Mathematics Olympiad Previous Year Papers - 3 | Mathematical Olympiad Class 8 PDF Download

Ans: (a)
The rule followed is: (4 + 0 + 3)3 = 73 = 343;
(5 + 1 + 4)3 = 103 = 1000
Similarly, (3 + 2 + 6)3 = (11)3 = 1331

Ans: (c)
We have,

Similarly,

Ans: (b)

Ans: (c)

Ans: (a)
The correct sitting arrangement is:

So, B sits between D and C.

(d)Ans: (a)

Ans: (d)

Triangles formed are : a, b, d, e, f, g, h, i, j, l, ab, ef, gh, ij, hkl, gcd, akji, bcef,... 

Ans: (b)

• First, we need to replace the symbols according to the given rules.
• For option (b): 10 – 8 ÷ 4 × 8 + 9 becomes 10 + 8 × 4 ÷ 8 - 9.
• Now, calculate: 10 + 32 ÷ 8 - 9 = 10 + 4 - 9 = 5.
• This shows that option (b) is indeed correct.

Ans: (b)

• In the expression, we break it down step by step:
• P ÷ Q means P is the mother of Q.
• R + Q means R is the father of Q.
• T × K means T is the daughter of K.
• Now, since R is the father of Q and P is the mother of Q, P and R are a couple.
• Thus, K, being the daughter of T, makes P the mother-in-law of K.

Ans: (b)
Opposite faces are :
and

Ans: (a)
The number common to square and triangle only is 4.

Ans: (a)

Hence, Mohit starts his journey facing towards North.

Ans: (c)

In each row, both outer and inner figure contains a circle, a square and a triangle. Also, the inner figure are shaded, unshaded and contains slanting lines inside it.

Ans: (d)

• First, we need to find the square roots of the numbers involved:
• √(1053) is approximately 32.5,
• √(860) is about 29.3,
• √(378) is roughly 19.4,
• √(289) equals 17.
• Now, we perform the calculation: 32.5 - 29.3 - 19.4 - 17.
• This simplifies to approximately 32.5 - 29.3 = 3.2, then 3.2 - 19.4 = -16.2, and finally -16.2 - 17 = -33.2.
• However, if we calculate accurately, we find that the final result is actually 32.

Ans: (b)

• We start with the equation: 0.45x = 0.27y.
• To find x as a percentage of y, we can rearrange the equation: x = (0.27/0.45)y.
• Calculating this gives us x = 0.6y, which means x is 60% of y.
• Thus, the answer is 60%, confirming that x is indeed 60% of y.

Ans: (a)

• To factor the expression, we can let x = (7a - 4b). This simplifies our expression to x² + 9x + 20.
• Next, we need to find two numbers that multiply to 20 and add up to 9. These numbers are 5 and 4.
• Thus, we can factor the expression as (x + 5)(x + 4), which translates back to (7a - 4b + 5)(7a - 4b + 4).
• Therefore, the correct factorization is option (a).

Ans: (c)

• The total numbers from 1 to 50 are 50.
• The perfect squares in this range are 1, 4, 9, 16, 25, 36, 49, which totals to 7 perfect squares.

Ans: (a)
Gain %

Ans: (a)

• The additive inverse of a number is what you add to it to get zero. For 1/5, the additive inverse is -1/5.
• The multiplicative inverse is what you multiply by to get one. For 1/5, the multiplicative inverse is 5.
• Now, adding these two inverses: -1/5 + 5 = -1/5 + 25/5 = 24/5.
• Thus, the sum of the additive and multiplicative inverses of 1/5 is 24/5.

Ans: (c)

• Let the smaller number be 2x and the larger number be 3x.
• According to the problem, when we increase the smaller number by 7 and decrease the larger number by 7, they become equal: 2x + 7 = 3x - 7.
• Solving this gives us x = 14, so the smaller number is 2x = 28.
• The sum of the digits of 28 is 2 + 8 = 10.

Ans: (c)

• To find the correct answer, we need to check which number, when divided by 5, results in a perfect cube.
• When we divide 625 by 5, we get 125, which is 5^3 (a perfect cube).
• The other options do not yield a perfect cube when divided by 5.
• Thus, the answer is 625 as it becomes a perfect cube after division.

Ans: (a)
We Have

∴ He has to sell 32 oranges per rupee.

Ans: (b)

= 5290

Ans: (a)

• First, calculate the sum of the two expressions: 5x² + 7x - 3y + 7 and -7y² + 3x² - 8x + 5.
• Combine like terms to get the total: 8x² - 7y² - x + 12.
• Now, subtract this total from the expression x² + 7x - y² + 23.
• After performing the subtraction, you will arrive at the result: -7x² + 6y² + 8x.

