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We know that the area of triangle \\( BEF \\) is given as \\( 1 \\) and the area of triangle \\( ADE \\) is given as \\( 9 \\).

To find the area of parallelogram \\( ABCD \\), we can first consider the relationship between the areas of the triangles and the parallelogram they form.

Let \\( [X] \\) denote the area of figure \\( X \\).

The area of parallelogram \\( ABCD \\) can be represented in terms of the areas of triangles \\( ADE \\) and \\( BCE \\).

Since \\( ABCD \\) is a parallelogram, triangles \\( ADE \\) and \\( BEC \\) are equal in area, as they share the same base \\( DE \\) and lie between the same two parallels \\( AB \\) and \\( CD \\).

Thus, we can express: \\[ [BEC] = [ADE] = 9 \\]

Now, considering triangle \\( BEF \\), we can find the area of triangle \\( BEC \\): \\[ [BEC] = [BEF] + [BCE] = 1 + [BCE] \\]

This gives us: \\[ 9 = 1 + [BCE] \\]

By simplifying, we find: \\[ [BCE] = 9 - 1 = 8 \\]

Now, we've established the area of triangle \\( BCE \\) and know that the area of triangle \\( ADE = 9 \\).

Therefore, the total area of the parallelogram \\( ABCD \\) is the sum of the areas of triangles \\( ADE \\) and \\( BCE \\):

\\[ [ABCD] = [ADE] + [BCE] = 9 + 8 = 17 \\]

Thus, the area of parallelogram \\( ABCD \\) is: \\[ \\boxed{17} \\]

Let's analyze the equations one by one:

A + B = C: This tells us that C must be greater than both A and B.

AA - B = 2C: This equation can be rewritten as 10A + A - B = 2C, or 11A - B = 2C. Since C is greater than A and B, 2C must be greater than 11A.

This is only possible if A is a very small number.

C * B = AA + A: This equation can be rewritten as CB = 11A. Since 11A is a multiple of 11, CB must also be a multiple of 11.

Considering these three equations, we can deduce the following:

A must be 1: This is the only way to satisfy the second equation, 11A - B = 2C, with C being greater than A and B.

C must be 3: Since A is 1, the first equation becomes 1 + B = C. The only way to satisfy this with C being greater than B is if C is 3.

B must be 2: From the first equation, if A is 1 and C is 3, then B must be 2.

Therefore, A + B + C = 1 + 2 + 3 = 6.

Hi there!

I found the solutions y = -4 and y = 4, so the only possible value of y^2 = 16.

Hope this helps!

Let's analyze the problem step by step:

1. Understanding the Given Information:

EFGH is a trapezoid with EF parallel to GH.

P is a point on EH such that EP:PH = 1:2.

Area of triangle PEF = 6.

Area of triangle PGH = 6.

2. Visualizing the Situation:

trapezoid EFGH with point P dividing EH in a 1:2 ratio

3. Using the Ratio of Areas:

Since triangles PEF and PGH share the same height (the perpendicular distance between EF and GH), the ratio of their areas is equal to the ratio of their bases.

So, (Area of triangle PEF) / (Area of triangle PGH) = EP/PH = 1/2.

We know the area of both triangles, so we can confirm this ratio: 6/6 = 1/2.

4. Finding the Area of Triangle EGH:

Since EP:PH = 1:2, we can say that EH = 3 * EP.

Triangles EGH and PEF share the same height, so the ratio of their areas is equal to the ratio of their bases.

(Area of triangle EGH) / (Area of triangle PEF) = EH/EP = 3/1.

Therefore, the area of triangle EGH = 3 * (Area of triangle PEF) = 3 * 6 = 18.

5. Finding the Area of Trapezoid EFGH:

The area of trapezoid EFGH is the sum of the areas of triangles PEF and EGH.

Area of trapezoid EFGH = Area of triangle PEF + Area of triangle EGH = 6 + 18 = 24.

Therefore, the area of trapezoid EFGH is 24.

The probability is 4/55.

That simplifies to 2*3^(1/3).  Hope this helps!

To solve this problem, let's first understand the given information:

n = ABC_7: This means n is a 3-digit number in base 7, where A, B, and C are digits from 0 to 6.

n = CBA_11: This means n is a 3-digit number in base 11, where C, B, and A are digits from 0 to 10.

We need to find the largest possible value of n in base 10.

Approach:

Convert both representations to base 10:

n = ABC_7 = 7^2A + 7B + C

n = CBA_11 = 11^2C + 11B + A

Set the two expressions equal to each other:

7^2A + 7B + C = 11^2C + 11B + A

Rearrange the equation:

48A - 4B - 120*C = 0

Divide the equation by 4:

12A - B - 30C = 0

Analyze the equation:

We want to maximize n, so we want to maximize A and minimize C.

Since A, B, and C are digits, A must be at least 1 and C must be at most 5 (otherwise, A would have to be greater than 6).

Trying different values:

A = 6, C = 1:

126 - B - 301 = 0

B = 36

A = 6, C = 2:

126 - B - 302 = 0

B = 24

A = 6, C = 3:

126 - B - 303 = 0

B = 12

A = 6, C = 4:

126 - B - 304 = 0

B = 0

A = 6, C = 5:

126 - B - 305 = 0

B = -12 (not possible)

The largest possible value of n in base 10 is when A = 6, B = 36, and C = 1:

n = 7^26 + 736 + 1 = 49*6 + 252 + 1 = 294 + 252 + 1 = 547

Therefore, the largest possible value of n in base 10 is 547.

There are 312 ways.

Here is part (c).

Analyzing the Cube and the Viewing Point

Understanding the Problem:

We have a 7x7x7 cube with alternating colors on adjacent faces.

We want to maximize the number of cubes of a single color visible in a diagonal.

Key Points:

The maximum number of cubes visible in a diagonal depends on the orientation of the viewing point.

We need to find the optimal viewing point to maximize the number of visible cubes of a single color.

Determining the Maximum Visible Cubes

Optimal Viewing Point:

To see the maximum number of cubes of a single color in a diagonal, we need to position the viewing point directly in front of a corner of the cube. This will ensure that we can see the longest diagonal.

Visualizing: Imagine the cube with a corner facing you. You'll see a diagonal going from the top corner to the bottom corner of the opposite face

.

Counting the Cubes:

In a 7x7x7 cube, the longest diagonal consists of 7 cubes.

Since the colors alternate on adjacent faces, every other cube in this diagonal will be the same color.

Calculation:

Number of cubes in the diagonal = 7

Half of the cubes = 7/2 = 3.5

Rounding down, we get 3.

Therefore, the maximum number of cubes of each color visible in a diagonal from a single point is 3.