International Mathematics Olympiad (IMO) - Class 7 Solved Paper
Class 7 IMO championship paper focusing on rational numbers, exponents and powers, algebraic expressions, linear equations in one variable, triangle congruence, angles, simple interest, data handling, and higher order logic.
Evaluate the expression: (-1)^101 + (-1)^102 + (-1)^103 + ... + (-1)^200.
Detailed Step-by-Step Verified Solution
- Step 1: Total terms from 101 to 200 = (200 - 101 + 1) = 100 terms.
- Step 2: Notice consecutive pairs: (-1)^101 + (-1)^102 = (-1) + (+1) = 0.
- Step 3: Since there are exactly 100 terms, there are 50 pairs, each summing to 0.
- Step 4: Total sum = 50 × 0 = 0.
If (2/3)^3 × (2/3)^-6 = (2/3)^(2x - 1), find the value of x.
Detailed Step-by-Step Verified Solution
- Step 1: Apply product law on LHS: (2/3)^(3 + (-6)) = (2/3)^(-3).
- Step 2: Since bases are identical and non-zero, equate exponents: -3 = 2x - 1.
- Step 3: 2x = -3 + 1 = -2 => x = -1.
Two supplementary angles are in the ratio 7 : 11. What is the measure of the complementary angle of the smaller angle?
Detailed Step-by-Step Verified Solution
- Step 1: Supplementary angles sum to 180°: 7x + 11x = 180° => 18x = 180° => x = 10°.
- Step 2: Smaller angle = 7 × 10° = 70° (larger angle = 110°).
- Step 3: The complementary angle of the smaller angle is: 90° - 70° = 20°.
Pointing to a gentleman, Deepak said, "His only brother is the father of my daughter's father." How is the gentleman related to Deepak?
Detailed Step-by-Step Verified Solution
- Step 1: Break down Deepak's statement from the end: "my daughter's father" is Deepak himself.
- Step 2: "Father of my daughter's father" means father of Deepak.
- Step 3: "His only brother is Deepak's father" means the gentleman is the brother of Deepak's father.
- Step 4: The brother of one's father is one's paternal uncle.
In a class of 50 students, 30 play Cricket, 25 play Football, and 12 play both games. How many students play neither Cricket nor Football?
Detailed Step-by-Step Verified Solution
- Step 1: Total students playing at least one game = n(Cricket) + n(Football) - n(Both).
- Step 2: n(Cricket ∪ Football) = 30 + 25 - 12 = 55 - 12 = 43 students.
- Step 3: Students playing neither game = Total students - 43 = 50 - 43 = 7 students.
Find the value of x such that: (3x - 2)/4 - (2x + 3)/3 = 2/3 - x.
Detailed Step-by-Step Verified Solution
- Step 1: Multiply every term by 12:
- 3(3x - 2) - 4(2x + 3) = 4(2) - 12x.
- Step 2: Expand the parentheses: 9x - 6 - 8x - 12 = 8 - 12x.
- Step 3: Combine like terms: x - 18 = 8 - 12x.
- Step 4: Transpose variables: x + 12x = 8 + 18 => 13x = 26 => x = 2.
In a triangle, an exterior angle measures 115° and one of the interior opposite angles measures 45°. What is the measure of the other interior opposite angle?
Detailed Step-by-Step Verified Solution
- Step 1: Recall the Exterior Angle Theorem: The exterior angle of a triangle is equal to the sum of its two interior opposite angles.
- Step 2: 115° = 45° + x.
- Step 3: x = 115° - 45° = 70°.
At what rate percent per annum simple interest will a sum of money double itself in 8 years?
Detailed Step-by-Step Verified Solution
- Step 1: Let the principal be P. When it doubles, Amount = 2P.
- Step 2: Simple Interest SI = Amount - Principal = 2P - P = P.
- Step 3: Use SI formula: SI = (P × R × T) / 100.
- P = (P × R × 8) / 100 => 1 = 8R / 100.
- Step 4: R = 100 / 8 = 12.5% per annum.
If triangle ABC is congruent to triangle PQR under the correspondence ABC ↔ PQR, and ∠A = 50°, ∠B = 60°, what is the measure of ∠R?
Detailed Step-by-Step Verified Solution
- Step 1: In triangle ABC, sum of angles is 180°: ∠C = 180° - (50° + 60°) = 180° - 110° = 70°.
- Step 2: Given ΔABC ≅ ΔPQR, corresponding angles are equal: ∠R = ∠C.
- Step 3: Therefore ∠R = 70°.
A wire is in the shape of a square of side 22 cm. If the wire is rebent into the shape of a circle, what is the radius of the circle formed? (Take π = 22/7)
Detailed Step-by-Step Verified Solution
- Step 1: Length of wire = Perimeter of square = 4 × 22 cm = 88 cm.
- Step 2: When rebent into a circle, Circumference = Length of wire: 2 × π × r = 88 cm.
- Step 3: 2 × (22/7) × r = 88 => (44/7) × r = 88.
- Step 4: r = (88 × 7) / 44 = 2 × 7 = 14 cm.
The mean of 5 numbers is 27. If one number is excluded, their mean becomes 25. What is the excluded number?
Detailed Step-by-Step Verified Solution
- Step 1: Total sum of original 5 numbers = Mean × Count = 27 × 5 = 135.
- Step 2: Total sum of remaining 4 numbers = 25 × 4 = 100.
- Step 3: Excluded number = (Sum of 5) - (Sum of 4) = 135 - 100 = 35.
If 2^(x - 1) + 2^(x + 1) = 320, find the value of the integer x.
Detailed Step-by-Step Verified Solution
- Step 1: Rewrite terms using base powers: 2^(x+1) = 2^(x-1) × 2^2 = 4 × 2^(x-1).
- Step 2: Substitute into equation: 2^(x-1) + 4 × 2^(x-1) = 5 × 2^(x-1).
- Step 3: 5 × 2^(x-1) = 320 => 2^(x-1) = 320 ÷ 5 = 64.
- Step 4: Express 64 as a power of 2: 64 = 2^6.
- Step 5: Equate exponents: x - 1 = 6 => x = 7.
If a + b + c = 0, what is the value of: (a²/bc) + (b²/ca) + (c²/ab)?
Detailed Step-by-Step Verified Solution
- Step 1: Combine the three fractions with a common denominator of abc:
- (a²·a + b²·b + c²·c) / (abc) = (a³ + b³ + c³) / (abc).
- Step 2: Standard algebraic theorem: When a + b + c = 0, then a³ + b³ + c³ = 3abc.
- Step 3: Substitute 3abc in the numerator: (3abc) / (abc) = 3.