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Sunday Afternoon Maths XXXIV

 Posted on 2014-11-09 

Find them all

Find all continuous positive functions, \(f\) on \([0,1]\) such that:
$$\int_0^1 f(x) dx=1\\ \mathrm{and }\int_0^1 xf(x) dx=\alpha\\ \mathrm{and }\int_0^1 x^2f(x) dx=\alpha^2$$

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Four points on a shape

Fiona wants to draw a 2-dimensional shape whose perimeter passes through the points A, B, C and D
Which of the following shapes can she draw?
(i) A circle
(ii) An equilateral triangle
(iii) A square

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Algebraic fractions

Given that
$$\frac{3x+y}{x-3y}=-1$$
what is the value of
$$\frac{x+3y}{3x-y}$$
?

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Tags: algebra

Four integers

\(a\), \(b\), \(c\) and \(d\) are four positive (and non-zero) integers.
$$abcd+abc+bcd+cda+dab+ab+bc+cd+da+ac+bd\\+a+b+c+d=2009$$
What is the value of \(a+b+c+d\)?

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Sum

What is
$$\sum_{i=1}^{\infty}\frac{1}{i 2^i}$$
?

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