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Find, with proof, all solutions of the equation

2^n = a! + b! + c!

in positive integers a, b, c and n.

2^n = a! + b! + c!

in positive integers a, b, c and n.

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**IrMO 2001 P1 Q1: Upper Secondary Mathematics Competition Question**

Find, with proof, all solutions of the equation 2 n = a! + b! + c! in positive integers a, b, c and n. (Here, ! means "factorial".) [IrMO 2001 Paper 1 Question 1] Feel free to comment, ask questions and even check your answer in the comments box below power...

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Two regular octagons with 4 red and 4 blue beads at their vertices. Given any initial layout and the ability to rotate the octagons, what is the highest number of vertices guaranteed to have the same colour? Full question at the link.

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**Matching Octagons: Middle Secondary Mathematics Competition Question**

The image shows two regular octagons, each has 4 red and 4 blue beads, one placed on each vertex. We say that there is a 'match' if a vertex has the same colour in both octagons. In the diagram, we can see that the upper-right vertices are both blue and the...

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Let N = a! + b!

Find all solutions (a,b) such that N is divisible by 11 and both a and b are positive integers less than 11, with a ≤ b.

How many solutions are there?

Find all solutions (a,b) such that N is divisible by 11 and both a and b are positive integers less than 11, with a ≤ b.

How many solutions are there?

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**Sums of Factorials: Middle Secondary Mathematics Competition Question**

Let N = a! + b! Find all solutions (a,b) such that N is divisible by 11 and both a and b are positive integers less than 11, with a ≤ b. How many solutions are there? You may leave your answers in the form n!, where n! = n.(n-1).(n-2)... 3.2.1. Feel free to...

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Six beads around a regular hexagon; 3 orange and 3 green.

Using all six beads, how many unique arrangements are there?

Using all six beads, how many unique arrangements are there?

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**Beads on a Hexagon: Lower Secondary Mathematics Competition Question**

Six beads are arranged at the corners of a regular hexagon; 3 are orange and 3 green. All arrangements that are rotational symmetries of each other count as one unique arrangement. Using all six beads, how many unique arrangements are there? Feel free to co...

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Prove that in each set of ten consecutive integers there is one which is coprime with each of the other integers.

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**IrMO 2000 P2 Q4: Upper Secondary Mathematics Competition Question**

Prove that in each set of ten consecutive integers there is one which is coprime with each of the other integers. For example, taking 114, 115, 116, 117, 118, 119, 120, 121, 122, 123 the numbers 119 and 121 are each coprime with all the others. [Two integer...

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