this month, Stage 3 & 4

July 2001

 

Problems

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Triangular Triples

Stage:3 Challenge Level:Challenge Level:1

Show that 8778, 10296 and 13530 are three triangular numbers and that they form a Pythagorean triple.

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Art and Mathematics

Stage:3 Challenge Level:Challenge Level:1

The art of tiling has been around since the beginning of early civilisation. This is evident in the floors and walls of ancient monuments and other religious buildings. Alhambra at Granada in Spain. . . .

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Take Ten

Stage:3 Challenge Level:Challenge Level:3Challenge Level:3Challenge Level:3

Is it possible to remove ten unit cubes from a 3 by 3 by 3 cube made from 27 unit cubes so that the surface area of the remaining solid is the same as the surface area of the original 3 by 3 by 3. . . .

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Mod 3

Stage:4 Challenge Level:Challenge Level:1

Prove that if a^2+b^2 is a multiple of 3 then both a and b are multiples of 3.

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Root to Poly

Stage:4 Challenge Level:Challenge Level:1

Find the polynomial p(x) with integer coefficients such that one solution of the equation p(x)=0 is 1+sqrt2+sqrt3.

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Long Short

Stage:4 Challenge Level:Challenge Level:1

A quadrilateral inscribed in a unit circle has sides of lengths s1, s2, s3 and s4 where s1 ≤ s2 ≤ s3 ≤ s4. Find a quadrilateral of this type for which s1= sqrt2 and show s1 cannot. . . .

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Quadarc

Stage:4 Challenge Level:Challenge Level:2Challenge Level:2

Given a square ABCD of sides 10 cm, and using the corners as centres, construct four quadrants with radius 10 cm each inside the square. The four arcs intersect at P, Q, R and S. Find the. . . .

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Iff

Stage:4 Challenge Level:Challenge Level:2Challenge Level:2

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.

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Parabolas Again

Stage:4 and 5 Challenge Level:Challenge Level:1

Here is a pattern composed of the graphs of 14 parabolas. Can you find their equations?