An `arithmetic' has five numbers 0, 2, 4, 6, 8. They are subjected to `digital addition' and `digital multiplication', which are ordinary addition and multiplication save that only the units digit is retained for the result. (Thus \(6 + 8 = 4\), \(2 \times 8 = 6\).) Establish the existence among them of a `unit' \(e\), that is, a number such that \(a \times e = a\) for all five numbers. Prove also that, if \(b \neq 0\), then \(b^4 = e\), for that value of \(e\). Suggest a formal definition for a process of `digital subtraction' and solve the equation $$x^2 - 4x + 8 = 0.$$
Determine the limitations, if any, on the value of \(p\) if the expression $$x^2(y^2 + 2y + 2) + 2x(y^2 + 2py + 2) + (y^2 + 2y + 2)$$ is greater than or equal to zero for all pairs of real values of \(x\) and \(y\).
A triangle \(ABC\) suffers two displacements in its plane: (i) a reflexion about a point \(O\) to a position \(UVW\), so that \(O\) is the middle-point of each of \(AU\), \(BV\), \(CW\); (ii) a parallel displacement to a position \(XYZ\), so that \(AX\), \(BY\), \(CZ\) are equal and parallel. The centres (intersections of diagonals) of the parallelograms \(BYZC\), \(CZXA\), \(AXYB\) are \(L\), \(M\), \(N\) respectively. Establish the existence of a point \(K\) such that \(L\), \(M\), \(N\) are the middle points of \(KU\), \(KV\), \(KW\) respectively.
Points \(P\), \(Q\), \(R\) are taken on the sides \(BC\), \(CA\), \(AB\) respectively of a triangle \(ABC\). Prove that the circles \(AQR\), \(BRP\), \(CPQ\) have a common point \(O\). The tangents at \(A\), \(B\), \(C\) to the circles \(AQR\), \(BRP\), \(CPQ\) form a triangle \(UVW\) with \(A\) on \(VW\), \(B\) on \(WU\), \(C\) on \(UV\). Identify the point of intersection of the circles \(BCU\), \(CAV\), \(ABW\).
The altitudes \(AP\), \(BQ\), \(CR\) of an acute-angled triangle \(ABC\) meet in the orthocentre \(H\) and \(U\) is an arbitrary point in the plane. The inverse of \(U\) with respect to the circle of centre \(A\) and radius \(\sqrt{(AH \cdot AP)}\) is \(V\); the inverse of \(V\) with respect to the circle of centre \(B\) and radius \(\sqrt{(BH \cdot BQ)}\) is \(W\); the inverse of \(W\) with respect to the circle of centre \(C\) and radius \(\sqrt{(CH \cdot CR)}\) is \(X\). Prove that \(UX\) passes through \(H\) and that \(HU \cdot HX = HA \cdot HP\).
In a tetrahedron \(ABCD\), the points \(P\), \(Q\), \(R\) are the feet of the perpendiculars to \(BC\), \(CA\), \(AB\) respectively. Prove that the lines in the plane \(ABC\) through \(P\), \(Q\), \(R\) perpendicular to \(BC\), \(CA\), \(AB\) respectively have a common point \(U\). In a particular case, \(U\) is the incentre of the triangle \(ABC\). Determine whether can be drawn to pass through the incentre and to touch each of the lines \(DR\), \(DQ\), \(DP\) and is the orthocentre of the triangle \(ABC\). How are the edges of the tetrahedron related if, alternatively, \(U\) is the orthocentre of \(ABC\)?
A point \(P\) is taken on the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) whose foci are \(S(ae, 0)\), \(S'(-ae, 0)\). The lines \(SP\), \(S'P\) meet the \(y\)-axis in \(U\), \(V\). If \(O\) is the centre of the ellipse, there is a position of \(P\) for which the line \(OP\) touches the \(y\)-axis and only if, \(e > 1/\sqrt{2}\).
The hyperbola \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) has focus \(S(ae, 0)\) and centre \(O(0, 0)\). A parabola with focus \(S\) and vertex \(O\) meets the hyperbola at \(P\) and \(Q\). Prove that, if the tangent at \(P\) to the parabola, then \(e = \sqrt{3}\).
Four points \(X\), \(Y\), \(Z\), \(U\) lie on a given conic; \(UX\), \(UY\), \(UZ\) meet \(YZ\), \(ZX\), \(XY\) respectively in \(P\), \(Q\), \(R\). Prove that a conic can be drawn to touch \(YZ\) at \(P\), \(ZX\) at \(Q\), and that, if \(X\), \(Y\), \(Z\) are kept fixed, the polar of \(U\) with respect to this conic passes through a point which is fixed for all positions of \(U\) on the given conic.
Prove the theorem of Pappus that, if \(ABC\) and \(PQR\) are two straight lines, then the points of intersection \(L \equiv (BP, CQ)\), \(M \equiv (CP, AR)\), \(N \equiv (AQ, BR)\) are collinear. Two circles meet in \(C\) and \(R\). A line through \(P\) meets the first circle in \(Q\) and the second in \(P\). The lines \(BR\), \(CQ\) meet in \(L\); the lines \(CP\), \(AR\) meet in \(M\). Prove that \(AQ\), \(BP\), \(LM\) are parallel lines.