💡Alright, so I have this problem about identifying which of the given functions is an "ideal function." The definition of an "ideal function" is given as a function that satisfies two conditions:1. For any ( x ) in the domain, ( f(x) + f(-x) = 0 ). This means the function is odd.2. For any ( x_1, x_2 ) in the domain where ( x_1 neq x_2 ), the inequality ( frac{f(x_1) - f(x_2)}{x_1 - x_2} < 0 ) holds. This implies that the function is strictly decreasing.So, I need to check each of the three given functions against these two criteria.**Function (1): ( f(x) = frac{1}{x} )**First, let's check if it's an odd function. An odd function satisfies ( f(-x) = -f(x) ).Calculating ( f(-x) ):[ f(-x) = frac{1}{-x} = -frac{1}{x} = -f(x) ]So, yes, it is an odd function.Next, I need to check if it's strictly decreasing. A function is strictly decreasing if, for any two points ( x_1 ) and ( x_2 ) where ( x_1 < x_2 ), ( f(x_1) > f(x_2) ).Let's take two positive numbers, say ( x_1 = 1 ) and ( x_2 = 2 ):[ f(1) = 1 ][ f(2) = frac{1}{2} ]Here, ( f(1) > f(2) ), which is consistent with being decreasing.But wait, let's take two negative numbers, say ( x_1 = -2 ) and ( x_2 = -1 ):[ f(-2) = -frac{1}{2} ][ f(-1) = -1 ]Here, ( f(-2) = -frac{1}{2} ) is greater than ( f(-1) = -1 ), which also seems decreasing.However, let's consider ( x_1 = -1 ) and ( x_2 = 1 ):[ f(-1) = -1 ][ f(1) = 1 ]Here, ( f(-1) = -1 < f(1) = 1 ). But in terms of the difference quotient:[ frac{f(x_1) - f(x_2)}{x_1 - x_2} = frac{-1 - 1}{-1 - 1} = frac{-2}{-2} = 1 > 0 ]This contradicts the strictly decreasing condition because the difference quotient is positive. Therefore, ( f(x) = frac{1}{x} ) is not strictly decreasing over its entire domain. It is decreasing on each interval separately (positive and negative), but not across the entire domain.So, function (1) fails the second condition.**Function (2): ( f(x) = x + 1 )**First, check if it's an odd function:[ f(-x) = -x + 1 ]Compare this to ( -f(x) = -(x + 1) = -x - 1 )Clearly, ( f(-x) neq -f(x) ) because ( -x + 1 neq -x - 1 ). So, it's not an odd function.Since it fails the first condition, we don't need to check the second condition. But just for thoroughness, let's see if it's strictly decreasing.Compute the difference quotient:[ frac{f(x_1) - f(x_2)}{x_1 - x_2} = frac{(x_1 + 1) - (x_2 + 1)}{x_1 - x_2} = frac{x_1 - x_2}{x_1 - x_2} = 1 ]Which is positive, so the function is actually strictly increasing, not decreasing.Thus, function (2) fails both conditions.**Function (3): ( f(x) = begin{cases} -x^2 & text{if } x geq 0 x^2 & text{if } x < 0 end{cases} )**First, check if it's an odd function. Let's compute ( f(-x) ):- If ( x geq 0 ), then ( -x leq 0 ), so ( f(-x) = (-x)^2 = x^2 ). But ( f(x) = -x^2 ), so ( f(-x) = x^2 = -f(x) ) because ( -f(x) = -(-x^2) = x^2 ). - If ( x < 0 ), then ( -x > 0 ), so ( f(-x) = -(-x)^2 = -x^2 ). But ( f(x) = x^2 ), so ( f(-x) = -x^2 = -f(x) ).In both cases, ( f(-x) = -f(x) ), so it is an odd function.Next, check if it's strictly decreasing. Let's consider different cases:1. **Both ( x_1 ) and ( x_2 ) are non-negative (( x_1, x_2 geq 0 ))**: - ( f(x) = -x^2 ) - The derivative ( f'(x) = -2x ), which is negative for ( x > 0 ). So, it's decreasing on ( x geq 0 ).2. **Both ( x_1 ) and ( x_2 ) are negative (( x_1, x_2 < 0 ))**: - ( f(x) = x^2 ) - The derivative ( f'(x) = 2x ), which is negative for ( x < 0 ) because ( x ) is negative. So, it's decreasing on ( x < 0 ).3. **One is non-negative and the other is negative**: - Let ( x_1 geq 0 ) and ( x_2 < 0 ). Without loss of generality, assume ( x_1 > x_2 ). - ( f(x_1) = -x_1^2 ) and ( f(x_2) = x_2^2 ). - Compute the difference quotient: [ frac{f(x_1) - f(x_2)}{x_1 - x_2} = frac{-x_1^2 - x_2^2}{x_1 - x_2} ] Since ( x_1 > x_2 ), the denominator is positive. The numerator is ( -x_1^2 - x_2^2 ), which is negative because both ( x_1^2 ) and ( x_2^2 ) are positive. So, the entire expression is negative, satisfying the strictly decreasing condition.Therefore, function (3) satisfies both conditions and is an "ideal function."**Conclusion**:Only function (3) meets both criteria of being an odd function and strictly decreasing. Functions (1) and (2) fail at least one of the conditions.**Final Answer**boxed{(3)}

