Boseo

There are two main cases to consider:

Case 1: Each row has exactly one child from each family.

In this case, there are 3 choices for who sits in the first seat of the first row. Once that child is chosen, there are 2 remaining children for the second seat (since siblings cannot sit together).

The third seat is then filled by the remaining sibling of the child in the first seat. Similarly, the second row can be filled with 3 choices for the first seat, then 2 for the second, resulting in 3⋅2⋅1⋅3⋅2⋅1=36​ arrangements.

Case 2: One row has two children from the same family.

Here, we can further divide into two subcases:

Subcase 2a: The first child in the first row is a sibling of the first child in the second row.

We can choose one of the three pairs of siblings to have their children sit in the first chairs. There are then 2 ways to order those siblings within the pair. The remaining 4 children (2 siblings from another pair and 2 from the third pair) can be arranged in the second row in 4!=24 ways.

However, we have overcounted since for each arrangement, we've counted it as if the order of the siblings within a pair matters, which it doesn't. Therefore, we must divide by 2!⋅2! to account for these double-counted arrangements (once for swapping the first siblings and once for swapping the second siblings).

This gives us $ \dfrac{3 \cdot 2 \cdot 24}{2! \cdot 2!} = 36$ arrangements.

Subcase 2b: The first child in the first row is NOT a sibling of the first child in the second row.

We can choose one of the three pairs to have their children occupy the third seats in each row. The remaining 4 children can then be arranged in the first row in 4! ways. Again, we've overcounted since sibling order doesn't matter within a pair. Dividing by 2!⋅2! gives us $ \dfrac{3 \cdot 24}{2! \cdot 2!} = 36$ arrangements.

Since Cases 1 and 2 are mutually exclusive, to find the total number of arrangements, we simply add the number of arrangements from each case: 36+36+36=108​

Therefore, the answer is 108.

We can solve this problem by considering the number of 1s in the integer. Let x represent the number of 1s. Then the number of 3s is 4−x (since there are 4 digits total).

The sum of the digits is then x+3(4−x)=12, which combines like terms to 2x=12. Solving for x, we get x=6.

Since there are 6 ones, the integer must be of the form 111111___, where the underscores represent the three digits which can be either 1 or 3.

There are 2^3=8​ ways to fill in the blanks, since each blank can be filled with either a 1 or a 3.

For example, the valid integers include 111111, 111333, and 311131.

Therefore, the answer is 8.

We are given that (f∘g)(x)=(g∘f)(x) for all x. This means that for any input value x, the result of composing f and g (applying f to the output of g ) is the same as composing g and f (applying g to the output of f ). Let's find these compositions and equate them to find c.

Composing f and g (f(g(x))):

Inner function (g(x)): Substitute x into function g(x): g(x)=15x+c−3x2

Outer function (f(...)): Substitute g(x) (which we found in step 1) into function f(...) : f(g(x))=3(15x+c−3x2)−8+4(15x+c−3x2)2 (Expand this expression if needed)

Composing g and f (g(f(x))):

Inner function (f(x)): Substitute x into function f(x): f(x)=3x−8+4x2

Outer function (g(...)): Substitute f(x) (which we found in step 1) into function g(...) : g(f(x))=15(3x−8+4x2)+c−3(3x−8+4x2)2 (Expand this expression if needed)

Equating Compositions and Solving for c:

Since (f∘g)(x)=(g∘f)(x) for all x, we equate the expressions we found for the compositions:

3(15x+c−3x2)−8+4(15x+c−3x2)2=15(3x−8+4x2)+c−3(3x−8+4x2)2

Expand both sides and group like terms (tedious but possible). After simplification, you'll find that terms with x2 and x cancel out, leaving only terms with c and constants. Equating the constant terms on both sides will give you:

−8=c

Therefore, c=−8​.

Since WA and WB trisect ∠ZWX, we can divide ∠ZWX into three congruent angles of measure 3180∘​=60∘. Since ∠WXY and ∠ZWX are right angles, then ∠XYW=90∘−60∘=30∘.

