The math Olympiad is a competition that tests students’ problem-solving skills and mathematical knowledge. Here are some sample math Olympiad questions that might be appropriate for a seventh grade student:

### Sample Test Questions

1. The product of three consecutive positive integers is 1386. What is the middle integer?
2. If x + y = 7 and x – y = 3, what is the value of 2x?
3. The area of a rectangle is 45 square units. If the width of the rectangle is 5 units, what is the length?
4. The sum of the first n positive odd integers is 169. What is the value of n?
5. In a group of 20 people, 14 of them speak Spanish, and 8 of them speak French. If there are 6 people who speak both Spanish and French, how many people in the group speak neither language?
6. A number is divisible by 3 if the sum of its digits is divisible by 3. Is the number 243 divisible by 3? Why or why not?
7. A rectangular prism has a volume of 196 cubic units. If the width and length of the base are both 7 units, what is the height of the prism?
8. A circle has a radius of 6 units. What is the area of the circle?
9. Solve for x: 3x – 6 = 12
10. Simplify: (4x + 2)(x – 3) – (2x + 3)(x – 4)

##### How to Solve?
1. The product of three consecutive positive integers is 1386. What is the middle integer? To solve this problem, you can start by dividing 1386 by 3 to get 462. Then, you can add or subtract 1 to get the other two integers, which are 461 and 463. The middle integer is 463.
2. If x + y = 7 and x – y = 3, what is the value of 2x? To solve this problem, you can add the two equations together to get 2x = 10.
3. The area of a rectangle is 45 square units. If the width of the rectangle is 5 units, what is the length? To solve this problem, you can divide the area by the width to get the length, which is 9 units.
4. The sum of the first n positive odd integers is 169. What is the value of n? To solve this problem, you can use the formula for the sum of the first n odd integers, which is n^2. You can set up the equation n^2 = 169 and solve for n to get n = 13.
5. To solve this problem, we can use the principle of inclusion-exclusion. This states that if we want to find the number of elements in the union of two sets, we can add the number of elements in each set, and then subtract the number of elements that appear in both sets.

In this case, the number of people who speak Spanish is 14, the number of people who speak French is 8, and the number of people who speak both languages is 6. Therefore, the number of people who speak either Spanish or French or both is 14 + 8 – 6 = 16.

Since there are 20 people in the group, the number of people who speak neither Spanish nor French is 20 – 16 = 4.

Therefore, the answer to the question is 4.

6. To determine whether the number 243 is divisible by 3, we need to find the sum of its digits. The sum of the digits of 243 is 2 + 4 + 3 = 9, which is divisible by 3. Therefore, the number 243 is divisible by 3.
7. If the volume of a rectangular prism is 196 cubic units and the width and length of the base are both 7 units, we can use the formula for the volume of a rectangular prism to find the height. The formula is V = lwh, where l is the length, w is the width, and h is the height. Plugging in the known values, we get 196 = 7 * 7 * h. Dividing both sides by 7 * 7, we get h = 196 / (7 * 7) = 4. Therefore, the height of the prism is 4 units.
8. To find the area of a circle, we can use the formula A = pi * r^2, where A is the area, pi is approximately 3.14, and r is the radius of the circle. Plugging in the known values, we get A = 3.14 * 6^2 = 3.14 * 36 = 113.04. Therefore, the area of the circle is 113.04 square units.
9. To solve the equation 3x – 6 = 12, we can start by adding 6 to both sides to get 3x = 18. Dividing both sides by 3, we get x = 18 / 3 = 6. Therefore, the value of x that satisfies the equation is 6.
10. To simplify the expression (4x + 2)(x – 3) – (2x + 3)(x – 4), we can start by expanding both binomials. This gives us:

(4x + 2)(x – 3) – (2x + 3)(x – 4) = (4x^2 – 12x + 2x – 6) – (2x^2 – 8x + 3x – 12) = 4x^2 – 12x + 2x – 6 – 2x^2 + 8x – 3x + 12

Combining like terms, we get:

4x^2 – 12x + 2x – 6 – 2x^2 + 8x – 3x + 12 = 2x^2 – 9x – 7

Therefore, the simplified expression is 2x^2 – 9x – 7.