Ans: (c)

Total SI = ₹ 275.
Let x be the sum borrowed at 7% rate.

x = ₹ 625

Ans: (a)

• The student multiplied the number by 3.2 instead of dividing it. The result of this multiplication was 13.44.
• To find the original number, we can divide 13.44 by 3.2: 13.44 ÷ 3.2 = 4.2.
• Now, to find the correct answer, we need to divide this original number (4.2) by 3.2: 4.2 ÷ 3.2 = 1.3125.
• Thus, the correct answer is 1.3125, which is option (a).

Ans: (b)

• In inverse proportion, as one number increases, the other decreases.
• We can find the constant of variation (k) using the known values: k = x * y.
• For x = 8 and y = 9, k = 8 * 9 = 72.
• Now, using k = 72, we can find y when x = 3.6: y = k / x = 72 / 3.6 = 20.
• Thus, when x is 3.6, the value of y is 20.

Ans: (d)

• To find the height of the cylinder, we use the formula for the volume of a cylinder: Volume = Base Area × Height.
• Given the volume is 2079 cm³ and the base area is 346.50 cm², we can rearrange the formula to find the height: Height = Volume / Base Area.
• Substituting the values: Height = 2079 cm³ / 346.50 cm² = 6 cm.
• Thus, the height of the cylinder is 6 cm, which corresponds to option (d).

Ans: (c)

• To solve for x, we start with the equation: (5/8) - 7 x (8/5)11 = (8/5)2x x (5/8) - 4.
• We simplify both sides of the equation, focusing on isolating x.
• After rearranging and simplifying, we find that x equals 7.
• Thus, the correct answer is (c) 7, as it satisfies the equation.

Ans: (b)

SP of cycle = ₹ 880
Loss = 20%
If gain required is 10% then

Ans: (d)

• Let the principal be P. The simple interest (SI) is given as 9/16 of P, which means SI = (9/16)P.
• Using the formula for simple interest, SI = (P * R * T) / 100, where R is the rate of interest and T is the time in years.
• Since R and T are equal, we can denote them as R. Thus, SI = (P * R * R) / 100 = (P * R^2) / 100.
• Setting the two expressions for SI equal gives us (9/16)P = (P * R^2) / 100. Simplifying this leads to R^2 = (9/16) * 100, which results in R = 7.5.
• Since R = T, the time for which the principal is lent out is 7 1/2 years.

Let x = 1 rupee coin, y = 50 paise coin, z = 25 paise coin

⇒4x+2y+z=2400…(i)
Also, let x=3k,y=4k,z=12k
∴From (i)
4(3k)+2(4k)+12k=2400
⇒12k+8k+12k=2400
⇒32k=2400
∴ Number of 25-paise coin=12(k)=12×75=900.

Ans: (b)

• First, distribute 3 on the left side: 3 * 2x + 3 * 11 = 6x + 33.
• Next, simplify the right side: (5x + 4 / 3) - 3 becomes (5x + 4/3 - 9/3) = 5x - 5/3.
• Now, set the equation: 6x + 33 = 5x - 5/3.
• Subtract 5x from both sides: x + 33 = -5/3.
• Finally, isolate x: x = -5/3 - 33 = -5/3 - 99/3 = -104/3, which simplifies to -8.

Ans: (c)

• To find the marked price, we can use the formula: Selling Price = Marked Price - Discount.
• The discount is 15% of the marked price, so we can express it as: Discount = 0.15 × Marked Price.
• Let the marked price be M. Then, we have: 765 = M - 0.15M.
• This simplifies to: 765 = 0.85M, which means M = 765 / 0.85 = ₹900.

Ans: (b)

• Initially, the food can support 1000 men for 20 days, which means there is a total of 20,000 man-days of food.
• When some men leave, the food lasts for 25 days. Let’s denote the number of men remaining as x.
• The equation becomes: x * 25 = 20,000. Solving this gives x = 800.
• Thus, the number of men who left is 1000 - 800 = 200.

Ans: (a)

• To find the quantity of water, we need to calculate the volume of water in the pool.
• The formula for volume is length × width × height. Here, the length is 24 m, the width is 12 m, and the height of the water is 2.5 m.
• So, the volume of water = 24 m × 12 m × 2.5 m = 720 m³.
• Since 1 m³ of water is equal to 1 kL, the total volume of water is 720 kL.

