Problems

Establish necessary and sufficient conditions that \(ax^2+2bx+c\) shall be positive for all real values of \(x\). \newline If the \(a\)'s, \(b\)'s and \(c\)'s are real numbers such that \begin{gather*} a_1 > 0, \quad a_2 > 0, \quad a_3 > 0, \\ a_1c_1-b_1^2 > 0, \quad a_2c_2-b_2^2 > 0, \quad a_3c_3-b_3^2 > 0, \end{gather*} prove that \[ (a_1+a_2+a_3)(c_1+c_2+c_3) - (b_1+b_2+b_3)^2 > 0. \]

Prove that, if \(a, b, c\) are the sides of a triangle of area \(\Delta\), \[ \begin{vmatrix} (b-c)^2 & b^2 & c^2 & 1 \\ a^2 & (c-a)^2 & c^2 & 1 \\ a^2 & b^2 & (a-b)^2 & 1 \\ 1 & 1 & 1 & 0 \end{vmatrix} = -16\Delta^2. \]

Sum to \(N\) terms, and where possible to infinity, the series whose \(n\)th terms are \[ \text{(i)} \ (n+2)n, \quad \text{(ii)} \ (n+2)x^n, \quad \text{(iii)} \ (n+2)\cos n\theta. \]

Lines drawn from the vertices \(A, B, C\) of a triangle through a variable point \(O\) within the triangle meet the opposite sides at \(D, E, F\) respectively.

1. [(i)] Prove that \(\frac{AO}{AD} + \frac{BO}{BE} + \frac{CO}{CF}\) is constant.
2. [(ii)] If \(p, q, r\) are fixed positive numbers with \(p>q>r\), between what limits does \[ p \frac{AO}{AD} + q \frac{BO}{BE} + r \frac{CO}{CF} \] lie, as \(O\) varies within the triangle?

A variable circle through two fixed points \(A\) and \(B\) cuts a fixed circle at \(P\) and \(Q\). Prove that the ratio of the areas of the triangles \(APQ, BPQ\) is constant.

A fixed point \(A\) and a variable point \(P\) are taken on a given sphere. \(AP\) is produced to \(Q\) so that \(PQ\) is of constant length. Prove that the plane through \(Q\) perpendicular to \(PQ\) touches a fixed sphere.

Prove that the locus \[ x=a_1 t^2 + 2b_1 t, \quad y=a_2 t^2 + 2b_2 t, \] where \(t\) is a parameter, is, in general, a parabola. \newline Find the condition that the line \[ y-y_0 = m(x-x_0) \] may touch the parabola, and prove that the directrix is \[ a_1x + a_2y + b_1^2 + b_2^2 = 0. \]

The lines joining a variable point \(P\) on the ellipse \(x^2/a^2+y^2/b^2=1\) to the fixed points \((ka,0)\) and \((k'a,0)\) cut the ellipse again at \(Q\) and \(R\). Prove that, in general, the line \(QR\) envelops the conic \[ \frac{x^2}{a^2} + \frac{(1-kk')^2}{(1-k^2)(1-k'^2)} \frac{y^2}{b^2} = 1. \] Examine the case in which \(kk'=1\).

Equilateral triangles \(BCP, CAQ, ABR\) are drawn outward on the sides of an acute-angled triangle \(ABC\). Prove that the triangles \(ABC, PQR\) have the same centroid.

\(A, B, C, D\) are four distinct points on a given circle. A variable circle is drawn through \(B, C\) and another variable circle is drawn through \(A, D\). Prove that the radical axis of these variable circles cuts the given circle in pairs of points belonging to an involution. \newline Prove or disprove the following statement: ``The line joining two given corresponding points of the involution just obtained is the radical axis of two circles (one through \(B, C\) and the other through \(A, D\)) which are uniquely determined by those two given points.''