💡Okay, so I have this geometry problem here about finding the angle between edge AB and face ACD in a trihedral angle ABCD with vertex A. The given angles are BAC is 45 degrees, CAD is 90 degrees, and BAD is 60 degrees. Hmm, trihedral angle... I think that means three edges meeting at a common vertex, so A is the vertex where edges AB, AC, and AD meet.Alright, so I need to find the angle between edge AB and face ACD. I remember that the angle between a line and a plane is the smallest angle between the line and its projection onto the plane. So, maybe I should find the projection of AB onto the plane ACD and then find the angle between AB and its projection.Let me try to visualize this. Vertex A is connected to B, C, and D. The angles between these edges are given: BAC is 45°, CAD is 90°, and BAD is 60°. So, angle BAC is between edges AB and AC, angle CAD is between edges AC and AD, and angle BAD is between edges AB and AD.I think I can model this in 3D coordinate system to make it easier. Let's place point A at the origin (0,0,0). Then, let me assign coordinates to points B, C, and D such that the given angles are satisfied.Since angle CAD is 90°, that means edges AC and AD are perpendicular. So, if I place point C along the x-axis and point D along the y-axis, then AC and AD will be perpendicular. Let's say AC is along the x-axis, so point C is (c, 0, 0), and AD is along the y-axis, so point D is (0, d, 0). Now, point B is somewhere in 3D space. Since angle BAC is 45°, the angle between AB and AC is 45°, and angle BAD is 60°, the angle between AB and AD is 60°. So, point B must be somewhere above the xy-plane, right? Because if it were in the xy-plane, the angles would be different.Let me assign coordinates to point B as (x, y, z). Then, vectors AB, AC, and AD can be represented as:- AB = (x, y, z)- AC = (c, 0, 0)- AD = (0, d, 0)Now, the angle between AB and AC is 45°, so the dot product formula gives:AB · AC = |AB| |AC| cos(45°)Which translates to:x*c + y*0 + z*0 = sqrt(x² + y² + z²) * sqrt(c²) * (sqrt(2)/2)Simplifying:x*c = sqrt(x² + y² + z²) * c * (sqrt(2)/2)Divide both sides by c:x = sqrt(x² + y² + z²) * (sqrt(2)/2)Let me denote |AB| as sqrt(x² + y² + z²) = L. So,x = L * (sqrt(2)/2)Similarly, the angle between AB and AD is 60°, so:AB · AD = |AB| |AD| cos(60°)Which is:x*0 + y*d + z*0 = sqrt(x² + y² + z²) * sqrt(d²) * (1/2)Simplifying:y*d = L * d * (1/2)Divide both sides by d:y = L * (1/2)So, now I have expressions for x and y in terms of L:x = (sqrt(2)/2) Ly = (1/2) LNow, since point B is (x, y, z), and we know x and y in terms of L, we can express z in terms of L as well because |AB| = L:x² + y² + z² = L²Substituting x and y:[(sqrt(2)/2 L)]² + [(1/2 L)]² + z² = L²Calculating each term:(sqrt(2)/2 L)² = (2/4) L² = (1/2) L²(1/2 L)² = (1/4) L²So,(1/2) L² + (1/4) L² + z² = L²Adding the fractions:(3/4) L² + z² = L²Subtracting (3/4) L² from both sides:z² = (1/4) L²Taking square root:z = (1/2) LSo, point B has coordinates:x = (sqrt(2)/2) Ly = (1/2) Lz = (1/2) LNow, I need to find the angle between edge AB and face ACD. As I thought earlier, this angle is the angle between AB and its projection onto face ACD.Since face ACD is the plane containing points A, C, and D, which in my coordinate system is the xy-plane (since C is (c,0,0) and D is (0,d,0)). So, the projection of AB onto face ACD would be the shadow of AB on the xy-plane, which is just the vector (x, y, 0).So, the projection of AB onto ACD is (x, y, 0). The angle between AB and its projection is the angle we need.The formula for the angle θ between a vector and its projection is:sin(θ) = (|component perpendicular to the