Show that angles in the same segment of a circle are equal. A rod \(PQ\) slides with its ends \(P, Q\) on the two straight arms of a bent rod. At each position of \(P\) and \(Q\) lines \(PR, QR\) are drawn perpendicular respectively to the arms on which \(P\) and \(Q\) move. Show that, when the bent rod is fixed and \(PQ\) moves, the locus of \(R\) is a circle, and that, when \(PQ\) is fixed and the bent rod is moved, the locus of \(R\) is again a circle, of radius half the former circle and touching it at \(R\).
Prove that the number of combinations of \(n\) things \(r\) at a time is \(n!/\{r!(n-r)!\}\). A pack of cards is dealt (in the usual way) to four players. One player has just 5 cards of a particular suit; prove that the chance that his partner has the remaining 8 cards of that suit is \(1/(4.17.19.37)\).
Small errors \(\delta a, \delta b, \delta c\) are made in measuring the sides of a triangle; prove that the consequent error in reckoning the radius of the circumcircle is \[ \frac{1}{2}\cot A \cot B \cot C\left(\frac{\delta a}{\cos A} + \frac{\delta b}{\cos B} + \frac{\delta c}{\cos C}\right). \]
Prove that the radius of curvature of a curve, which is given by the equations \[ x=\phi(t), y=\psi(t), \] is \[ \frac{\left\{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2\right\}^{\frac{3}{2}}}{\left\{\frac{d^2y}{dt^2}\frac{dx}{dt} - \frac{d^2x}{dt^2}\frac{dy}{dt}\right\}}, \] and find the radii of curvature at the origin of the two branches of the curve given by the equations \[ y=t-t^3, \quad x=1-t^2. \]
Trace the curve \[ y^2(a+x) = x^2(3a-x), \] and show that the area of the loop and the area included between the curve and the asymptote are both equal to \(3\sqrt{3}a^2\).
Show that the series \[ 1 - \frac{1}{1+x} + \frac{1}{2} - \frac{1}{2+x} + \frac{1}{3} - \frac{1}{3+x} + \dots \] is convergent, provided that \(x\) is not a negative integer.
Two equal heavy cylinders of radius \(a\) are placed in contact in a smooth fixed cylinder of radius \(b\) (\(>2a\)); a third equal cylinder is placed gently on top of them, the axes of all the cylinders being horizontal. Show that the two lower cylinders will not separate if \[ b < a(1+2/\sqrt{7}). \]
A string \(ABC\) (\(AB=BC=a\)) is stretched out straight on a smooth table with masses \(m\) tied at \(A, B, C\). Impulses each equal to \(I\) perpendicular to the string are applied at \(A\) and \(C\). Find the tensions just before \(A\) and \(C\) collide.
A particle projected with speed \(u\) strikes at right angles a plane through the point of projection inclined at \(\theta\) to the horizon; find the time of flight and the vertical height of the point struck above the point of projection.