Increased Difficulty for Naitik, 1/10/2024

Evaluate the ratio: (2²⁰⁰¹ ⋅ 3²⁰⁰³)/(6²⁰⁰²)

• 2/3

• 3/2

• 1/6

• 1/3

For the nonzero numbers a, b, and c:

D(a,b,c) = (abc)/(a + b + c)

Find D(2,4,6)

• 24

• 2

• 4

• 6

Let a, b, and c be real numbers such that a - 7b + 8c = 4 and 8a + 4b - c = 7.

Determine the value of a² - b² + c².

• 0

• 1

• 4

• 7

Circles of r = 2 and r = 3 are externally tangent and circumscribed by a third circle as shown in the figure. Find the area of the shaded region.

• 4π

• 6π

• 9π

• 12π

The first term of a sequence is 2005. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the 2005th term of the sequence?

• 55

• 85

• 133

• 250

On the 9th grade math final, 10% of the students got 70 points, 25% of students got 80 points, 20% got 85 points, 15% got 90 points, and the rest got 95 points.

What is the difference between the mean and median scores on the exam

• 1

• 2

• 5

• 0

How many numbers between 1 and 2005 are integer multiples of 3 or 4 but not 12?

• 501

• 668

• 835

• 1169

A right circular cylinder with its diameter equal to its height is inscribed in a right circular cone. The cone has diameter 10 and altitude 12, and the axes of the cylinder and cone coincide. Find the radius of the cylinder.

• 3

• 30/11

• 8/3

• 7/2

When 7 fair standard 6-sided dice are thrown, the probability that the sum of the numbers on the top faces is 10 can be written as:

n/6⁷

where n is a positive integer. What is n?

• 49

• 56

• 84

• 63

Chloe, Chelsea, and Tyasia listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those two girls but disliked by the third. In how many different ways is this possible?

• 108

• 671

• 132

• 846