A party of seven people arrives at a tavern which has six vacant rooms. In how many ways can they be accommodated if each room can take only two persons?
Obtain the condition for the equation \(ax^2 + 2bx + c = 0\) to have real roots, where \(a\), \(b\) and \(c\) are real numbers. The real numbers \(p\), \(q\) and \(r\) are such that none has unit modulus, and \(p^2 + q^2 + r^2 + 2pqr = 1.\) Prove that \(p\), \(q\), and \(r\) either all lie between \(+1\) and \(-1\), or all lie outside this range.
Prove that \(\sum_{r=1}^n r(r+1)(r+2)\ldots(r+s-1) = n(n+1)\ldots(n+s)/(s+1).\) Evaluate \(\sum_{r=1}^n r^4\).
A zero of the polynomial \(f(x) = a_0 x^n + a_1 x^{n-1} + \ldots + a_n\) is \(p/q\), where \(p/q\) is a fraction in its lowest terms, and each \(a_r\) is an integer. Show that
In a group with identity \(e\), an element \(g\) is said to have order \(n\) if \(n\) is the least positive integer such that \(g^n = e\).
Solution:
Let \(N\) denote the non-negative integers. A subset \(S \subseteq N\) is called convex if \(x \in S\), \(y \in N\), \(x < y < z\), implies that \(y \in S\). Let \(*\) be a composition on \(N\) defined by \(x * y = \max(x, y)\). Prove that if an equivalence relation \(R\) on \(N\) has convex equivalence classes then \((x * y) R (x' * y')\) whenever \(x R x'\) and \(y R y'\). Is the converse true?
\(ABC\) is a non-isosceles triangle, with \(M\) the mid-point of \(BC\). A line passes through \(A\), \(B\) in \(P\), \(Q\) respectively, and \(AP = AQ\). Prove that \(BP = CQ\). If \(ABC\) is a non-isosceles triangle with points \(B\), \(C\), \(P\), \(Q\) cannot be concyclic, and that the triangles \(BMP\), \(CMQ\) cannot have equal areas.
A straight line meets the sides \(BC\), \(CA\), \(AB\) of a triangle \(ABC\) in \(L\), \(M\), \(N\) respectively. Prove that \(\frac{BL \cdot CM \cdot AN}{LG \cdot MA \cdot NB} = -1,\) due regard being paid to sign. The mid-points of the sides \(PQ\), \(RS\) of a parallelogram \(PQRS\) are \(X\), \(Y\) respectively. \(H\) is a point on the diagonal \(PR\) and \(HX\), \(HY\) meet \(QR\), \(PS\) respectively in \(U\), \(V\). Prove that \(UV\) is parallel to \(PQ\). If \(UV\) cuts \(PR\) in \(W\) prove that \(\frac{2}{HW} = \frac{1}{HP} + \frac{1}{HR},\) due regard being paid to sign.
A fixed point \(K\) lies inside a triangle \(ABC\) and a circle through \(A\) and \(K\) meets \(AB\), \(AC\) again in \(R\), \(Q\) respectively. Prove that the circles \(BRK\) and \(CQK\) meet again in a point \(P\) of \(BC\). Show further that all triangles \(PQR\) obtained in this way are similar, and that the one of smallest area has its vertices at the feet of the perpendiculars from \(K\) to the sides of \(ABC\).