Activities
P.o.t.W. #4 Solution
Other points in the figure have been labelled as shown in Figure 1. Since a line tangent to a circle is perpendicular to the radius drawn to the point of tangency, then \(\Delta {\textrm{ABC}} and \(\Delta...
Activities
P.o.t.W. #13 Solution
If the three angles of the triangle form an arithmetic sequence, then the middle angle must be \(60^\circ . This can be explained as follows. Let the three angles be \({\theta _1},{\theta _2} and \({\theta...
Activities
P.o.t.W. #18 Solution
Let \(a be the total number of autographs that Alicia has in her collection; and let the fraction of autographs in box #2 be represented by \(\dfrac{x}{7}. Clearly, \(a and \(x must both be positive...
Activities
P.o.t.W. #31 Solution
2a{x^2} - 4ax + a + 1 Let {r_1} and {r_2} be the two roots. Given {r_2} = 3{r_1} Sum and product of the roots of a polynomial of the form {a_n} + {a_{n - 1}} + {a_{n - 2}} + \; \cdots \; + {a_1} + {a_0}...
Activities
P.o.t.W. #28 Solution
Drop perpendicular line segments from P and Q to side SR. The distance from the foot of each perpendicular to the nearest vertex of the trapezium (either S or R) is \(\dfrac{{b - a}}{2}. Let \(d be the...
Activities
P.o.t.W. #11 Solution
Let \({u_1},\;{u_1}r and \({u_1}{r^2} be the three consecutive terms of the geometric sequence. If \({u_1} and \({u_1}r are the first and fourth terms, respectively, of an arithmetic sequence then \({u_1}r...
Activities
P.o.t.W. #16 Solution
Let A represent a win for Anika and B represent a win for Bella. , There are 4 ways that Anika wins 3 matches out of 4. 1. AAAB 2. AABA 3. ABAA 4. BAAA Can also be computed...
Activities
P.o.t.W. #7 Solution
site user Dan Buckley contributed an efficient solution and simpler than Solution #1 - see Dan's Solution #2read Dan's comment at the bottom of this page Solution #1 (below) Let B and T be the bottom...
Activities
P.o.t.W. #22 Solution
\(g\left( x \right) = \sqrt[3]{x}\left( {x - 1} \right) = {x^{\frac{1}{3}}}\left( {x - 1} \right) = {x^{\frac{4}{3}}} - {x^{\frac{1}{3}}} \(g'\left( x \right) = \dfrac{4}{3}{x^{\frac{1}{3}}} - \dfrac{1}{3}{x^{...
Activities
P.o.t.W. #9 Solution
gradient of line, \(m = \frac{{a{c^2} - a{b^2}}}{{c - b}} = \frac{{a\left( {c + b} \right)\left( {c - b} \right)}}{{c - b}} = ac + ab equation of line: \(y - a{b^2} = \left( {ac + ab} \right)\left( {x...