IMO Practice Test — Linear Inequalities
6 Questions • 20 min • Olympiad level
20:00
Question 1 of 6
hard
Solve the inequality: x/(x − 5) > 1/2.
(−∞, −5) ∪ (5, ∞)
(−5, 5)
(−∞, 5) ∪ (5, ∞)
(5, ∞)
Explanation: x/(x−5) − 1/2 > 0 → (2x − (x−5))/(2(x−5)) > 0 → (x+5)/(x−5) > 0, which holds for x < −5 or x > 5, i.e. (−∞, −5) ∪ (5, ∞).
Question 2 of 6
hard
Find all pairs of consecutive odd positive integers both of which are smaller than 10, such that their sum is more than 11.
(5, 7) and (7, 9)
(3, 5) and (5, 7)
(7, 9) only
(5, 7) only
Explanation: Let integers be x and x+2. x+2 < 10 → x < 8. Sum: 2x+2 > 11 → x > 4.5. Odd integers between 4.5 and 8 are 5, 7. Pairs are (5, 7) and (7, 9).
Question 3 of 6
hard
Find all pairs of consecutive even positive integers, both of which are larger than 5, such that their sum is strictly less than 23.
(6, 8), (8, 10)
(8, 10), (10, 12)
(6, 8), (8, 10), (10, 12)
(4, 6), (6, 8)
Explanation: x > 5 and x + x + 2 < 23 → 2x < 21 → x < 10.5. Valid even x: 6, 8, 10. The pairs are (6, 8), (8, 10), and (10, 12).
Question 4 of 6
hard
Find the solution set of (x + 1)/(x + 2) ≥ 1.
(−∞, −2)
(−2, ∞)
(−∞, −2]
No solution
Explanation: (x + 1)/(x + 2) − 1 ≥ 0 → (x + 1 − x − 2)/(x + 2) ≥ 0 → −1/(x + 2) ≥ 0. For this to hold, x + 2 < 0, so x < −2. (−∞, −2).
Question 5 of 6
hard
If A = {x : x² − 5x + 6 < 0} and B = {x : x² − 3x + 2 < 0}, then A ∩ B is:
Empty set
(1, 3)
(2, 3)
(1, 2)
Explanation: For A: (x−2)(x−3) < 0 → 2 < x < 3. For B: (x−1)(x−2) < 0 → 1 < x < 2. Their intersection is empty.
Question 6 of 6
hard
The longest side of a triangle is 3 times the shortest side and the third side is 2 cm shorter than the longest side. If the perimeter is at least 61 cm, find the minimum length of the shortest side.
7 cm
8 cm
9 cm
10 cm
Explanation: Let shortest side be x. Longest = 3x. Third = 3x − 2. x + 3x + 3x − 2 ≥ 61 → 7x − 2 ≥ 61 → 7x ≥ 63 → x ≥ 9.