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 #1
avatar+116 
+1

Analyzing the function f(x) = floor((2 - 3x) / (x + 3)):

 

The denominator (x + 3) is 0 when x = -3. This means f(x) is undefined at x = -3.

 

We will need to consider different cases based on the sign of the denominator (x + 3) and the relative values of 2 - 3x compared to 0.

 

Cases for f(x):

 

x < -3: In this case, both denominator (x + 3) and numerator (2 - 3x) are negative. Dividing two negative numbers results in a positive value.

 

Since we take the floor (greatest integer less than or equal to), f(x) will be -1.

 

-3 < x < 2/3: Here, the denominator (x + 3) is positive, but the numerator (2 - 3x) is negative.

 

Dividing a positive by a negative results in a negative number.

 

Taking the floor of a negative number keeps it negative, so f(x) will be -2.

 

x = 2/3: At this specific point, the numerator becomes 0, and the result of the division is 0. The floor of 0 is 0, so f(x) = 0.

 

x > 2/3: In this case, both the numerator (2 - 3x) and denominator (x + 3) are positive. Dividing two positive numbers results in a positive value.

 

Taking the floor doesn't change the positive sign, so f(x) will be 1.

 

Evaluating the sum:

 

The key to evaluating the sum efficiently is to recognize that for a large range of x values (between -3 and 2/3), f(x) will be -2.

 

We can exploit this by calculating the number of terms that fall into this range and summing the contributions from the remaining terms separately.

 

Number of terms where f(x) = -2:

 

We know x = -3 falls outside this range (f(x) is undefined).

 

The range ends when x = 2/3, which is between terms 1000 and 1001 (1000th term is x = 999 and 1001st term is x = 1000).

 

Therefore, there are 1000 - (-3) + 1 = 1004 terms where f(x) = -2.

 

Contribution from terms where f(x) = -2:

 

Each term contributes -2 to the sum.

 

Total contribution = -2 * (number of terms) = -2 * 1004 = -2008

 

Remaining terms:

 

We need to consider terms for x < -3 (f(x) = -1), x = 2/3 (f(x) = 0), and x > 2/3 (f(x) = 1).

 

There are very few terms less than -3 (all negative x values), and they can be ignored for a large sum like this (their contribution will be negligible).

 

There's only one term for x = 2/3, contributing f(2/3) = 0.

 

The remaining terms from x slightly greater than 2/3 to x = 1000 will all have f(x) = 1. The exact number of these terms depends on the specific values, but there will be significantly fewer compared to the 1004 terms with f(x) = -2.

 

Overall Sum:

 

Sum from terms with f(x) = -2: -2008

 

Contribution from f(2/3) (x = 2/3): 0

 

Contribution from remaining terms with f(x) = 1 (positive but less than those with -2): + (positive value)

 

Since the number of terms with f(x) = 1 is significantly less than those with -2, and there's a negligible contribution from terms less than -3, the positive value from the remaining terms will be much smaller than 2008.

 

Therefore, the overall sum f(1) + f(2) + ... + f(999) + f(1000) is equal to -2008.

Mar 24, 2024
 #1
avatar+116 
0

Let's denote lengths:

 

AP=x (since it's a multiple of PB)

 

PB=x/3

 

CP=y (since information about this segment isn't given, we denote it with a variable)

 

PD=y/3 (similar logic as for PB)

 

AB=a (what we're solving for)

 

CD=c (not directly needed, but can be helpful for visualization)

 

Apply the Power of a Point Theorem:

 

The Power of a Point Theorem states that for any point P inside a circle, the product of the lengths of the two segments created by drawing secants from that point to the circle is equal. In our case, point P is inside the circle (since chords intersect within the circle), and we can apply the theorem to both secants AB and CD.

 

For secant AB:

 

AP⋅PB=x⋅3x​=3x2​

 

For secant CD:

 

CP⋅PD=y⋅3y​=3y2​

 

Since both expressions represent the same power of point P, they must be equal:

 

3x2​=3y2​

 

Utilize the given information:

 

We are given that AP=3⋅PB, which translates to x=3⋅3x​ (substituting the values we defined). This simplifies to x=3x​, which implies x=0. However, a chord cannot have zero length. Therefore, our initial assumption (that x represents a positive length) must be incorrect.

 

Here's the correction: We can rewrite the given information as x=3PB=3⋅3x​. Solving for x, we get x2=9. Taking the square root of both sides (remembering positive for lengths), we have x=3.

 

Substitute and solve for AB:

 

Since we found x=3, we can substitute this value back into the equation we obtained from the Power of a Point Theorem:

 

3x2​=3y2​

 

332​=3y2​ (substitute x with 3)

 

3=y2

 

Taking the square root (positive for lengths), we have y=3​.

 

Now, consider segment AP : its total length is x+PB=3+33​=4. Since AP=x=3, segment PB must have a length of PB=14−3​=1.

 

Finally, to find the length of AB, we add the lengths AP and PB:

 

AB=AP+PB=3+1=4​.

Mar 20, 2024
 #1
avatar+116 
+1

Analyzing the Cube Cuts

 

Imagine the large cube is 3 units on each side. When we cut it into smaller cubes, each side of the larger cube will be made of 3 smaller cubes.

 

(a) Cubes with One Black Face

 

A cube will have exactly one black face only if it's on the edge of the larger cube but not on a corner.

 

Edges: There are 12 edges on a cube.

 

Corner exceptions: On each edge, there are 2 smaller cubes that touch a corner.

 

Since corner cubes will have 3 black faces, we subtract these exceptions from the total edge cubes. There are 8 corners, so there are 8×2=16 corner exception cubes.

 

Therefore, the number of cubes with one black face is the total number of edge cubes minus the corner exceptions: 12 edges - 16 corner exceptions = -4 cubes

 

This seems like a negative number of cubes, which doesn't make sense. The mistake lies in assuming all the edge cubes have one black face.

 

Here's the correction: We only counted the  edges once, but each edge actually has 2 cubes that qualify (one on each side). So, we need to multiply the number of edges by 2:

 

Total one-black-face cubes = (2 cubes/edge) x (12 edges) - 16 corner exceptions = 24 - 16 = 8 cubes

 

(b) Cubes with No Black Faces

 

These cubes must be completely inside the larger cube, not touching any of the faces.

 

Inner core: Since each side of the larger cube is made of 3 smaller cubes, the inner core will be a cube with sides of length 1 unit less (3 - 2 = 1 unit). The volume of this inner core is therefore 1 x 1 x 1 = 1 cube.

 

Inner cubes: This inner core cube is itself made of smaller cubes. Each side has 1 cube, so there are a total of 1 x 1 x 1 = 1 smaller cube inside.

Therefore, there is only 1 cube with no black faces.

 

(c) Probability of Top Face Black

 

When a small cube is rolled, there are 6 possible faces that could land on top.

 

Out of these 6 faces, only 1 face is painted black (since we're considering the cubes with at least one black face).

 

Therefore, the probability of the top face being black is the number of black faces divided by the total number of faces: Probability = 1 black face / 6 total faces = 1/6.

Mar 19, 2024