💡First, I need to determine the total number of integers from 1 to 121, which is straightforwardly 121.Next, I'll find how many of these numbers are divisible by 5. By dividing 121 by 5 and taking the floor of the result, I get 24 numbers divisible by 5.Similarly, I'll calculate the numbers divisible by 7 by dividing 121 by 7 and taking the floor, resulting in 17 numbers.However, some numbers are divisible by both 5 and 7, which means they are divisible by 35. Dividing 121 by 35 and taking the floor gives me 3 such numbers.Using the principle of inclusion and exclusion, I can find the total number of numbers divisible by either 5 or 7 by adding the numbers divisible by 5 and 7 and then subtracting those divisible by both: 24 + 17 - 3 = 38.Finally, to find the numbers not divisible by either 5 or 7, I subtract the 38 numbers from the total 121, which gives me 83.

💡First, calculate the original discount with six friends: 6 friends × 50 discount per person = 300 discount.Subtract the discount from the base price: 1700 - 300 = 1400.Subtract the car wash earnings: 1400 - 500 = 900.Divide the remaining amount equally among the six friends: 900 ÷ 6 = 150 per person.Now, with Brad not participating, there are five friends. Calculate the new discount: 5 friends × 50 discount per person = 250.Subtract the new discount from the base price: 1700 - 250 = 1450.Subtract the car wash earnings: 1450 - 500 = 950.Divide the remaining amount equally among the five friends: 950 ÷ 5 = 190 per person.Finally, find the difference in what each friend has to pay: 190 - 150 = 40.

💡Alright, so I have this problem about three brothers—A, B, and C—who each donated different amounts to the "Light Angel" fund. The donations were 240,000, 250,000, and 260,000 yuan. Each brother made a statement during an interview, and I need to figure out how much B donated based on their statements.First, let me list out the donations and the statements:Donations: 240,000; 250,000; 260,000.Statements:- A says: "B did not donate the least."- B says: "A donated more than C."- C says: "If I donated the least, then A did not donate the most."Okay, so I need to figure out who donated which amount. Since each brother donated a different amount, there are six possible permutations of who donated what. But maybe I can narrow it down using their statements.Let me consider each statement one by one and see how they can help me eliminate possibilities.Starting with A's statement: "B did not donate the least." So, B is not the one who donated 240,000. That means B either donated 250,000 or 260,000.Next, B's statement: "A donated more than C." So, A's donation is greater than C's donation. That means A is not the one who donated the least, and C is not the one who donated the most.Lastly, C's statement: "If I donated the least, then A did not donate the most." This is a conditional statement. In logical terms, it's like saying "If P, then Q," where P is "C donated the least" and Q is "A did not donate the most." So, if C donated the least (240,000), then A did not donate the most (260,000). But if C didn't donate the least, the statement doesn't necessarily give us any information.Okay, so let's try to piece this together.From A's statement, B is not the least, so B is either 250,000 or 260,000.From B's statement, A > C. So, A is not the least, and C is not the most.From C's statement, if C is the least, then A is not the most. So, if C is 240,000, then A is not 260,000. That would mean A is 250,000, and B would have to be 260,000.But let's see if that works.Assume C donated the least: 240,000.Then, according to C's statement, A did not donate the most, so A is 250,000.Then, B would have to be 260,000.Now, let's check A's statement: "B did not donate the least." In this case, B donated 260,000, which is not the least, so A's statement holds.B's statement: "A donated more than C." A donated 250,000 and C donated 240,000, so yes, A donated more than C. That works.C's statement: "If I donated the least, then A did not donate the most." Since C donated the least, A did not donate the most, which is true because A donated 250,000, not 260,000. So, all statements hold in this scenario.But let's check if there are other possibilities.What if C didn't donate the least? Then C's statement doesn't give us any information, because the "if" part is false, and the statement is automatically true regardless of the "then" part.So, if C didn't donate the least, then someone else donated the least. From A's statement, B is not the least, so the least must be A or C. But if C didn't donate the least, then A must have donated the least.So, if A donated the least: 240,000.Then, from B's statement: "A donated more than C." But if A donated 240,000, which is the least, then C must have donated less than A, which is impossible because A already donated the least. So, this leads to a contradiction.Therefore, C must have donated the least, which is 240,000.So, the only possible scenario is:C: 240,000A: 250,000B: 260,000Let me double-check all statements:A: "B did not donate the least." B donated 260,000, which is not the least. True.B: "A donated more than C." A donated 250,000, C donated 240,000. True.C: "If I donated the least, then A did not donate the most." C donated the least, and A did not donate the most (A donated 250,000, not 260,000). True.Everything checks out. So, B donated 260,000 yuan.

