China Team Mathematical Olympiad

Description: China Team Mathematical Olympiad is a prestigious mathematics competition for high school students in China. It is held annually and selects the top six students to represent China at the International Mathematical Olympiad (IMO).
Number of Questions: 15
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Tags: china team mathematical olympiad mathematical competitions high school mathematics number theory geometry algebra combinatorics
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What is the sum of the first 100 positive integers?

  1. 5050

  2. 5150

  3. 5250

  4. 5350


Correct Option: A
Explanation:

The sum of the first n positive integers is given by the formula n(n+1)/2. Substituting n = 100, we get 100(101)/2 = 5050.

What is the area of a circle with radius 5?

  1. 25π

  2. 50π

  3. 75π

  4. 100π


Correct Option: A
Explanation:

The area of a circle is given by the formula πr^2. Substituting r = 5, we get π(5)^2 = 25π.

What is the volume of a cube with side length 3?

  1. 9

  2. 18

  3. 27

  4. 36


Correct Option: C
Explanation:

The volume of a cube is given by the formula s^3. Substituting s = 3, we get 3^3 = 27.

What is the slope of the line passing through the points (1, 2) and (3, 4)?

  1. 1

  2. 2

  3. 3

  4. 4


Correct Option: B
Explanation:

The slope of a line passing through two points (x1, y1) and (x2, y2) is given by the formula (y2 - y1)/(x2 - x1). Substituting the given points, we get (4 - 2)/(3 - 1) = 2.

What is the equation of the line passing through the point (2, 3) and having a slope of -2?

  1. y = -2x + 7

  2. y = -2x + 5

  3. y = -2x + 3

  4. y = -2x + 1


Correct Option: A
Explanation:

The equation of a line passing through a point (x1, y1) and having a slope m is given by the formula y - y1 = m(x - x1). Substituting the given point and slope, we get y - 3 = -2(x - 2), which simplifies to y = -2x + 7.

What is the solution to the equation x^2 - 4x + 3 = 0?

  1. x = 1, x = 3

  2. x = 2, x = 4

  3. x = 3, x = 5

  4. x = 4, x = 6


Correct Option: A
Explanation:

We can solve this equation using the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. Substituting a = 1, b = -4, and c = 3, we get x = (-(-4) ± √((-4)^2 - 4(1)(3))) / 2(1) = (4 ± √(16 - 12)) / 2 = (4 ± √4) / 2 = (4 ± 2) / 2. Therefore, the solutions are x = 1 and x = 3.

What is the value of the expression 2^3 + 3^3 + 4^3?

  1. 99

  2. 100

  3. 101

  4. 102


Correct Option: B
Explanation:

We can evaluate this expression using the formula for the sum of cubes: a^3 + b^3 + c^3 = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca). Substituting a = 2, b = 3, and c = 4, we get 2^3 + 3^3 + 4^3 = (2 + 3 + 4)(2^2 + 3^2 + 4^2 - 2(3)(4) - 2(3)(4) - 2(4)(2)) = 9(4 + 9 + 16 - 24 - 24 - 16) = 9(100) = 100.

What is the probability of getting a head when flipping a fair coin?

  1. 1/2

  2. 1/3

  3. 1/4

  4. 1/5


Correct Option: A
Explanation:

A fair coin has two possible outcomes: head and tail. Since each outcome is equally likely, the probability of getting a head is 1/2.

What is the probability of getting a sum of 7 when rolling two fair dice?

  1. 1/6

  2. 1/5

  3. 1/4

  4. 1/3


Correct Option: A
Explanation:

There are 36 possible outcomes when rolling two dice. The outcomes that sum to 7 are (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1). Therefore, the probability of getting a sum of 7 is 6/36 = 1/6.

What is the next term in the sequence 1, 4, 9, 16, ...?

  1. 21

  2. 25

  3. 28

  4. 36


Correct Option: B
Explanation:

This sequence is the square of the natural numbers. The next term is the square of 5, which is 25.

What is the sum of the first 100 prime numbers?

  1. 24133

  2. 24233

  3. 24333

  4. 24433


Correct Option: A
Explanation:

The sum of the first 100 prime numbers is 24133.

What is the largest prime factor of 1001?

  1. 31

  2. 37

  3. 41

  4. 43


Correct Option: C
Explanation:

The largest prime factor of 1001 is 41.

How many positive integers less than 100 are divisible by 3 or 5?

  1. 31

  2. 33

  3. 35

  4. 37


Correct Option: B
Explanation:

There are 33 positive integers less than 100 that are divisible by 3 or 5.

What is the value of the expression 1^3 + 2^3 + 3^3 + ... + 10^3?

  1. 3025

  2. 3125

  3. 3225

  4. 3325


Correct Option: A
Explanation:

The value of the expression 1^3 + 2^3 + 3^3 + ... + 10^3 is 3025.

What is the sum of the first 50 positive even integers?

  1. 2550

  2. 2650

  3. 2750

  4. 2850


Correct Option: A
Explanation:

The sum of the first 50 positive even integers is 2550.

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