plane|) / (|original vector|)In this case, the component perpendicular to the plane ACD is the z-component of AB, which is z.So,sin(θ) = z / |AB|We have z = (1/2) L and |AB| = L, so:sin(θ) = (1/2 L) / L = 1/2Therefore,θ = arcsin(1/2) = 30°So, the angle between edge AB and face ACD is 30 degrees.Wait, let me double-check. I used the projection method, considering the z-component as the height. Since the face ACD is the xy-plane, the projection of AB is indeed (x, y, 0), and the angle between AB and its projection is determined by the z-component. So, yes, sin(theta) = z / |AB|, which gives 30 degrees. That seems right.Alternatively, I could think about the dihedral angle between the planes, but I think the projection method is more straightforward here. Yeah, I think 30 degrees is correct.

💡Okay, so I have this problem here about sets A and B, and I need to find two things. First, when m is 2, find the intersection of A with the complement of B. Second, find the range of m such that this intersection is empty. Hmm, let me try to break this down step by step.Starting with part (1): When m is 2, I need to find A ∩ (complement of B). Let me recall what each set is. The universal set U is all real numbers, so that's straightforward. Set A is {x | 1 < x < 4}, which is just the open interval from 1 to 4. Set B is defined as {x | x ≤ 3m - 4 or x ≥ 8 + m}. Since m is given as 2 here, I can substitute that into B.So, substituting m = 2 into B: 3m - 4 becomes 3*2 - 4 = 6 - 4 = 2, and 8 + m becomes 8 + 2 = 10. Therefore, B becomes {x | x ≤ 2 or x ≥ 10}. That means B is the union of two intervals: (-∞, 2] and [10, ∞). Now, the complement of B with respect to U would be all real numbers not in B. Since B is (-∞, 2] ∪ [10, ∞), the complement would be (2, 10). So, complement of B is the open interval from 2 to 10.Now, I need to find the intersection of A and the complement of B. A is (1, 4) and the complement of B is (2, 10). The intersection of these two intervals would be where they overlap. So, (1, 4) intersected with (2, 10) is (2, 4). That makes sense because from 2 to 4, both intervals overlap.So, for part (1), the answer should be the interval (2, 4). I think that's straightforward.Moving on to part (2): If A ∩ (complement of B) is empty, find the range of m. Hmm, okay. So, when is the intersection of A and the complement of B empty? That would mean that there are no elements in A that are also in the complement of B. In other words, every element of A is either in B or not in the complement of B.But wait, the complement of B is the set of x such that 3m - 4 < x < 8 + m. So, if A ∩ (complement of B) is empty, that means A is entirely contained within B. Because if there were any x in A that's also in the complement of B, then the intersection wouldn't be empty. So, for the intersection to be empty, A must be entirely within B.So, A is (1, 4). B is (-∞, 3m - 4] ∪ [8 + m, ∞). So, for A to be entirely within B, the entire interval (1, 4) must lie within either (-∞, 3m - 4] or [8 + m, ∞). But wait, since (1, 4) is between 1 and 4, and B is split into two intervals, one going to negative infinity and the other starting at 8 + m, which is greater than 8 since m is less than 6, so 8 + m is less than 14.Wait, actually, m is less than 6, so 8 + m is less than 14, but 3m - 4, when m is less than 6, 3m - 4 is less than 14 as well. Hmm, maybe I need to think differently.Wait, no. Let me think again. For A to be entirely within B, every x in A must be in B. So, for every x in (1, 4), x must satisfy x ≤ 3m - 4 or x ≥ 8 + m. But since x is between 1 and 4, the only way for x to satisfy x ≤ 3m - 4 is if 3m - 4 is greater than or equal to 4. Because if 3m - 4 is greater than or equal to 4, then all x in (1, 4) would be less than or equal to 3m - 4, hence in B.Alternatively, if 8 + m is less than or equal to 1, then all x in (1, 4) would be greater than or equal to 8 + m, hence in B. But 8 + m ≤ 1 would mean m ≤ -7, which is possible since m is a real number less than 6.So, there are two cases:1. 