Suppose that \(a\), \(b\) and \(c\) are real numbers such that the equation \[x^3-ax^2+bx-c=0\] has three distinct real roots, which are in geometric progression. Prove that \(abc > 0\) and that \[\left|\frac{a}{c}-1\right| > 2.\]
Express \((a^2+b^2+c^2)(x^2+\beta^2+\gamma^2)-(a\alpha+b\beta+c\gamma)^2\) as the sum of three squares. Deduce that if \(\alpha\), \(\beta\), \(\gamma\) are real numbers then \[(a^4+\beta^4+\gamma^4)(a^2+\beta^2+\gamma^2) \geq (a^2+\beta^2+\gamma^2)^2.\] Give necessary and sufficient conditions on \(\alpha\), \(\beta\), \(\gamma\) for equality to hold.
Suppose that \(n\), \(x\) and \(y\) are positive integers such that \(n+x\) is a square and \(n+y\) is the next larger square. Show that \(n+xy\) and \(n+xy+x+y\) are adjacent squares. Hence show that \(n+x+y+xy(2+x+y+xy)\) is a square.
Solution: \begin{align*} && n+x &= k^2\\ && n+y &= (k+1)^2 = n+x+2k+1 \\ \Rightarrow && y &= x + 2k+1 \\ && n +xy &= n + x(x+2k+1) \\ &&&= n+x^2+2kx+x \\ &&&= (k^2-x)+x^2+2kx+x \\ &&&= k^2+x^2+2kx \\ &&&=(k+x)^2 \\ &&n+xy+x+y &= k^2-x+x(x+2k+1)+x+x+2k+1 \\ &&&= k^2-x+x^2+2kx+x+2x+2k+1 \\ &&&= k^2+x^2+2kx+2x+2k+1 \\ &&&=(k+x+1)^2 \\ \\ && n+x+y+xy(2+x+y+xy) &= (k+x)^2+x+y+xy+xy(x+y+xy) \\ &&&= (k+x)^2+(x+y+xy)(1+xy) \\ &&&= (k+x)^2+((k+x+1)^2-n)(1+(k+x)^2-n) \\ &&&= (k+x)^2+((k+x)^2+2(k+x)+1-n)((k+x)^2+1-n) \\ &&&= (k+x)^2+(k+x)^4+(1-n)^2+(k+x)^2(2(k+x)+2(1-n))+2(k+x)(1-n) \\ &&&= (k+x)^4+(k+x)^2( \end{align*}
Show that the number of ways of arranging \(N\) indistinguishable oranges and \(M\) indistinguishable pencils in a line is \[\frac{(N+M)!}{N!M!}.\] Hence or otherwise calculate the number of ways of putting \(N\) indistinguishable oranges into \(P\) boxes numbered \(1, \ldots, P\).
By considering \((1-1)^n\), prove that \[\binom{n}{0}-\binom{n}{1}+\binom{n}{2}- \ldots + (-1)^n\binom{n}{n} = 0,\] for \(n = 1, 2, \ldots\). Hence or otherwise prove by induction that \[1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots + \frac{1}{n} = \binom{n}{1}\frac{1}{1} - \binom{n}{2}\frac{1}{2} + \ldots + (-1)^{n-1}\binom{n}{n}\frac{1}{n}\] for \(n = 1, 2, \ldots\). [You may assume without proof that \(\displaystyle \binom{n}{r} = \binom{n-1}r + \binom{n-1}{r-1}\)]
Decompose \[\frac{3x^2+2ax+2bx+ab}{x^3+(a+b)x^2+abx}\] into partial fractions. By considering the smallest denominator or otherwise, show that this expression takes the value 1 for only a finite number of positive integral values of \(x\), \(a\) and \(b\) (You are not required to find all such values.)
Whenever possible, solve the following simultaneous equations (in which \(\lambda\) is a real number). \begin{align*} \lambda x + y &= 1\\ x + (\lambda - 1)y &= 2\\ x + y + (\lambda - 2)z &= \lambda \end{align*} For what values of \(\lambda\) are there no solutions?
The equation \(x^4-8x^3+ax^2-28x+12\) has the property that the sum of a certain pair of roots is equal to the sum of the remaining two roots. Determine \(a\) and find all the roots.
Show that if \(n\) straight lines are drawn in a plane in such a way that no two are parallel and no three meet in a point, then the plane is divided into \(\frac{1}{2}(n^2+n+2)\) regions. How many of these regions stretch to infinity?
Show that angles subtended by a chord of a circle at the circumference and in the same segment are equal. A rod is bent so as to form an acute angle at \(X\). Another rod \(PQ\) slides with its ends \(P\) and \(Q\) on the two straight arms of the bent rod. At each position of \(P\) and \(Q\) lines \(PR\), \(QR\) are drawn perpendicular to the arms on which respectively \(P\) and \(Q\) move. Show that, when the bent rod is fixed and \(PQ\) moves, \(R\) moves on a circle. Show further that, when \(PQ\) is fixed and the bent rod is moved, \(R\) again moves on a circle, of radius half that of the former circle.