If \(x+y+z+t=0\), prove that
If \(a, b, c\) are unequal non-zero numbers, solve the simultaneous equations \begin{align*} x+y+z &= a+b+c, \\ \frac{x}{a} + \frac{y}{b} + \frac{z}{c} &= 1, \\ \frac{x}{a^3} + \frac{y}{b^3} + \frac{z}{c^3} &= 0, \end{align*} distinguishing the various cases that may arise.
Prove that, if the roots of the equation \[ x^n - \binom{n}{1}p_1 x^{n-1} + \dots + (-)^r \binom{n}{r} p_r x^{n-r} + \dots + (-)^n p_n = 0, \] where \(\binom{n}{r} = \frac{n!}{r!(n-r)!}\) and \(p_n \ne 0\), are all real and positive, so are the roots of the equation \[ x^{n-1} - \binom{n-1}{1}p_1 x^{n-2} + \dots + (-)^r \binom{n-1}{r} p_r x^{n-r-1} + \dots + (-)^{n-1} p_{n-1} = 0. \] Deduce that \[ p_{r-1}p_{r+1} < p_r^2 \quad (1< r< n; p_0=1), \] stating when equality occurs. Prove also that \[ p_1 \ge p_2^{1/2} \ge p_3^{1/3} \ge \dots \ge p_n^{1/n}. \]
Find the equation whose roots are less by 2 than the squares of the roots of \[ x^3+qx+r=0. \] Examine the particular case \[ x^3-3x+1=0, \] and interpret the result.
\(P\) is a point inside a triangle \(ABC\), at distances \(a', b', c'\) from \(A, B, C\) respectively; the angles \(BPC, CPA, APB\) are \(A', B', C'\). By producing \(AP, BP, CP\) to meet the circumcircle of the triangle again, or otherwise, prove that \[ \frac{aa'}{\sin(A'-A)} = \frac{bb'}{\sin(B'-B)} = \frac{cc'}{\sin(C'-C)}. \]
Prove that \(\tan^2(\pi/11)\) is a root of the equation \[ t^5 - 55t^4 + 330t^3 - 462t^2 + 165t - 11 = 0. \] What are the other roots? Hence, by expressing \[ u = \tan(3\pi/11) + 4\sin(2\pi/11) \] as a rational function of \(\tan(\pi/11)\), or otherwise, prove that \(u\) is equal to \(\sqrt{11}\).
If \(y^2 = ax^2+2bx+c\), prove that \[ y^3 \frac{d^2y}{dx^2} = ac-b^2. \] Prove that, if \(n\) is a positive integer, \[ y^{2n+1} \frac{d^{2n}}{dx^{2n}}(y^{2n-1}) = 1^2 \cdot 3^2 \cdot 5^2 \dots (2n-1)^2 (ac-b^2)^n. \]
Discuss the maxima and minima of the function \[ \sin mx \csc x, \] where \(m\) is a positive integer, finding how many there are between \(-\frac{1}{2}\pi\) and \(\frac{1}{2}\pi\). Deduce that \[ \sin^2 mx \le m^2 \sin^2 x. \]
Integrate \[ \frac{1}{(6x^2-7x+2)\sqrt{(x^2+x+1)}}, \quad \frac{1}{(a+b\tan\theta)^2}. \] Prove that, if \(m\) is a number not less than 2, and \(n\) is a positive integer, \[ \int_0^{\frac{1}{2}\pi} \sin^m x \cos 2nx dx = k_{m,n} \int_0^{\frac{1}{2}\pi} \sin^{m-2}x \cos 2nx dx, \] evaluating the number \(k_{m,n}\). Find the value of \[ \int_0^{\frac{1}{2}\pi} \sin^8 x \cos 2nx dx. \]
Give a rough sketch of the curve \[ y^2 - 2x^3 y + x^7 = 0, \] and find the area of the loop.