\(\Sigma\) is a conic, and \(ABC, A'B'C'\) are triangles such that the lines \(B'C', C'A', A'B'\) are the polars with respect to \(\Sigma\) of \(A, B, C\) respectively. Show that \(AA', BB', CC'\) are concurrent.
If \(A, B\) are points in the plane, the part of the line \(AB\) between \(A\) and \(B\) is the segment \(AB\). Points \(P_1, P_2, \ldots, P_6\) in the plane are such that no three are collinear and no three segments \(P_iP_j, P_kP_l, P_mP_n\) are concurrent. A crossing is a point common to two distinct segments \(P_iP_j, P_kP_l\). Prove that \(P_1, P_2, \ldots, P_6\) always have three crossings, and find six points with exactly three crossings.
Prove that, for any four points \(A, B, C, D\) in a plane, \[\begin{vmatrix} 0 & 2AB^2 & AB^2+AC^2-BC^2 & AB^2+AD^2-BD^2 \\ 2AC^2 & 0 & 2BC^2 & AC^2+AD^2-CD^2 \\ AC^2+AB^2-CB^2 & 2AB^2 & 0 & 2CD^2 \\ AD^2+AB^2-DB^2 & AD^2+AC^2-DC^2 & 2CD^2 & 0 \end{vmatrix} = 0\]
Let \(l_1, l_2, l_3, l_4\) be lines in the plane and let \(C_i\) be the circumcircle of the triangle obtained by omitting \(l_i\). Prove that
Let \[f(x) = \sum_{n=1}^{\infty} \frac{x}{n(n+x)}\] for real positive \(x\). Prove that \[2f(2x) - f(x) - f(x+\frac{1}{2}) = 2\log 2 - \frac{1}{(x+\frac{1}{2})}.\]
Find the most general solution of the 'differential equation' \[f'(x) = \lambda f(1-x),\] where \(\lambda\) is a real constant.
A fair coin is tossed successively until either two heads occur in a row or three tails occur in a row. What is the probability that the sequence ends with two heads?
Solution: Suppose \(A\) bets on \(H\) every time (until \(HH\) appears) and \(B\) bets on \(T\) every time (until \(TTT\) appears). When either of them get their desired string then we stop betting. Since each team's score is a martingale we must have \begin{align*} \mathbb{E}(A\text{ winnings}) &= 0 = (4+2)p - \mathbb{E}(\tau)\\ &= 6p - \mathbb{E}(\tau) \\ \mathbb{E}(B\text{ winnings}) &= 0 = (8+4+2)(1-p) - \mathbb{E}(\tau)\\ &= 14(1-p) - \mathbb{E}(\tau) \\ 6p &= 14(1-p) \\ p &= \frac{14}{20} = 0.7 \end{align*}
My house lies between two bus stops, one of which lies 90 yards to the right and one 270 yards to the left. If I catch a bus at the left-hand stop it costs me 6 pence, and if I catch it at the right-hand stop it will cost me 7 pence and if I miss the bus I must take a taxi which will cost 20 pence. The bus comes from the the right and comes to a stop at a point 90 yards further away from my house than the bus stop. I reckon to walk 2 yards a second until I see the bus and then to run at 6 yards a second. The bus travels at 15 yards a second until it reaches the first bus stop where it waits for 3 seconds and then goes round a corner out of sight. When I leave the house there is no bus in sight, and I reckon that it does not matter which stop I go to. How frequent are the buses?
A farmer wishes to provide his cattle with three nutrients \(A, B\) and \(C\), for which the minimum requirements of 21, 9 and 12 units respectively. Two animal foods \(F_1\) and \(F_2\) are available; their content for unit cost are given in the following table.
A heavy horizontal carriageway of uniform weight \(w\) per unit length is suspended from a heavy flexible wire attached to two pillars a distance \(2d\) apart. The weight of the wire per unit length at any point is chosen to be \(k\) times the tension it has to sustain. Assuming that the carriageway acts as a continuous vertical load on the wire, and that \(kd < \pi\), show that the vertical load on each pillar is given by \(T_0\beta\tan\beta kd\) where \(T_0\) is the minimum tension in the wire and \(\beta^2 = (w+T_0k)/T_0k\).