CMO 1973 Problem 1

(i) Solve the simultaneous inequalities, $x<\frac{1}{4x}$ and $x<0$ ; i.e. find a single inequality equivalent to the two simultaneous inequalities.

(ii) What is the greatest integer that satisfies both inequalities $4x+13 < 0$ and $x^{2}+3x > 16$ .

(iii) Give a rational number between $11/24$ and $6/13$ .

(iv) Express 100000 as a product of two integers neither of which is an integral multiple of 10.

(v) Without the use of logarithm tables evaluate $$\frac{1}{\log_{2}36}+\frac{1}{\log_{3}36}.$$