💡First, I need to determine the total sum of the original 50 observations. Since the mean was 40, the total sum is 50 multiplied by 40, which equals 2000.Next, after correcting the error, the new mean is 40.66. Therefore, the new total sum of the observations is 50 multiplied by 40.66, which equals 2033.The difference between the new sum and the original sum is 2033 minus 2000, which equals 33. This difference indicates how much the incorrect observation was increased to correct the error.Since the correct observation should have been 45, the wrong value must have been 45 minus the difference of 33. Therefore, the wrong value of the observation was 12.

💡First, I need to understand the dimensions of the rectangular wooden block, which are 3 inches long, 4 inches wide, and 2 inches high. The block is painted blue on all six faces and then cut into 1-inch cubes. My goal is to determine how many of these small cubes have an even number of blue faces.To approach this, I'll categorize the cubes based on their positions within the larger block:1. **Corner Cubes**: These are the cubes located at the eight corners of the block. Each corner cube will have three blue faces because they are exposed on three sides.2. **Edge Cubes (excluding corners)**: These cubes are along the edges of the block but not at the corners. Each edge cube will have two blue faces. I'll calculate the number of edge cubes along each dimension: - Along the length (3 inches): There are (3 - 2) × 2 = 2 edge cubes. - Along the width (4 inches): There are (4 - 2) × 2 = 4 edge cubes. - Along the height (2 inches): There are (2 - 2) × 2 = 0 edge cubes.3. **Face Cubes (excluding edges and corners)**: These cubes are on the faces of the block but not on the edges or corners. Each face cube will have one blue face. I'll calculate the number of face cubes on each face: - Top and bottom faces: There are (3 - 2) × (4 - 2) × 2 = 4 face cubes. - Front and back faces: There are (4 - 2) × (2 - 2) × 2 = 0 face cubes. - Left and right faces: There are (3 - 2) × (2 - 2) × 2 = 0 face cubes.4. **Interior Cubes**: These cubes are completely inside the block and not on any face. They will have no blue faces. I'll calculate the number of interior cubes: - There are (3 - 2) × (4 - 2) × (2 - 2) = 0 interior cubes.Now, I'll summarize the counts:- **Corner Cubes**: 8 cubes with 3 blue faces (odd).- **Edge Cubes**: 6 cubes with 2 blue faces (even).- **Face Cubes**: 4 cubes with 1 blue face (odd).- **Interior Cubes**: 0 cubes with 0 blue faces (even).Finally, I'll count the cubes with an even number of blue faces:- Edge Cubes: 6 cubes.- Interior Cubes: 0 cubes.Therefore, the total number of cubes with an even number of blue faces is 6.

💡Okay, so I have this problem where there's a point inside an equilateral triangle with side length ( d ), and the distances from this point to the three vertices are 3, 4, and 5 units. I need to find the side length ( d ). Hmm, that seems a bit tricky, but let me try to figure it out step by step.First, I know that in an equilateral triangle, all sides are equal, and all angles are 60 degrees. The point inside the triangle is not one of the vertices, so it's somewhere in the middle. The distances from this point to each vertex are given as 3, 4, and 5. That reminds me of the Pythagorean theorem because 3, 4, 5 is a Pythagorean triple. Maybe that can help somehow.I remember that in some geometry problems, especially those involving equilateral triangles, rotating the triangle can help find relationships between points. Maybe I can try rotating the triangle and see if that gives me any useful information.Let me visualize the equilateral triangle ( ABC ) with side length ( d ). Let's say the point