If the equations \[ \frac{x}{y+z}=a, \quad \frac{y}{z+x}=b, \quad \frac{z}{x+y}=c, \] have a solution \(x, y, z\), find the relation that must be satisfied by \(a, b, c\). If \(bc, ca\) and \(ab\) are all unequal to 1, show that \[ \frac{x^2}{a(1-bc)} = \frac{y^2}{b(1-ca)} = \frac{z^2}{c(1-ab)}. \]
Find the condition on the coefficients \(p, q, r, s\) of the equation \[ x^4+px^3+qx^2+rx+s=0 \] for two of its roots \(\alpha, \beta\) to satisfy the equation \(\alpha+\beta=0\). Show that the equation \[ x^4 - 2x^3 + 4x^2 + 6x - 21 = 0 \] satisfies this condition, and solve it completely.
If \(a,b,c,x,y,z\) are all real numbers, and \[ a+b \ge c, \quad b+c \ge a, \quad c+a \ge b, \] show that the expression \[ a^2(x-y)(x-z) + b^2(y-x)(y-z) + c^2(z-x)(z-y) \] can never be negative.
(i) Show that every number of the form \(n^5-n\), where \(n\) is an integer, is divisible by 30, and that, if \(n\) is odd, it is divisible by 240. (ii) Show that, if an odd number has an even digit in the tens' place, all its integral powers have an even digit in the tens' place.
Prove the identity \[ \begin{vmatrix} x_1 & y_1 & a & b \\ \lambda_1 x_2 & x_2 & y_2 & c \\ \lambda_1 \lambda_2 x_3 & \lambda_2 x_3 & x_3 & y_3 \\ \lambda_1 \lambda_2 \lambda_3 x_4 & \lambda_2 \lambda_3 x_4 & \lambda_3 x_4 & x_4 \end{vmatrix} = (x_1 - \lambda_1 y_1)(x_2 - \lambda_2 y_2)(x_3 - \lambda_3 y_3)x_4. \] Hence, or otherwise, prove that \[ \begin{vmatrix} a_1b_1 & a_1b_2 & a_1b_3 & a_1b_4 \\ a_1b_2 & a_2b_2 & a_2b_3 & a_2b_4 \\ a_1b_3 & a_2b_3 & a_3b_3 & a_3b_4 \\ a_1b_4 & a_2b_4 & a_3b_4 & a_4b_4 \end{vmatrix} = a_1 b_4 (a_2b_1 - a_1b_2)(a_3b_2-a_2b_3)(a_4b_3-a_3b_4), \] and evaluate \[ \begin{vmatrix} 1 & 1 & 1 & 1 \\ b_1 & a_1 & a_1 & a_1 \\ b_1 & b_2 & a_2 & a_2 \\ b_1 & b_2 & b_3 & a_3 \end{vmatrix}. \]
A man on a hill observes that three vertical towers standing on a horizontal plane subtend equal angles at his eye, and that the angles of depression of their bases are \(\alpha_1, \alpha_2, \alpha_3\); prove that, if \(c_1, c_2, c_3\) are the heights of the towers, \[ \frac{\sin(\alpha_2-\alpha_3)}{c_1\sin\alpha_1} + \frac{\sin(\alpha_3-\alpha_1)}{c_2\sin\alpha_2} + \frac{\sin(\alpha_1-\alpha_2)}{c_3\sin\alpha_3} = 0. \]
If the angles \(\theta_1, \theta_2, \dots, \theta_n\) all lie between \(0\) and \(\frac{1}{2}\pi\), and \(\theta_1+\theta_2+\dots+\theta_n=\alpha\), where \(\alpha\) is fixed, show that \[ \sin\theta_1+\sin\theta_2+\dots+\sin\theta_n \] attains its maximum value when all the angles \(\theta_r\) are equal. State and prove the corresponding result for \[ \tan\theta_1+\tan\theta_2+\dots+\tan\theta_n. \]
A leaf of a book is of width \(a\) in. and height \(b\) in., where \(3a \le 2\sqrt{2}b\); the lower corner of the leaf is folded over so that the corner just reaches the inner edge of the page. Find the minimum length of the resulting crease. Explain why the condition \(3a \le 2\sqrt{2}b\) has to be imposed.
Sketch roughly the curve \[ y^2(a^2+x^2) = x^2(a^2-x^2), \] and find the area of one of its loops.
Find the relation between \(p\) and \(\alpha\) in order that the straight line \[ x\cos\alpha+y\sin\alpha=p \] should cut the circles \[ (x-a)^2+y^2=b^2, \quad (x+a)^2+y^2=c^2, \] in chords of equal length. Prove that the envelope of the lines satisfying this condition is a parabola, and find its equation.