Amc test usamo Math Challenge

1. If a, b, and c are positive real numbers such that a + b + c = 1, then what is the maximum value of (a^(1/3)) * (b^(1/3)) * (c^(1/3))?

A. 1/3 B. 1/27 C. 1 D. (1/3)^(1/3)

Solution

Correct: D

Complete explanation available in the mobile app.

2. The function f(x) = (x^3) - (6 * (x^2)) + (9 * x) + 2 has a graph that passes through which of the following points?

A. (0, 0) B. (0, 1) C. (0, 2) D. (0, -2)

Solution

Correct: C

Complete explanation available in the mobile app.

3. Let P(x) = (x^2) - 4. For how many real numbers a such that 0 < a < 3 does the graph of y = P(x) intersect the graph of y = a * x when x > 0?

A. 0 B. 1 C. 2 D. 3

Solution

Correct: C

Complete explanation available in the mobile app.

4. Let T be a triangle with side lengths 12, 13, and 15. If T' is the triangle formed by removing a triangle of side length 1 from each vertex of T, then what is the area of triangle T'?

A. 65 B. 60 C. 55 D. 50

Solution

Correct: B

Complete explanation available in the mobile app.

5. For each positive integer n, let a_n be the sum of the distinct positive divisors of n that are less than n and relatively prime to n. What is the maximum value of a_n for n between 1 and 20?

A. 5 B. 11 C. 12 D. 7

Solution

Correct: C

Complete explanation available in the mobile app.

6. For what value of m will the lines with equations y = 3x + 2 and y = mx + 4 intersect at the point (2, 8)?

A. 2 B. 1 C. 3 D. -1

Solution

Correct: A

Complete explanation available in the mobile app.

7. The positive integers x, y, and z satisfy the equation (x^2) + (y^2) + (z^2) = 30. How many solutions (x, y, z) are there such that x ≤ y ≤ z?

A. 2 B. 3 C. 4 D. 6

Solution

Correct: D

Complete explanation available in the mobile app.

8. If the sum of the first 20 terms of an arithmetic sequence is 240 and the sum of the first 10 terms is 120, then what is the 20th term of the sequence?

A. 4 B. 2 C. 6 D. 8

Solution

Correct: A

Complete explanation available in the mobile app.

9. Given an integer n > 1, let g(n) be the smallest integer k > 1 that is relatively prime to n. What is the sum of g(n) for n = 2 to 10?

A. 17 B. 20 C. 22 D. 24

Solution

Correct: A

Complete explanation available in the mobile app.

10. What is the largest integer that is a factor of every number of the form (a^2) + (b^2) + (c^2) where a, b, and c are positive integers?

A. 2 B. 3 C. 5 D. 1

Solution

Correct: D

Complete explanation available in the mobile app.

11. Two circles are tangent to each other and to a line as shown. If the radius of the larger circle is 4 and that of the smaller circle is 1, then what is the length of the line segment connecting the two points of tangency?

A. 4 B. 6 C. 8 D. 3

Solution

Correct: B

Complete explanation available in the mobile app.

12. A function f(x) satisfies f(0) = 2, f(1) = 3, and for x > 0, f(x + 1) = 2 * f(x) - 1. What is f(5)?

A. 22 B. 23 C. 32 D. 33

Solution

Correct: C

Complete explanation available in the mobile app.

13. The equation x^3 + 2x + 1 = 0 has one real root. If r is this real root, then what is the value of r^2 + 3r + 2?

A. -2 B. -3 C. -1 D. 1

Solution

Correct: A

Complete explanation available in the mobile app.

14. In how many ways can the string 0010 be formed from the letters A, B, C, and D, where A = 0, B = 1, C = 2, and D = 3, such that C and D are never adjacent?

A. 4 B. 6 C. 8 D. 12

Solution

Correct: B

Complete explanation available in the mobile app.

15. A parabola with equation y = ax^2 + bx + c opens downward and has its vertex at (1, 2). If the parabola passes through (0, 0), then what is the value of a + b + c?

A. -2 B. -1 C. 0 D. 1

Solution

Correct: A

Complete explanation available in the mobile app.

