A point P is selected anywhere inside an equilateral triangle. What can you say about the sum of the perpendicular distances from P to the sides of the triangle? Can you prove your conjecture?
We are given a regular icosahedron having three red vertices. Show that it has a vertex that has at least two red neighbours.
Can you make sense of the three methods to work out the area of the kite in the square?
Biren Patel, Heathland School, Hounslow, England; Andrei, School no. 205, Bucharest, Romania and Ang Zhi and Chai from River Valley High School, Singapore sent in very good solutions. Here is Chai solution:
Prove that if n is a triangular number then 8n+1 is a square number.
Prove, conversely, that if 8n+1 is a square number then n is a triangular number.
Solution:
Substitute n = K/2 (K+1) into 8n + 1
Therefore, if n is a triangular number then 8n+1 is a square number.
To prove the converse, let X 2 = 8n + 1 be a square number.
As 8 is an even number, 8n will always be an even number. If 8n is an even number, then 8n + 1 will always be an odd number. X cannot be even because the square of an even number is even.
Hence X= (2k + 1) represents an odd number where k can be any whole number.
(2k + 1) 2 = 8n + 1
4k 2 + 4k+ 1 = 8n + 1
4k 2 + 4k + 1 - 1 = 8n
4/8(k 2 + k) = n
(k 2 + k)/2 = n
Therefore, n = k/2 (k+1) so n is a triangular number.