Find the range [a,b] of f(x)=1/(2+sin3x+cos3x), then find α/β where α, β are the AM and GM of a and b, using the fact that sin3x+cos3x ranges over [−√2,√2].
Given |z+2|=1 and Im((z+1)/(z+2))=1/5, find |Re(z+2)| by extracting y directly from the imaginary-part condition and using the circle equation for x.
Given the real part of (z−2i)/(z+2i) is zero, show z traces a circle of radius 2 about the origin, then find the maximum distance from that circle to (6,8).
Find the number of complex numbers z satisfying |z−i|=|z+i|=|z−1| simultaneously, by recognising z as the circumcenter of the triangle formed by i, −i, and 1.
Sets A, B have m, n elements, with A’s subset count 56 more than B’s. Find m and n via prime factorisation of 2^m−2^n=56, then the distance from P(m,n) to Q(−2,−3).
An ellipse centred at (1,0) has latus rectum 1/2 and its minor axis subtends 60° at the foci. Find the square of the sum of its major and minor axis lengths.
If z1, z2 are distinct complex numbers with |z1−2z2|/|1/2−z1z̄2|=2, determine which circle condition z1 and z2 must satisfy, by expanding both sides.
If z is a complex number with |z|≤1, find the minimum value of |z+1/2(3+4i)| by interpreting it as the distance from a fixed point to the unit disk.
In triangle ABC, y=x bisects angle B, AC is 2x−y=2, and 2AB=BC. Given A=(4,6) and B=(α,β), find α+2β using the angle bisector’s side-ratio property.
Given 3×3 matrices A and P, find the sum of the prime factors of |P⁻¹AP−2I| using the property that similar matrices share the same determinant after shifting by 2I.