APMO 1992 Q5: Number Sequences: Upper Secondary Mathematics Competition Question

Find a sequence of maximal length consisting of non-zero integers in which the sum of any seven consecutive terms is positive and that of any eleven consecutive terms is negative.


[APMO 1992 Q5]

APMO 1991 Q4: Children Round a Circle: Upper Secondary Mathematics Competition Question

During a break, n children at school sit in a circle around their teacher to play a game. The teacher walks clockwise close to the children and hands out candies to some of them according to the following rule. He selects one child and gives him a candy, then he skips the next child and gives a candy to the next one, then he skips 2 and gives a candy to the next one, then he skips 3, and so on. Determine the values of n for which eventually, perhaps after many rounds, all children will have at least one candy each.

[APMO 1991 Q4]

APMO 1995 Q2: Upper Secondary Mathematics Competition Question

Let a₁, a₂, . . . , a_n be a sequence of integers with values between 2 and 1995 such that:

(i) Any two of the a_i’s are relatively prime,

(ii) Each a_i is either a prime or a product of primes.

Determine the smallest possible values of n to make sure that the sequence will contain a prime number.


[APMO 1995 Q2][with minor edit of typo]

I am posting this question as I think it is interesting, however, I also feel it needs some interpretation. I have copied it as written (apart from one typo correction), but I'm not sure why it asks for "values of n" in the plural. I assume the question is asking us to firstly find the maximum number of terms such that conditions (i) and (ii) are satisfied but without any primes appearing; then by adding one unused prime to such a sequence we would have found the minimum number that guarantees that a prime be present. I also assume that the integers are 2 to 1995 inclusive. I leave these assumptions open for discussion; it may be a case of lost in translation as APMO is the Asian Pacific MO.

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.

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!

Amicable Numbers: Lower Secondary Mathematics Competition Question

The sum of the proper divisors of a positive integer n, spd(n), is the sum of the factors that includes 1 but excludes n itself.

A pair of numbers, m and n, is said to be amicable if spd(n)=m and spd(m)=n. The smallest such pair of numbers is 220 and 284.

Which of the following numbers form another pair of amicable numbers?

1174, 1184, 1210, 2394, 2924.

An Odd Sequence: Middle Secondary Mathematics Competition Question

A function T(x) is defined as follows, where x is a positive integer:

If x is even, divide x by 2;

If x is odd, calculate the sum of all the factors of x (including 1 and x itself).

Repeat these rules, thereby creating a sequence of numbers. Let S₀=x, S₁=T(x), S₂=T(T(x)) and so on, so that S_n=Tⁿ(x). Also, let m be the first iteration at which S_m=T^m(x)=1. Note that T(1)=1, so we terminate the sequence at the first 1 we encounter.

For example, if x=5, S₀=5, S₁=6, S₂=3, S₃=4, S₄=2, S₅=1. This results in the sequence {5, 6, 3, 4, 2, 1}, so that for x=5, m=5.

a) Find the value of m for x=121.

b) Find the values of x for which m=7.


One open question to ponder is whether such a sequence terminates for every starting value of x.

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?

Prize Maths Quiz: Points in a Triangle (PMQ28)

An equilateral triangle lies in the plane with two of its vertices at the points (0, 0) and (n, 0), where n is an integer. Determine the number of points (x, y) with integer coordinates that lie in the interior of the triangle.

Your final answer should be a formula that relates the total number of lattice points (x, y), call it N, to the x-coordinate, n. Note also that the lattice points must lie within the triangle and not along its perimeter.

Have fun!

Number Puzzle 11-14: Upper Primary Mathematics Competition Question

This is easier than the number puzzle I posted earlier today - or at least I think so. However, as you can see, they share some genetic material! Have fun!

Find the missing number. There is one solution. If you find different ingenious solutions then feel free to share them below.

Number Puzzle 60-42: Lower Secondary Mathematics Competition Question

Find the missing number and include your reasoning.

General Diagonals in Squares: Lower Secondary Mathematics Competition Question

This is the general case of the question posted earlier.

A NxN grid is drawn so that it has N^2 unit squares where N is an even number. A diagonal within a unit square is a straight line with endpoints at opposite corners of the square - one example is drawn below. You are required to draw such diagonals in such a way that none of them have a point in common; that is, two diagonals cannot overlap and they cannot touch at an endpoint.

What is the maximum number of such diagonals of unit squares that can be drawn within a general NxN grid where N is even?

What is the formula if N is an odd number?