IMO Practice Test — Expansions
6 Questions • 15 min • Olympiad level
15:00
Question 1 of 6
hard
If x - (1/x) = 3, calculate the exact numerical value of the cubic expression x³ - (1/x³).
27
36
18
30
Explanation: Cube both sides: (x - 1/x)³ = 27 -> x³ - 1/x³ - 3(x - 1/x) = 27 -> x³ - 1/x³ - 3(3) = 27 -> 27 + 9 = 36.
Question 2 of 6
hard
Given that x² + y² + z² = 14 and xy + yz + zx = 11, find the value of the expression (x + y + z).
6
±6
36
±5
Explanation: (x+y+z)² = 14 + 2(11) = 36. Taking the square root gives both positive and negative 6.
Question 3 of 6
hard
Simplify the complex algebraic product fraction: ((a + b)² - c²) / (a + b + c).
a - b - c
a + b - c
a + b + c
1
Explanation: Treat (a+b) as a single term and apply difference of squares to the numerator, then cancel the common term.
Question 4 of 6
hard
If a + b + c = 6 and a³ + b³ + c³ - 3abc = 18, calculate the value of the paired variable sum ab + bc + ca.
11
13
9
15
Explanation: Identity: 18 = 6 × (a²+b²+c² - (ab+bc+ca)). Thus 3 = (6² - 3(ab+bc+ca)). Solving gives 13.
Question 5 of 6
hard
Find the value of (x - y)³ + (y - z)³ + (z - x)³ divided by the product term 3(x - y)(y - z)(z - x).
0
1
3
-1
Explanation: Let a=x-y, b=y-z, c=z-x. Notice a+b+c=0. Therefore, the numerator simplifies to 3abc, resulting in a ratio of 1.
Question 6 of 6
hard
If x⁴ + (1/x⁴) = 47, find the positive value of the simple linear expression x + (1/x).
7
9
3
5
Explanation: Add 2 to both sides to find x² + 1/x² = 7. Add 2 again to find (x + 1/x)² = 9, so the square root is 3.