Given any four points on the surface of a sphere of unit radius, prove that it is possible to find two of them whose distance apart is at most \(\sqrt{3}\).
Let \(O\), \(U\), \(A\), \(B\) be distinct points on a line \(l\); \(a\), \(b\), \(u\) lines through \(A\), \(B\), \(U\) in the plane and in general position. The line \(u\) meets \(a\) in \(P\) and \(b\) in \(Q\), \(OQ\) meets \(a\) in \(R\), \(UR\) meets \(b\) in \(S\), and \(PS\) meets \(l\) in \(T\). Prove that \(T\) depends only on \(O\), \(U\), \(A\), \(B\), not on the choice of \(a\), \(b\), \(u\).
Prove the cross-axis theorem for homography on a proper conic locus. Show, by giving a geometrical construction for it, that there is one and only one homography with given cross-axis and a given pair of corresponding points (of which neither lies on the cross-axis). For which given pairs is it an involution?
The lines \(a\), \(b\), \(c\), \(d\) form a plane quadrilateral, and the diagonals \((ab, cd)\), \((ac, bd)\) are denoted by \(e\), \(f\), \(g\) respectively. A transversal meets the sides and diagonals of the quadrilateral in \(A\), \(B\), \(C\), \(D\), \(E\), \(F\), \(G\). Prove that \(AB\), \(CD\), \(FG\), belong to an involution \(I_1\), that \(AC\), \(BD\), \(GE\) belong to an involution \(I_2\), that \(AD\), \(BC\), \(EF\) belong to an involution \(I_3\), and that the mates of \(E\) in \(I_1\), of \(F\) in \(I_2\) and of \(G\) in \(I_3\) all coincide.
Prove that the arithmetic mean of \(n\) positive numbers is not less than their geometric mean. Prove that the area of a triangle which has a given perimeter \(l\) cannot exceed \(l^2/12\sqrt{3}\), and that this value of the area is attained if and only if the triangle is equilateral.
Prove that the only positive integers \(x\) and \(y\) satisfying the conditions \(x < y\) and \(x^y = y^x\) are \(x = 2\), \(y = 4\).
The positive numbers \(p\) and \(q\) are such that \(\frac{1}{p} + \frac{1}{q} = 1\). Prove that $$ab = \frac{a^p}{p} + \frac{b^q}{q}$$ and \(a\) and \(b\) are positive numbers. Prove that $$ab = \frac{a^p}{p} + \frac{b^q}{q}.$$ Prove also that if \(a^p \neq b^q\), then $$ab < \frac{a^p}{p} + \frac{b^q}{q}.$$ (Notice that the relation \(a^p = b^q\) can also be written in the form \(b = a^{p-1}\), and in the form \(a = b^{q-1}\).) If \(a_1, a_2, \ldots, a_n\) and \(b_1, b_2, \ldots, b_n\) are two sets of positive numbers such that \(\sum a_i^p = \sum b_i^q = 1\), where the symbol \(\sum\) implies summation from \(r = 1\) to \(r = n\), prove that \(\sum a_i b_i \leq 1\). Hence prove that if \(a_1, a_2, \ldots, a_n\) and \(b_1, b_2, \ldots, b_n\) are any two sets of positive numbers, then $$\sum a_i b_i \leq (\sum a_i^p)^{1/p} (\sum b_i^q)^{1/q}.$$
A uniform circular loop of weight \(W\) rests on a rough horizontal table, the coefficient of friction between the loop and the table being \(\mu\). The pressure of the hoop on the table is uniform over the area of contact. A gradually increasing horizontal force \(P\) is applied in a tangential direction at a point \(A\) on the rim of the loop. Show that the hoop begins to turn about a point on the rim diametrically opposite to \(A\), and find the value of \(P\) at which slipping commences.
A circular flywheel of radius \(a\) and moment of inertia \(I\) is rotating about a fixed axis with angular velocity \(\Omega\). A frictional couple of constant magnitude \(C\) is suddenly applied to the flywheel, and at the same instant a point on the rim of the flywheel is attached to one end of a smooth elastic string of length \(L\) and modulus \(E\) whose other end is fixed. Initially the string is just taut and lies tangential to the circumference and in the plane of the wheel; in the subsequent motion the string winds around its circumference. Show that the wheel never changes its direction of rotation if \(C \geq \mu\Omega L/3\), where \(\mu = (a^2 E/L)^{\frac{1}{2}}\).
A ladder stands on rough horizontal ground and leans against a rough vertical wall, in a vertical plane perpendicular to the wall. The coefficient of friction between the ladder and the ground is 1. A man stands on the ladder. Find the minimum slope the ladder may have if it is not to slip, no matter how much the man weighs nor where he stands. Justify your answer.