Summer Problem Solving Marathon Solution #37

FLOOR(log₂N) will always be equal to the exponent of the largest power of 2 less than or equal to N. So:

FLOOR(log₂N) = 0 for N = 1
FLOOR(log₂N) = 1 for 2 ≤ N ≤ 3
FLOOR(log₂N) = 2 for 4 ≤ N ≤ 7
FLOOR(log₂N) = 3 for 8 ≤ N ≤ 15
FLOOR(log₂N) = 4 for 16 ≤ N ≤ 31
FLOOR(log₂N) = 5 for 32 ≤ N ≤ 63
FLOOR(log₂N) = 6 for 64 ≤ N ≤ 127
FLOOR(log₂N) = 7 for 128 ≤ N ≤ 255
FLOOR(log₂N) = 8 for 256 ≤ N ≤ 511
FLOOR(log₂N) = 9 for 512 ≤ N ≤ 1023
FLOOR(log₂N) = 10 for N = 1024

So the total we want is 2*1 + 4*2 + 8*3 + 16*4 + 32*5 + 64*6 + 128*7 + 256*8 + 512*9 + 10 = 8204.

Summer Problem Solving Marathon Question #39

[Value = 4 points]

A bug is at the origin of the Cartesian coordinate plane. It travels one unit right, to the point (1,0). It then makes a 90 degree turn counterclockwise, and travels .5 units to (1,.5). It continues in this manner -- making 90 degree turns counterclockwise, and each time travelling half as far as on the previous move. In the limit, what point does the bug approach?

Summer Problem Solving Marathon Question #38

[Value = 5 points]

Suppose that z⁴ + az³ + bz² + cz + d is a fourth-degree polynomial with real-valued coefficients. Suppose also that each root of this polynomial is a complex number lying on the circle in the complex plane centered at 0 + 0i and with radius 1. What is the sum of the reciprocals of the roots of the polynomial?

Summer Problem Solving Marathon Question #37

[Value = 6 points]

Let FLOOR(x) be the greatest integer that is less than or equal to x. What is the sum from N = 1 to N = 1024 of FLOOR(log₂ N)?

Summer Problem Solving Marathon Solution #36

The least common multiple of 6 and 8 is 24. Every multiple of 24 thus has a remainder of 0 when divided by 6 and when divided by 8.

For similar reasons, numbers that are 1 to 5 more than a multiple of 24 have remainders of 1 to 5 when divided by either 6 or 8.

We thus find numbers that are within 5 of a multiple of 24, between 101 and 199. There are four multiples of 24 in that range -- 120, 144, 168, and 192. That gives us 24 numbers with the same remainder upon division by 6 or 8. In addition, 101 is 5 more than 96, a multiple of 24. It thus works as well. This gives a total of 25 numbers.

Summer Problem Solving Marathon Solution #35

The volume of a sphere is (4/3)π r³, so the volume of the hemisphere of ice cream is (2/3)π 2³ = (16/3)π.

The volume of a cone is (1/3)hB, where B is the area of the circular base of the cone. Here the height is 8, and the circular base has radius 2 and hence area 4π. So the volume of the cone is (32/3)π.

Thus the total volume of ice cream is (16/3)π + (32/3)π = (48/3)π = 16π.

Summer Problem Solving Marathon Solution #34

When a right triangle is inscribed in a circle, its hypotenuse is a diameter of the circle. Thus the hypotenuse of the triangle is 14. Since the triangle is a 3:4:5 right triangle, its legs are then 42/5 and 56/5. The area of the triangle is then 1/2(42/5)(56/5) = 1176/25 = 47.04.