
weeks?
Solution
Year 12 Problem-Solving Summer School
Question 3
In a sequence of numbers the nth term is given by n2 + 2n.
(a)
Write the first six numbers in the sequence;
(b) Explain why the numbers in the sequence alternates between odd
and even.
Solution
(a)
(b)
Year 12 Problem-Solving Summer School
Question 4
The number
(102 – 82)(92 – 72)(82 – 62)(72 – 52)(62 – 42)(52 – 32)(42 – 22)(32 – 12)
can be written in the form k × 2n, where k and n are positive integers and
k is odd. What are the values of k and n?
Solution
Year 12 Problem-Solving Summer School
Questions to be considered in the next online session
Question 1
Five numbers are arranged in order from least to greatest:
x, x3, x4, x2, x0
Where does –x–1 belong in the list above?
Solution
Year 12 Problem-Solving Summer School
Question 2
Is n2 + n + 41 a prime number for all natural numbers n? If your answer
is yes, prove it. If your answer is no, give a counter-example.
Solution
Year 12 Problem-Solving Summer School
Question 3
How many two digit numbers N have the property that the sum of N and
the number formed by reversing the digits of N is a square?
Solution
Year 12 Problem-Solving Summer School
Question 4
Show that, for any natural number n, n(n +1)(2n + 1) is a multiple of 6
Solution
Year 12 Problem-Solving Summer School
Question 4a
Show that every cube number is either a multiple of 9, is one more than
a multiple of 9 or is one less than a multiple of 9.
Solution
Year 12 Problem-Solving Summer School
Question 5
The expressions
௫ାଷ
ଷ
and
ଶ௫ିଵ
ହ
represent a pair of consecutive
whole numbers.
The expressions
ଶ௬ିଵ
ହ
and
whole numbers.
What is the value of |x – y|?
Solution
୷ାଷ
ଷ
represent a pair of consecutive
Year 12 Problem-Solving Summer School
Question 6 (AEA, part)
Define
f(x) = x – [x], x ≥ 0
where [x] is the largest integer that is less than or equal to x.
For example, f(3.7) = 3.7 – 3 = 0.7; f(3) = 3 – 3 = 0.
(a)
Sketch the graph of y = f(x) for 0 ≤ x < 4.
(b)
Find the value of p for which ‫׬‬0 fሺxሻdx= 0.18
Solution
(a)
(b)
p
Year 12 Problem-Solving Summer School
Question 7 (STEP, part)
How many integers greater than or equal to zero and less than a million
are not divisible by 2 or 5? What is the average value of these integers?
Solution