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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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?