16. An equilateral triangle has a perimeter of 36. What is the length of the altitude of the triangle?

A. 6 * √3 B. 9 * √3 C. 12 * √3 D. 3 * √3

Solution

Correct: B

Complete explanation available in the mobile app.

17. A set of 20 distinct positive integers has the property that the square of any one of them is the sum of the squares of the other 19. What is the largest possible value in this set?

A. 20 B. 19 C. 18 D. 17

Solution

Correct: D

Complete explanation available in the mobile app.

18. For any integer n ≥ 2, let H(n) = 1 + 1/2 + 1/3 + ... + 1/n. What is the smallest integer value of n for which H(n) > 3?

A. 10 B. 11 C. 12 D. 13

Solution

Correct: C

Complete explanation available in the mobile app.

19. The diagonals of a rectangle intersect at (1, 2). If the diagonals have length 4, then what are the coordinates of the vertices of the rectangle?

A. (0, 0), (2, 4), (0, 4), (2, 0) B. (0, 1), (3, 1), (0, 3), (3, 3) C. (2, 1), (0, 3), (2, 3), (0, 1) D. (1, 0), (3, 2), (1, 4), (3, 2)

Solution

Correct: A

Complete explanation available in the mobile app.

1. If a, b, and c are positive real numbers such that a + b + c = 1, then what is the maximum value of (a^(1/3)) * (b^(1/3)) * (c^(1/3))?

2. The function f(x) = (x^3) - (6 * (x^2)) + (9 * x) + 2 has a graph that passes through which of the following points?

3. Let P(x) = (x^2) - 4. For how many real numbers a such that 0 < a < 3 does the graph of y = P(x) intersect the graph of y = a * x when x > 0?

4. Let T be a triangle with side lengths 12, 13, and 15. If T' is the triangle formed by removing a triangle of side length 1 from each vertex of T, then what is the area of triangle T'?

5. For each positive integer n, let a_n be the sum of the distinct positive divisors of n that are less than n and relatively prime to n. What is the maximum value of a_n for n between 1 and 20?

6. For what value of m will the lines with equations y = 3x + 2 and y = mx + 4 intersect at the point (2, 8)?

7. The positive integers x, y, and z satisfy the equation (x^2) + (y^2) + (z^2) = 30. How many solutions (x, y, z) are there such that x ≤ y ≤ z?

8. If the sum of the first 20 terms of an arithmetic sequence is 240 and the sum of the first 10 terms is 120, then what is the 20th term of the sequence?

9. Given an integer n > 1, let g(n) be the smallest integer k > 1 that is relatively prime to n. What is the sum of g(n) for n = 2 to 10?

10. What is the largest integer that is a factor of every number of the form (a^2) + (b^2) + (c^2) where a, b, and c are positive integers?

11. Two circles are tangent to each other and to a line as shown. If the radius of the larger circle is 4 and that of the smaller circle is 1, then what is the length of the line segment connecting the two points of tangency?

12. A function f(x) satisfies f(0) = 2, f(1) = 3, and for x > 0, f(x + 1) = 2 * f(x) - 1. What is f(5)?

13. The equation x^3 + 2x + 1 = 0 has one real root. If r is this real root, then what is the value of r^2 + 3r + 2?

14. In how many ways can the string 0010 be formed from the letters A, B, C, and D, where A = 0, B = 1, C = 2, and D = 3, such that C and D are never adjacent?

15. A parabola with equation y = ax^2 + bx + c opens downward and has its vertex at (1, 2). If the parabola passes through (0, 0), then what is the value of a + b + c?

16. An equilateral triangle has a perimeter of 36. What is the length of the altitude of the triangle?

17. A set of 20 distinct positive integers has the property that the square of any one of them is the sum of the squares of the other 19. What is the largest possible value in this set?

18. For any integer n ≥ 2, let H(n) = 1 + 1/2 + 1/3 + ... + 1/n. What is the smallest integer value of n for which H(n) > 3?

19. The diagonals of a rectangle intersect at (1, 2). If the diagonals have length 4, then what are the coordinates of the vertices of the rectangle?

Related Practice Tests

Test payment 2 math olympiad Math Challenge

Test payment 1 math olympiad Math Challenge

IAS Prelims test Math for IAS Prelims Math Challenge

Discussion & Comments