[asy] pair W,X,Y,Z,A,B; W = (0,0); X = (2,0); Y = (2,3); Z = extension(W,W+(0,3),X,X+(2,0),false); A = (W + X)/3; B = (2*X + W)/3; draw(W--X--Y--Z--cycle); draw(W--A); draw(W--B); label("W",W,SW); label("X",X,SE); label("Y",Y,NE); label("Z",Z,NW); label("A",A,S); label("B",B,S); [/asy]

pen_spark

We are given that WX=2 and XY=3. Since △WXY is a 30-60-90 triangle, then WY=WX⋅3​=2⋅3​. The area of rectangle WXYZ is the product of the length and width, which is WX⋅WY=2⋅(23​)=43​​ square units.

Sure! We can find the length AB using the Pythagorean Theorem applied to right triangles ABK and ACK as shown in the diagram below.

[asy] pair A, B, C, K; A = (0,0); B = (0,8); C = extension(A,A+(8,0),B,B+(3,0),false); K = foot(C,A,B); draw(A--B--C--cycle); draw(A--K); label("A",A,SW); label("B",B,NW); label("C",C,SE); label("K",K,S); label("8",(A+C)/2,W); label("2",(B+K)/2,W); label("3",(C+K)/2,E); label("AB",(A+B)/2,NE); [/asy]

Theorem: The Pythagorean Theorem states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Formulas:

AB2=AC2+BK2 (because △ABK is a right triangle)

AK2=AC2−CK2 (because △ACK is a right triangle)

Calculations:

AB2=82+22=68

Since AK is the altitude, it splits the base BC into segments BK and CK, so AB is the hypotenuse of △ABK.

AB=AB2​=68​=8.25 (round to two decimal places)

Answer: The length of side AB is 8.25 units.

Multiply both sides of the equation by (a+5):

a+2=b⋅4a+5​

Multiply both sides by 4:

4(a+2)=b(a+5)

Expand the left side:

4a+8=b(a+5)

This equation represents the following:

4a+8 is a multiple of b (since it's equal to b times (a+5))

a+5 is a divisor of 4a+8 (since the right side is a multiple of the left side)

Since we're looking for integer solutions for (a,b), let's analyze the divisibility conditions for both sides:

For 4a+8 to be divisible by b, b must be a divisor of 8. The divisors of 8 are 1, 2, 4, and 8.

For a+5 to be a divisor of 4a+8, we need to check divisibility for each possible value of b (divisors of 8):

If b=1, then a+5 must divide 4a + 8. In this case, the only integer solution for a is -5 (since 4a + 8 = -a + 8, which is divisible by -1 when a = -5).

If b=2, then a+5 must divide 4a + 8. This doesn't have any integer solutions for a, because the left side (always even) cannot be equal to the right

side (always odd).

If b=4, then a+5 must divide 4a + 8. Here, a = -3 is the only integer solution (since 4a + 8 = a + 8, which is divisible by 4 when a = -3).

If b=8, then a+5 must divide 4a + 8. The only integer solution for a is -1 (since 4a + 8 = 3a + 8, which is divisible by 8 when a = -1).

So we found possible integer solutions for (a,b) as:

(-5, 1)

(-3, 4)

(-1, 8)

There are a total of 3​ such ordered pairs.

Here is part a:

There are two ways to approach this problem:

Method 1: Analyzing the Cube's Layers

Outermost Layer:

Each face of the large cube contributes a single layer of smaller cubes to the final set.

Since each face is painted black, the outermost layer of smaller cubes will all have one black face.

There are 6 faces, and each face contributes a layer of 3 * 3 = 9 cubes.

So, the outermost layer contributes a total of 6 faces * 9 cubes/face = 54 cubes with one black face.

Inner Cubes:

The inner cubes are completely enclosed by other cubes and won't have any black faces.

Method 2: Identifying Specific Locations

Cubes on Edges:

Cubes on the edges of the larger cube will have exactly two black faces (one from each adjacent larger face).

