IMO Practice Test — Complex Numbers and Quadratic Equations
6 Questions • 20 min • Olympiad level
20:00
Question 1 of 6
hard
If z = 1 + √3i, then its argument is:
60°
30°
45°
90°
Explanation: tan θ = √3.
Question 2 of 6
hard
If x = a+b, y = aω+bω², and z = aω²+bω, find the product xyz.
a³ - b³
a³ + b³
3ab
a² + b²
Explanation: xyz = (a+b)(a²ω³ + abω² + abω⁴ + b²ω³). Substituting ω³=1 and ω⁴=ω gives (a+b)(a² - ab + b²), which is the factorization of a³ + b³.
Question 3 of 6
hard
What is the minimum value of |z - 1| + |z - 5| for any complex number z?
2
4
6
0
Explanation: By the triangle inequality, |z - 1| + |5 - z| ≥ |(z - 1) + (5 - z)| = |4| = 4. The minimum distance occurs on the line segment between 1 and 5.
Question 4 of 6
hard
Evaluate (1+i)⁶ + (1-i)⁶.
16
0
-16
8i
Explanation: (1+i)⁶ = ((1+i)²)³ = (2i)³ = -8i. (1-i)⁶ = ((1-i)²)³ = (-2i)³ = 8i. Sum is -8i + 8i = 0.
Question 5 of 6
hard
(3 + 4i)/(3 − 4i) has modulus:
1
5
25
0
Explanation: Numerator and denominator have equal modulus.
Question 6 of 6
hard
If x + iy = (2+3i)/(1-i), find the values of real numbers x and y.
x = -1/2, y = 5/2
x = 1/2, y = -5/2
x = 5/2, y = -1/2
x = -5/2, y = 1/2
Explanation: (2+3i)/(1-i) = (2+3i)(1+i)/2 = (2 + 2i + 3i - 3)/2 = (-1 + 5i)/2. Hence, x = -1/2 and y = 5/2.