inside is ( P ), and the distances are ( PA = 3 ), ( PB = 4 ), and ( PC = 5 ). If I rotate the triangle 60 degrees around one of its vertices, say ( C ), then point ( A ) will move to a new position ( A' ), and point ( P ) will move to a new position ( P' ). Since it's a 60-degree rotation, the distance from ( P ) to ( C ) remains the same, so ( CP = CP' = 5 ).Now, triangle ( PCP' ) should also be equilateral because all sides are equal (each is 5 units) and all angles are 60 degrees. That means ( PP' = 5 ) as well. So, the distance between ( P ) and ( P' ) is 5 units.Looking at triangle ( PAP' ), I know ( PA = 3 ), ( PP' = 5 ), and since ( A ) rotated to ( A' ), the distance ( PA' ) should be equal to ( PB = 4 ). Wait, is that right? Let me think. When I rotate ( A ) to ( A' ), the distance from ( P ) to ( A' ) should be the same as the distance from ( P ) to ( B ), which is 4. So, ( PA' = 4 ).So, triangle ( PAP' ) has sides of length 3, 4, and 5. That's a right triangle! Because ( 3^2 + 4^2 = 5^2 ). So, triangle ( PAP' ) is a right triangle with the right angle at ( A ). That means angle ( PAP' ) is 90 degrees.Now, considering the rotation, the angle between ( PA ) and ( PA' ) is 60 degrees because we rotated the triangle by 60 degrees. But wait, in triangle ( PAP' ), angle ( PAP' ) is 90 degrees, and the rotation angle is 60 degrees. How does that fit together?Maybe I need to consider the angles around point ( A ). The original angle at ( A ) in the equilateral triangle is 60 degrees. After rotation, the angle between ( PA ) and ( PA' ) is 60 degrees, but in triangle ( PAP' ), the angle at ( A ) is 90 degrees. Hmm, that seems conflicting. Maybe I need to think about the overall configuration.Alternatively, perhaps I should use coordinates to model this problem. Let me place the equilateral triangle in a coordinate system to make it easier. Let me set point ( A ) at ( (0, 0) ), point ( B ) at ( (d, 0) ), and point ( C ) at ( left( frac{d}{2}, frac{dsqrt{3}}{2} right) ). That way, the triangle is nicely positioned with base ( AB ) on the x-axis.Let the coordinates of point ( P ) be ( (x, y) ). Then, the distances from ( P ) to ( A ), ( B ), and ( C ) are given by:1. ( PA = sqrt{(x - 0)^2 + (y - 0)^2} = 3 )2. ( PB = sqrt{(x - d)^2 + (y - 0)^2} = 4 )3. ( PC = sqrt{left(x - frac{d}{2}right)^2 + left(y - frac{dsqrt{3}}{2}right)^2} = 5 )So, I have three equations:1. ( x^2 + y^2 = 9 ) (from PA = 3)2. ( (x - d)^2 + y^2 = 16 ) (from PB = 4)3. ( left(x - frac{d}{2}right)^2 + left(y - frac{dsqrt{3}}{2}right)^2 = 25 ) (from PC = 5)Now, I can try to solve these equations step by step.First, subtract equation 1 from equation 2:( (x - d)^2 + y^2 - (x^2 + y^2) = 16 - 9 )Simplify:( x^2 - 2xd + d^2 + y^2 - x^2 - y^2 = 7 )Which simplifies to:( -2xd + d^2 = 7 )Let me write that as:( -2xd + d^2 = 7 ) --> Equation 4Similarly, subtract equation 1 from equation 3:( left(x - frac{d}{2}right)^2 + left(y - frac{dsqrt{3}}{2}right)^2 - (x^2 + y^2) = 25 - 9 )Simplify:( x^2 - dx + frac{d^2}{4} + y^2 - dsqrt{3}y + frac{3d^2}{4} - x^2 - y^2 = 16 )Simplify further:( -dx + frac{d^2}{4} - dsqrt{3}y + frac{3d^2}{4} = 16 )Combine like terms:( -dx - dsqrt{3}y + d^2 = 16 )Let me write that as:( -dx - dsqrt{3}y + d^2 = 16 ) --> Equation 5Now, from Equation 4: ( -2xd + d^2 = 7 ), I can solve for ( x ):( -2xd = 7 - d^2 )( x = frac{d^2 - 7}{2d} )So, ( x = frac{d^2 - 7}{2d} ). Let's keep that in mind.Now, let's plug this expression for ( x ) into Equation 5:( -d left( frac{d^2 - 7}{2d} right ) - dsqrt{3}y + d^2 = 16 )Simplify term by term:First term: ( -d times frac{d^2 - 7}{2d} = -frac{d^2 - 7}{2} )Second term: ( -dsqrt{3}y )Third term: ( +d^2 )So, putting it all together:( -frac{d^2 - 7}{2} - dsqrt{3}y + d^2 = 16 )Multiply through by 2 to eliminate the denominator:( -(d^2 - 7) - 2dsqrt{3}y + 2d^2 = 32 )Simplify:( -d^2 + 7 - 2dsqrt{3}y + 2d^2 = 32 )Combine like terms:( d^2 + 7 - 2dsqrt{3}y = 32 )Subtract 7 from both sides:( d^2 - 2dsqrt{3}y = 25 )Let me solve for ( y ):( -2dsqrt{3}y = 25 - d^2 )( y = frac{d^2 - 25}{2dsqrt{3}} )So, ( y = frac{d^2 - 25}{2dsqrt{3}} ). Got that.Now, I have expressions for both ( x ) and ( y ) in terms of ( d ). Let's plug these back into Equation 1, which is ( x^2 + y^2 = 9 ).So, substitute ( x = frac{d^2 - 7}{2d} ) and ( y = frac{d^2 - 25}{2dsqrt{3}} ) into ( x^2 + y^2 = 9 ):First, compute ( x^2 ):( left( frac{d^2 - 7}{2d} right )^2 = frac{(d^2 - 7)^2}{4d^2} )Next, compute ( y^2 ):( left( frac{d^2 - 25}{2dsqrt{3}} right )^2 = frac{(d^2 - 25)^2}{12d^2} )So, adding them together:( frac{(d^2 - 7)^2}{4d^2} + frac{(d^2 - 25)^2}{12d^2} = 9 )To combine these fractions, let's find a common denominator, which is 12d^2:Multiply the first term by 3/3 and the second term by 1/1:( frac{3(d^2 - 7)^2}{12d^2} + frac{(d^2 - 25)^2}{12d^2} = 9 )Combine the numerators:( frac{3(d^2 - 7)^2 + (d^2 - 25)^2}{12d^2} = 9 )Multiply both sides by 12d^2:( 3(d^2 - 7)^2 + (d^2 - 25)^2 = 108d^2 )Now, let's expand the terms in the numerator:First, expand ( (d^2 - 7)^2 ):( (d^2 - 7)^2 = d^4 - 14d^2 + 49 )Multiply by 3:( 3d^4 - 42d^2 + 147 )Next, expand ( (d^2 - 25)^2 ):( (d^2 - 25)^2 = d^4 - 50d^2 + 625 )Now, add these two results together:( 3d^4 - 42d^2 + 147 + d^4 - 50d^2 + 625 = 4d^4 - 92d^2 + 772 )So, the equation becomes:( 4d^4 - 92d^2 + 772 = 108d^2 )Bring all terms to the left side:( 4d^4 - 92d^2 + 772 - 108d^2 = 0 )Combine like terms:( 4d^4 - 200d^2 + 772 = 0 )Divide the entire equation by 4 to simplify:( d^4 - 50d^2 + 193 = 0 )Hmm, this is a quartic equation, but it's quadratic in terms of ( d^2 ). Let me set ( u = d^2 ), so the equation becomes:( u^2 - 50u + 193 = 0 )Now, solve for ( u ) using the quadratic formula:( u = frac{50 pm sqrt{2500 - 4 times 1 times 193}}{2} )Calculate the discriminant:( 2500 - 772 = 1728 )So,( u = frac{50 pm sqrt{1728}}{2} )Simplify ( sqrt{1728} ):( sqrt{1728} = sqrt{144 times 12} = 12sqrt{12} = 12 times 2sqrt{3} = 24sqrt{3} )So,( u = frac{50 pm 24sqrt{3}}{2} = 25 pm 12sqrt{3} )Since ( u = d^2 ) must be positive, both solutions are positive, but we need to check which one makes sense in the context.If ( d^2 = 25 + 12sqrt{3} ), then ( d = sqrt{25 + 12sqrt{3}} ).If ( d^2 = 25 - 12sqrt{3} ), then ( d = sqrt{25 - 12sqrt{3}} ). But ( 12sqrt{3} ) is approximately 20.78, so ( 25 - 20.78 = 4.22 ), which is positive, so ( d ) would be about 2.05. However, considering the distances from the point inside the triangle are 3, 4, and 5, which are all larger than 2.05, that doesn't make sense because the side length must be larger than the distances from any interior point to the vertices. Therefore, ( d^2 = 25 + 12sqrt{3} ) is the correct solution.So, the side length ( d ) is ( sqrt{25 + 12sqrt{3}} ).Let me just double-check my steps to make sure I didn't make any mistakes. Starting from setting up the coordinates, subtracting equations to eliminate variables, solving for ( x ) and ( y ), plugging back into the first equation, expanding, and solving the quartic equation. It seems consistent. The key was realizing that the point inside the triangle creates a right triangle when considering the rotation, which helped set up the equations correctly. Also, recognizing that the side length must be larger than the given distances was crucial in choosing the correct solution from the quadratic.Yeah, I think that makes sense. So, the side length ( d ) is ( sqrt{25 + 12sqrt{3}} ).**Final Answer**The side length ( d ) is boxed{sqrt{25 + 12sqrt{3}}}.