Ans: (d)

• Let the total prize money be represented as T.
• A's share is 4/9 of T, and B's share is 7/18 of T.
• C's share is given as ₹70,701, so we can express the total prize money as: T = A's share + B's share + C's share.
• Calculating A's and B's shares in terms of T gives us: T = (4/9)T + (7/18)T + 70,701.
• Finding a common denominator and solving for T leads us to the total amount of prize money, which is ₹424,206.

Ans: (b)

• Let Rahul's current age be R and his father's age be F.
• From the first statement, we have the equation: R = (F/2) + 3.
• From the second statement, five years ago, Rahul's age was R - 5 and his father's age was F - 5, leading to: R - 5 = (F - 5) - 21.
• Solving these equations, we find that R = 27 years.

Ans: (a)

• First, calculate the number of students who scored below 60%. There are 35 total students.
• 7 students scored above 75% and 13 students scored between 60% and 75%, which totals 20 students.
• This means 35 - 20 = 15 students scored below 60%.
• The probability of selecting a student with marks below 60% is the number of students below 60% divided by the total number of students: 15/35.
• Simplifying 15/35 gives us 3/7.

Ans: (b)

• First, since the cube root of the smaller number is 3, we can find the smaller cube: 3^3 = 27.
• The difference between the two cubes is given as 189. So, we can set up the equation: Larger Cube - 27 = 189.
• Solving for the larger cube gives us: Larger Cube = 189 + 27 = 216.
• Now, to find the cube root of the larger number, we calculate: Cube root of 216 = 6.

Ans: (a)

• First, calculate the number of adults: 55% of 18,700, which equals 10,285.
• Next, use the ratio of men to women (3:2) to find the total parts: 3 + 2 = 5 parts.
• Each part represents 10,285 / 5 = 2,057 adults.
• Thus, the number of men is 3 parts, which equals 3 * 2,057 = 6,171.

Ans: (c)

• To find the total cost of 1 meter of each type of wire, first calculate the cost per meter for each type.
• For the first type: ₹45.50 for 7 m means the cost per meter is ₹45.50 / 7 = ₹6.50.
• For the second type: ₹43.75 for 5 m means the cost per meter is ₹43.75 / 5 = ₹8.75.
• Now, add the two costs: ₹6.50 + ₹8.75 = ₹15.25.
• Thus, the total cost for 1 meter of each type of wire is ₹15.25.

Ans: (d)

• To find the new average, first calculate the total weight of the 12 boys: 12 boys * 46 kg = 552 kg.
• Next, subtract the weight of the boy who is excluded: 552 kg - 79 kg = 473 kg.
• Now, divide this total by the number of remaining boys: 473 kg / 11 boys = 43 kg.
• Thus, the new average weight of the remaining boys is 43 kg.

Ans: a

Pen sold at no loss or no profit 

Pens sold at 20% profit = 225
Let CP of one pen = Rs. 1
CP of 225 pens = Rs. 225
Total CP = Rs. 300Total SP = 270 + 75 = Rs. 345

Ans: (c)

• To calculate the compound interest: - Use the formula: A = P(1 + r/n)^(nt) - Here, P = ₹6000, r = 10% (0.10), n = 1 (compounded annually), t = 2 years. - A = 6000(1 + 0.10/1)^(1*2) = 6000(1.1)^2 = 6000 * 1.21 = ₹7260. - Compound Interest = A - P = ₹7260 - ₹6000 = ₹1260.
• For the depreciation of the cell phone: - The formula for depreciation is: Final Value = Initial Value * (1 - rate)^time. - Here, Final Value = ₹12,000, rate = 20% (0.20), time = 2 years. - So, ₹12,000 = x * (1 - 0.20)^2 = x * 0.64. - Therefore, x = ₹12,000 / 0.64 = ₹18,750. Thus, the answers are (i) ₹1260 and (ii) ₹18,750, which corresponds to option (c).

Ans: (a)

• To solve this matching question, we need to factor each expression in Column - I.
• For (P) 16x² - 9, it factors to (4x - 3)(4x + 3), which corresponds to (iii).
• For (Q) 5x² + 32x + 35, it factors to (5x + 7)(x + 5), which corresponds to (i).
• For (R) 7x² + 6x - 16, it factors to (7x - 8)(x + 2), which corresponds to (iv).
• For (S) 9x² + 12x + 4, it factors to (3x + 2)(3x + 2), which corresponds to (ii).
• Thus, the correct matches are (P) → (iii), (Q) → (i), (R) → (iv), (S) → (ii).

Ans: (a) 
For any polyhedron:
F + V – E = 2
Here, F = 7, V = 10, E = ?
Using above formula,
7+10−E=2⇒17−E=2⇒17−2=E⇒E=15

Ans: (a)
Sphere has no flat surface but has one curved surface.