IMO Practice Test — Simultaneous Linear Equations
6 Questions • 15 min • Olympiad level
15:00
Question 1 of 6
hard
Solve for x and y in the fractional system: (4/x) + 3y = 14 and (3/x) - 4y = 23.
x=2, y=-1
x=1/2, y=-2
x=-1/2, y=2
x=1/5, y=-2
Explanation: Let 1/x = u. Equations become 4u+3y=14 and 3u-4y=23. Solving gives u=2 (so x=1/2) and y=-2.
Question 2 of 6
hard
A two-digit number is 4 times the sum of its digits. If 18 is added to the number, the digits reverse their order positions. Find the original number.
36
24
12
48
Explanation: Number is 10x+y. 10x+y = 4(x+y) -> 2x=y. Reversal: 10x+y+18 = 10y+x -> 9y-9x=18 -> y-x=2. Solving gives 24.
Question 3 of 6
hard
Find the value of the constant parameter k for which the system of equations (kx + 3y = 1) and (12x + ky = 2) has absolutely no solution.
k = 6
k = -6
k = ±6
k = 0
Explanation: For no solution, ratio a1/a2 = b1/b2 != c1/c2. Thus k/12 = 3/k -> k² = 36 -> k = ±6.
Question 4 of 6
hard
A boat travels 30 km upstream against the river current in 3 hours. It takes exactly 1 hour to complete the same distance downstream with the current. Find the speed of the boat in still water.
15 km/h
20 km/h
10 km/h
25 km/h
Explanation: Upstream speed = x-y = 30/3 = 10. Downstream speed = x+y = 30/1 = 30. Adding equations gives 2x = 40 -> x = 20.
Question 5 of 6
hard
If the pair of linear equations (2x + 3y = 7) and ((a + b)x + (2a - b)y = 21) has an infinite number of matching solutions, determine the values of a and b.
a=5, b=1
a=1, b=5
a=-5, b=-1
a=3, b=2
Explanation: For infinite solutions: 2/(a+b) = 3/(2a-b) = 7/21 = 1/3. This yields a+b=6 and 2a-b=9. Solving gives a=5, b=1.
Question 6 of 6
hard
In an examination containing 100 questions, a student scores 4 marks for every correct answer and loses 1 mark for every wrong answer. If the student answers all questions and scores a total of 250 marks, how many questions did they answer correctly?
60
70
75
80
Explanation: c+w=100 and 4c-w=250. Adding the two equations eliminates w, giving 5c = 350 -> c = 70.