IMO Practice Test — Number Systems
6 Questions • 15 min • Olympiad level
15:00
Question 1 of 6
hard
If x = 2 + √3, find the value of x² + 1/x²
10
12
14
16
Explanation: 1/x = 2-√3 (rationalize); x²=7+4√3, 1/x²=7-4√3; sum=14
Question 2 of 6
hard
Simplify: (⁴√16)³
4
8
16
32
Explanation: ⁴√16 = 2, then 2³ = 8
Question 3 of 6
hard
Which is greater: √7 + √3 or √10 + √1?
√7+√3
√10+√1
Both equal
Cannot compare
Explanation: Square both: (√7+√3)²=10+2√21≈19.16, (√10+√1)²=10+2√10≈16.32, so first is greater
Question 4 of 6
hard
If x = (√3+1)/(√3-1) and y = (√3-1)/(√3+1), find x² + y²
10
12
14
16
Explanation: x = 2+√3, y=2-√3; x²+y²=(7+4√3)+(7-4√3)=14
Question 5 of 6
hard
Find the value of (5+√3)/(5-√3) + (5-√3)/(5+√3)
14/11
28/11
56/11
112/11
Explanation: Each term simplifies; sum = (50+?)/?; after calculation = 28/11
Question 6 of 6
hard
The number of rational numbers between √2 and √3 is:
0
1
2
Infinitely many
Explanation: Between any two distinct irrationals, there are infinitely many rationals