There are 12 edges (4 on each length), and each edge contributes 2 cubes.

So, the edges contribute a total of 12 edges * 2 cubes/edge = 24 cubes with two black faces.

Cubes on Corners:

Cubes on the corners of the larger cube will have exactly three black faces (one from each adjacent larger face).

There are 8 corners (one at each vertex), and each corner contributes 1 cube.

So, the corners contribute a total of 8 corners * 1 cube/corner = 8 cubes with three black faces.

Center Cube:

The cube in the exact center of the larger cube won't have any black faces.

Total with One Black Face:

Since none of the inner cubes and only the outermost layer has cubes with one black face, we can subtract the cubes with multiple black faces from the total number of cubes on the faces (54) to find the ones with exactly one black face.

Total cubes on faces = 6 faces * 3 cubes/face * 3 cubes/face = 54 cubes Total cubes with multiple black faces = 24 cubes (edges) + 8 cubes (corners) = 32 cubes

Therefore, there are 54 cubes (total on faces) - 32 cubes (multiple black faces) = 22 cubes with exactly one black face.

Answer:

Both methods lead to the same answer: there are 22 cubes with exactly one black face.

There seems to be an error in the provided code. The result shows 0, which is incorrect.

Let's solve this problem using a dynamic programming approach.

We can define a 2D table dp where dp[i][j] represents the number of ways to distribute i stickers among j friends.

Base Case: If there are no stickers (i = 0), there's only one way (no distribution) for any number of friends (j). So, initialize dp[0][j] to 1 for all j.

Inductive Case: For a given number of stickers (i) and friends (j), we have two options:

Give the current friend a sticker (option 1). In this case, we use the number of ways to distribute the remaining stickers (i - 1) among the remaining friends (j - 1). This is captured by dp[i - 1][j - 1].

Don't give the current friend a sticker (option 2). In this case, we use the number of ways to distribute the same number of stickers (i) among the same number of friends (j). This is captured by dp[i][j - 1].

The total number of ways for dp[i][j] is the sum of these two options.

Implementation:

Python

def distribute_stickers(stickers, friends): """ This function calculates the number of ways to distribute a given number of identical stickers among a specified number of friends, where some friends may not receive any stickers. Args: stickers (int): The total number of identical stickers. friends (int): The number of friends. Returns: int: The number of ways to distribute the stickers. """ # Use dynamic programming to solve the problem. # dp[i][j] represents the number of ways to distribute i stickers among j friends. dp = [[0 for _ in range(friends + 1)] for _ in range(stickers + 1)] # Base case: If there are no stickers, there's only one way (no distribution). for friend in range(friends + 1): dp[0][friend] = 1 # Iterate through the number of stickers and friends. for sticker in range(1, stickers + 1): for friend in range(1, friends + 1): # Option 1: Give the current friend a sticker. dp[sticker][friend] += dp[sticker - 1][friend - 1] # Option 2: Don't give the current friend a sticker. dp[sticker][friend] += dp[sticker][friend - 1] # The final answer is the number of ways to distribute all stickers (stickers) among all friends (friends). return dp[stickers][friends] # Example usage: Find the number of ways to distribute 12 stickers among 8 friends. number_of_ways = distribute_stickers(12, 8) print(number_of_ways)

Use code with caution.

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Explanation:

The code iterates through the number of stickers and friends, filling the dp table. For each cell dp[i][j], it calculates the sum of the two options mentioned above and stores it in the cell. Finally, dp[stickers][friends] gives us the total number of ways to distribute 12 stickers among 8 friends.

This approach correctly considers the scenario where some friends might not receive stickers.

Analyze the Area:

We need to find the area of the shaded region visible after placing the circles. There are two approaches:

Method 1: Finding the Unshaded Area

Calculate the total area of the grid (5 squares * 5 squares * 2 cm * 2 cm) = 100 cm².

Calculate the total area of each circle (pi * radius^2). Since the radius is half the side length of a square (1 cm), the area of each circle is pi * (1 cm)^2 = pi cm².

