IrMO 2002 P2 Q2: Upper Secondary Mathematics Competition Question

Suppose n is a product of four distinct primes a, b, c, d such that

(a) a + c = d;

(b) a(a + b + c + d) = c(d - b);

(c) 1 + bc + d = bd.

Determine n.

[IrMO 2002, Paper 2, Question 2]

Primes in a Triangle: Lower Secondary Mathematics Competition Question

Place nine distinct prime numbers into the grid in such a way that the sums of the four numbers along each side are all equal to each other.

What is the smallest possible such sum?

Circled Primes: Upper Primary Mathematics Competition Question

The diagram shows the numbers 1 to 25 inserted into a table. Draw a circle round some of the prime numbers in such a way that every row and every column has only one circled prime.

a) What is the sum of your circled primes?

b) Prove that your solution is unique.

I am Thinking of a Prime: Upper Primary Mathematics Competition Question

I am thinking of a prime number less than 20.
If I reverse the digits, I get another prime number.
If I multiply these two numbers together, their product is a 4-digit number.

What is the number I was thinking of?

A Sequence of Prime Factors: Upper Secondary Mathematics Competition Question

Let f(x) be the sum of the prime factors of the positive integer x, including repeated factors. For example, f(20)=f(2x2x5)=2+2+5=9. Note that f(1)=0 and f(p)=p if p is prime.

Let g(x) be the function g(x)=f(ax+b), where a and b are positive integers. If we iterate g we obtain the sequence g₀=x, g₁=f(ag₀+b) and g_n=f(ag_n-1+b). Such sequences always end in a cycle of length L. Some of these sequences terminate at a fixed point P with a cycle length of 0.

For example, if g(x)=f(3x+1) and x=14 we get the sequence {14, 43, 20, 61, 29, 17, 17} which terminates at the fixed point 17.

Let's look specifically at the function g(x)=f(5x+3).

a) Calculate the sequence generated from x=40 and find its value of L.

b) Find the two fixed points of this sequence that are both less than 100.


Note that such sequences have not been exhaustively analysed and there are a number of open questions that I will discuss in another post.

IrMO 1995 P2 Q5: Upper Secondary Mathematics Competition Question

For each integer n such that n = p₁p₂p₃p₄, where p₁, p₂, p₃, p₄ are distinct primes, let

d₁ = 1 < d₂ < d₃ < ... < d₁₅ < d₁₆ = n

be the sixteen positive integers that divide n.
Prove that if n < 1995, then d₉ - d₈ ≠ 22.


[IrMO 1995 Paper 2 Question 5]

Terminating Primes: Middle Secondary Mathematics Competition Question

Let s(n) be the sum of the proper factors of a positive integer n; this is the sum of all the factors of n, including 1 but excluding n itself. Let s₀=n, s₁=s(n), s₂=s(s(n)) and so on, thereby creating the sequence {s₀, s₁, s₂, ...}.

If n is a prime number p, then s(p)=1 and s(s(p))=0, thus terminating the sequence. As most such sequences terminate in this way, it is normal to terminate the sequence at the first prime number.

a) Calculate the terminating prime number for the starting value of n=12.

b) Find all possible sequences such that s₆=7.

c) Prove that it is not possible for a sequence to terminate with a 5, unless s₀=5.


These types of sequences are still being researched and they do not all terminate in the manner described above. Try n=276 and see what happens. Have fun!

Fibonacci Locker Code: Middle Secondary Mathematics Competition Question

Larry’s locker has a 4-digit security code. To help him remember it, he chose digits from consecutive numbers in the standard Fibonacci sequence (0, 1, 2, 3, 5 and so on). He noticed that two such 4-digit codes were also prime numbers. Larry chose the larger of the two.

What is the 4-digit code to Larry’s locker?

Prime Numbers, Prime TV and a Prime Valentine’s Day

It is somewhat surreal to see the discovery of the largest prime number paraded on prime time broadcast media. Mathematicians around the world are asked to explain the significance of this discovery in layman’s terms, which is up there with physicists trying to explain what the Large Hadron Collider actually does.

Below you’ll see a pretty sparky Sky News interview by Eugenia Cheng, a senior lecturer at the University of Sheffield. Just notice at the very end how jolly pleased with themselves the newscasters seem to be.

The new record was discovered by Dr Curtis Cooper at the University of Central Missouri. This is the third time that Dr Cooper has entered the record books, and one must hope that he is using his university’s spare computing power to do so. Indeed, this discovery was not done in isolation but as part of a global internet project known as GIMPS (Great Internet Mersenne Prime Search). You too can join in; just go to the GIMPS website, sign up, download the software and you’re up and running; or rather, your computer will be.