Problems

Four elements \(a\), \(b\), \(c\), \(d\) are subject to a `multiplication table'

where the `product' \(xy\) of two elements \(x\) and \(y\) is that element which lies in the row through \(x\) and the column through \(y\). [For example, \(cd = a\), \(b^2 = d\), where \(b^2\) is written for the product \(bb\).] Prove that each of the four elements satisfies the relation \[x^3 = x,\] where \(x^3\) means \(x(x^2)\). Verify that the associative law \[x(yz) = (xy)z\] does not always hold, giving two instances of the failure. A `product' of the elements \(a\), \(b\), \(c\), \(d\) (in that order) is formed by successively multiplying two elements at a time: for example, in the sequences indicated by the groupings \[[(ab)c]d] \quad \text{or} \quad [(ab)(cd)],\] the order of the letters being unchanged. Find the values of all such `products'.

Three `ordered triplets' \[\mathbf{a} = (1, 1, 1), \quad \mathbf{b} = (1, 2, 3), \quad \mathbf{c} = (1, 3, 6)\] are given, each consisting of three numbers in an assigned order. [Thus the triplets \((3, 5, 7)\), \((3, 7, 5)\) are different.] By a `combination' \[\lambda\mathbf{a} + \mu\mathbf{b} + \nu\mathbf{c}\] is meant the triplet \[(\lambda + \mu + \nu, \lambda + 2\mu + 3\nu, \lambda + 3\mu + 6\nu).\] Prove that values \(\lambda\), \(\mu\), \(\nu\) can be found so that the combination is the given triplet \[\mathbf{x} = (p, q, r)\] and find \(\lambda\), \(\mu\), \(\nu\) in terms of \(p\), \(q\), \(r\). Express \(\mathbf{a}\) as a combination of \(\mathbf{b}\), \(\mathbf{c}\), \(\mathbf{x}\) in the form \(\alpha\mathbf{b} + \beta\mathbf{c} + \gamma\mathbf{x}\), or, in detail, \[(\alpha + \beta + \gamma p, 2\alpha + 3\beta + \gamma q, 3\alpha + 6\beta + \gamma r),\] stating any condition that may be necessary for this form of expression to be possible.

Two circles intersect in \(A\) and \(B\). [A convenient figure is obtained by taking the radii to be approximately 5 units and the distance between the centres to be approximately 6 units.] A line through \(A\) cuts the first circle again in \(P\) and the second in \(Q\). Prove that the triangle \(PBQ\) is constant in shape for all positions of the line through \(A\). The line through \(B\) parallel to \(PAQ\) cuts the first circle again in \(R\) and the second in \(S\). Prove that the points \(P\), \(Q\), \(S\), \(R\) are at the vertices of a parallelogram, two of whose sides are of length \(AB\). Give a construction for the chord \(PAQ\) if the parallelogram is to be a rhombus. [It may be assumed that the dimensions of the figure are such that this construction is possible.]

The altitudes \(AP\), \(BQ\), \(CR\) of an acute-angled triangle \(ABC\) meet in the orthocentre \(H\), where \(P\), for example, is the foot of the perpendicular from \(A\) to \(BC\). The points \(U\), \(V\), \(W\) are the fourth vertices of the parallelograms \(BHCU\), \(CHAV\), \(AHBW\) respectively. Points \(L\), \(M\), \(N\) outside the triangle \(ABC\) are selected such that \begin{align} AM &= AN = AH, \\ BN &= BL = BH, \\ CL &= CM = CH. \end{align} Prove that the nine points \(A\), \(B\), \(C\), \(U\), \(V\), \(W\), \(L\), \(M\), \(N\) lie on a circle.

A tetrahedron \(ABCD\) is given; \(L\), \(M\), \(N\) are the middle points of \(BC\), \(CA\), \(AB\) respectively and \(U\), \(V\), \(W\) are the middle points of \(AD\), \(BD\), \(CD\). Prove that the lines \(LU\), \(MV\), \(NW\) have a common middle point \(O\). Prove that, if \(LU\), \(MV\), \(NW\) are mutually perpendicular, then the four faces of the tetrahedron are congruent acute-angled triangles.

The points \(P\) and \(Q\) lie on different branches of a hyperbola whose foci are \(A\) and \(B\). Prove that \(A\) and \(B\) lie on an ellipse whose foci are \(P\) and \(Q\). Points \(X\) and \(Y\) are chosen on the same branch of the hyperbola in such a way that the quadrilateral \(AXBY\) is convex (all of its angles being less than two right angles). Prove that a circle can be drawn to touch the sides \(AX\), \(XB\), \(BY\), \(YA\).

Given the point \(P(ap^2, 2ap)\) on the parabola \(y^2 = 4ax\), prove that there are two circles which touch the parabola at \(P\) and also touch the \(x\)-axis \((y = 0)\). Verify that the middle point of the line joining the centres of these circles is at the same distance as \(P\) from the \(y\)-axis, but on the opposite side of it.

A point \(C\) is taken on the tangent to the rectangular hyperbola \(xy = k^2\) at its vertex \(A(k, k)\). The circle of centre \(C\) and radius \(CA\) cuts the hyperbola in four real points. Prove that there are just two parabolas through these four points and that one of them passes through the centre of the hyperbola while the other passes through the centre of the circle.

A conic \(S\) and two points \(U\), \(V\) not on it are given. A correspondence between two points \(P\), \(Q\) on the conic is set up as follows: \(UP\) meets the conic again in \(L\), \(VL\) meets the conic again in \(Q\). Determine the conditions under which the line \(PQ\) passes, for all positions of \(P\), through a fixed point \(W\) (to be identified).

A point \(P\) lies in the plane of a given triangle \(XYZ\). The lines \(XP\), \(YP\), \(ZP\) meet \(YZ\), \(ZX\), \(XY\) respectively in \(U\), \(V\), \(W\) and meet \(VW\), \(WU\), \(UV\) respectively in \(L\), \(M\), \(N\). Prove that \(VW\) meets \(MN\) on \(YZ\). The line \(VW\) is constrained to pass through a fixed point \(K\). Find the locus of (i) \(P\), (ii) \(L\). Discuss the particular cases that arise when \(K\) is on (a) \(YZ\), (b) \(XY\).