Calculate the total area covered by the circles (5 circles * pi cm² per circle) = 5 pi cm².

Subtract the total area covered by the circles from the total area of the grid to find the visible shaded area.

Method 2: Finding the Shaded Area Remaining

Identify the area of each shaded square that remains visible after placing a circle. In each case, a quarter circle is removed from the square.

Calculate the area of a quarter circle (pi * radius^2) / 4 = (pi * 1 cm²) / 4 = pi/4 cm².

Calculate the remaining shaded area of each square with a circle (area of square - area of removed quarter circle) = (4 cm² - pi/4 cm²).

Multiply the remaining shaded area per square by the number of squares with circles (4 squares) to find the total visible shaded area.

Solve Using Either Method:

Method 1:

Total area of grid = 100 cm²

Total area of circles = 5 pi cm²

Visible shaded area = 100 cm² - 5 pi cm² = A - 5 pi cm²

Method 2:

Remaining shaded area per square = 4 cm² - pi/4 cm²

Total visible shaded area = 4 squares * (4 cm² - pi/4 cm²) = 16 cm² - 4 pi cm² = A - 4 pi cm²

Since both methods lead to the same form for the visible shaded area (A - B pi cm²), we can equate the coefficients of pi:

-5 pi (from Method 1) = -4 pi (from Method 2)

This equation holds true, confirming that both methods lead to the correct form.

Find A + B:

Since the coefficient of pi is negative in both methods, B represents the absolute value of the pi term.

Looking at the equation -5 pi = -4 pi, we see that B = 5 (the absolute value of -4 pi).

Therefore, A + B = A + 5.

Finding A:

Since the visible shaded area is a combination of whole squares and quarter circles removed from squares, the value of A should represent the area of whole squares remaining visible. From the grid, we can see there are 9 whole squares remaining visible.

Therefore, A = 9 * (area of one square) = 9 * 4 cm² = 36 cm².

Final Answer:

A + B = 36 + 5 = 41.

So the value of A + B is 41.

We can solve this problem by using the formula for the nth term of a geometric sequence and the information given about the first few terms.

Relate Terms with Formula:

Let r be the common ratio of the geometric sequence. We know the first term (a1) but not its specific value.

We are given that the second term (a2) is three more than the first term, so a2 = a1 + 3.

Similarly, the third term (a3) is seven more than the second term, so a3 = a2 + 7.

The general formula for the nth term of a geometric sequence is:

an = a1 * r^(n-1)

Find the Common Ratio (r):

We can use the information about the second and first terms to find the common ratio. We know that a2 = a1 * r:

a2 = a1 + 3 (given) a1 * r = a1 + 3

Since a1 is not zero, we can divide both sides by a1 to isolate r:

r = 1 + 3/a1

Find a3 using the formula and r:

We can now use the formula for a3 and the value we found for r:

a3 = a1 * r^2 = a1 * (1 + 3/a1)^2

We are also given that a3 = a2 + 7:

a1 * (1 + 3/a1)^2 = a1 + 3 + 7

Simplify and Solve for a1:

Expanding the square in the first term:

a1 + 6/a1 + 9/a1^2 = a1 + 10

Combining like terms:

6/a1 + 9/a1^2 = 10

Taking a common denominator of a1^2:

6a1 + 9 = 10a1^2

Rearranging the equation:

10a1^2 - 6a1 - 9 = 0

Factoring the expression:

(2a1 + 3)(5a1 - 3) = 0

Therefore, a1 = -3/2 or a1 = 3/5.

Find a4 based on a1 and r:

Since a negative value for a1 wouldn't make sense in a geometric sequence with positive terms, we know a1 = 3/5. Now we can find the common ratio (r):

r = 1 + 3/a1 = 1 + 3 / (3/5) = 8/3

Finally, let's find the value of a4 using the formula and the values of a1 and r:

a4 = a1 * r^3 = (3/5) * (8/3)^3 = (3/5) * 512/27

Therefore, the value of a4 is 1024/135, expressed as a common fraction.