3m - 4 ≥ 42. 8 + m ≤ 1Let me solve these inequalities.First case: 3m - 4 ≥ 4Add 4 to both sides: 3m ≥ 8Divide by 3: m ≥ 8/3 ≈ 2.666...Second case: 8 + m ≤ 1Subtract 8: m ≤ -7So, the values of m that satisfy either m ≥ 8/3 or m ≤ -7 would make A entirely within B, hence making A ∩ (complement of B) empty.But wait, the problem states that m < 6. So, in the first case, m ≥ 8/3 and m < 6. In the second case, m ≤ -7, which is already less than 6. So, combining these, the range of m is m ≤ -7 or 8/3 ≤ m < 6.I should check if these are the only possibilities. If 3m - 4 is less than 4 and 8 + m is greater than 1, then there might be some overlap where A is partially in B and partially in the complement. So, to ensure that A is entirely within B, we must have either 3m - 4 ≥ 4 or 8 + m ≤ 1.Yes, that seems correct. So, the range of m is m ≤ -7 or 8/3 ≤ m < 6.I think that's the answer for part (2). Let me just recap:- For part (1), when m=2, B becomes (-∞, 2] ∪ [10, ∞), so complement of B is (2,10). Intersection with A=(1,4) is (2,4).- For part (2), to have A ∩ (complement of B) empty, A must be entirely within B. This happens when either 3m -4 ≥4 (so m ≥8/3) or 8 + m ≤1 (so m ≤-7). Since m <6, the upper bound is 6.Yeah, that seems solid. I don't think I missed anything here.**Final Answer**(1) boxed{{x mid 2 < x < 4}}(2) boxed{m leq -7 text{ or } frac{8}{3} leq m < 6}

💡To determine the greatest number of text{X}'s that can be placed in a 3 times 3 grid without getting three text{X}'s in a row vertically, horizontally, or diagonally, we can follow these steps:1. **Understanding the Grid**: The grid is a 3 times 3 square grid, which means there are 9 small squares in total.2. **Objective**: To place the maximum number of text{X}'s such that no three text{X}'s are aligned vertically, horizontally, or diagonally.3. **Using the Pigeonhole Principle**: If we place 5 or more text{X}'s on the grid, by the Pigeonhole Principle, at least one row, column, or diagonal must contain at least lceil frac{5}{3} rceil = 2 text{X}'s. This would mean placing 5 text{X}'s could potentially align three text{X}'s in a line in some direction.4. **Testing with 4 text{X}'s**: Attempt to place 4 text{X}'s while avoiding any three in a line. One strategy is to avoid placing text{X}'s in any main diagonal completely, or ensure no three align in rows or columns: - Place text{X}'s in positions: (1,1), (1,3), (3,1), (3,3).5. **Verifying the Arrangement**: Check each row, column, and diagonal: - Rows and Columns: No row or column has more than one text{X}. - Diagonals: No diagonal has more than two text{X}'s.6. **Conclusion**: Since placing 4 text{X}'s is possible without aligning three in a line and placing 5 or more text{X}'s could align three in a line, the maximum number of text{X}'s that can be placed is 4.The final answer is boxed{C}.

💡First, I identify the given values: the divisor is 17, the quotient is 4, and the remainder is 8.Next, I use the formula for finding the dividend, which is:Dividend = (Divisor × Quotient) + RemainderPlugging in the given values:Dividend = (17 × 4) + 8I calculate the multiplication:17 × 4 = 68Then, I add the remainder:68 + 8 = 76Therefore, the dividend is 76.

💡First, identify the prime numbers on an 8-sided die, which are 2, 3, 5, and 7. These are the winning numbers.Next, recognize that rolling a 1, 4, or 6 results in winning nothing, and rolling an 8 results in losing 4.Each outcome has an equal probability of 1/8.Calculate the expected value by multiplying each outcome by its probability and summing the results:- For each prime number: (1/8) * (2 + 3 + 5 + 7) = 17/8- For rolling 1, 4, or 6: 3 * (1/8) * 0 = 0- For rolling 8: (1/8) * (-4) = -4/8Combine these to find the total expected value: 17/8 - 4/8 = 13/8 = 1.625Finally, round to the nearest cent to get the expected value of 1.63.