💡Okay, so I've got this problem here: I need to figure out how many values of ( c ) in the interval ([0, 500]) make the equation ( 9 lfloor x rfloor + 3 lceil x rceil = c ) have a solution for ( x ). Hmm, let me break this down step by step.First, I remember that ( lfloor x rfloor ) is the floor function, which gives the greatest integer less than or equal to ( x ), and ( lceil x rceil ) is the ceiling function, which gives the smallest integer greater than or equal to ( x ). So, depending on whether ( x ) is an integer or not, these functions behave differently.Let me consider two cases: when ( x ) is an integer and when ( x ) is not an integer.**Case 1: ( x ) is an integer.**If ( x ) is an integer, then both ( lfloor x rfloor ) and ( lceil x rceil ) equal ( x ). So, the equation becomes:[9x + 3x = c]Simplifying that:[12x = c]So, ( c ) must be a multiple of 12. That means ( c ) can be 0, 12, 24, ..., up to the largest multiple of 12 less than or equal to 500.Let me figure out how many such multiples there are. The smallest multiple is 0, and the largest multiple less than or equal to 500 is 492 because ( 12 times 41 = 492 ) and ( 12 times 42 = 504 ), which is over 500. So, starting from 0, the number of multiples is 42 (since 41 intervals between 0 and 492, inclusive).Wait, actually, if we count 0 as the first term, then the number of terms is 42 because ( 492 / 12 = 41 ), so 41 intervals, meaning 42 terms. So, 42 values of ( c ) when ( x ) is an integer.**Case 2: ( x ) is not an integer.**If ( x ) is not an integer, then ( lceil x rceil = lfloor x rfloor + 1 ). Let me denote ( lfloor x rfloor ) as ( n ), where ( n ) is an integer. Then, ( lceil x rceil = n + 1 ).Substituting into the equation:[9n + 3(n + 1) = c]Simplify that:[9n + 3n + 3 = c][12n + 3 = c]So, ( c ) must be equal to ( 12n + 3 ). Now, ( n ) is an integer, so ( c ) will take values like 3, 15, 27, ..., up to the largest value less than or equal to 500.Let me find how many such values exist. The smallest ( c ) in this case is 3, and the largest is the largest number of the form ( 12n + 3 ) less than or equal to 500.Let me solve for ( n ) in ( 12n + 3 leq 500 ):[12n leq 497][n leq frac{497}{12} approx 41.4167]Since ( n ) must be an integer, the maximum ( n ) is 41. So, ( n ) can be from 0 to 41, inclusive. That gives 42 values (0 to 41).Therefore, in this case, there are also 42 values of ( c ).**Combining Both Cases:**So, from Case 1, we have 42 values of ( c ) that are multiples of 12, and from Case 2, we have 42 values of ( c ) that are of the form ( 12n + 3 ). These two sets of values don't overlap because one set is multiples of 12 and the other is 3 more than multiples of 12. So, there's no overlap between these two sets.Therefore, the total number of values of ( c ) is ( 42 + 42 = 84 ).Wait, but let me double-check. The first case gives ( c = 0, 12, 24, ..., 492 ), which is 42 terms. The second case gives ( c = 3, 15, 27, ..., 495 ). Let me confirm the last term in the second case: ( 12 times 41 + 3 = 492 + 3 = 495 ), which is indeed less than 500. So, that's correct.And the first case goes up to 492, which is less than 500, so that's also correct. So, no issues there.Therefore, combining both cases, the total number of ( c ) values is 84.**Wait a second**, but hold on. Let me think again about the first case. When ( x ) is an integer, ( c = 12x ). So, ( x ) can be 0, 1, 2, ..., up to ( x = 41 ) because ( 12 times 41 = 492 ), and ( 12 times 42 = 504 ) which is beyond 500. So, ( x ) can be 0 to 41, inclusive, which is 42 values.Similarly, in the second case, ( c = 12n + 3 ), where ( n ) is 0 to 41, inclusive, which is also 42 values. So, 42 + 42 = 84.But just to make sure, let me check if 495 is indeed within 500. Yes, 495 is less than 500, so that's fine. And 492 is also less than 500, so that's fine too.Is there any chance that some values of ( c ) might be counted twice? For example, is there a ( c ) that is both a multiple of 12 and of the form ( 12n + 3 )? Let's see: Suppose ( 12k = 12m + 3 ). Then, ( 12(k - m) = 3 ), which implies ( 4(k - m) = 1 ). But ( k ) and ( m ) are integers, so ( 4(k - m) = 1 ) has no integer solutions. Therefore, the two sets are disjoint. So, no overlap.Therefore, the total number is indeed 84.Wait, but hold on, let me think about the interval [0, 500]. So, 0 is included, and 500 is included. But in our calculations, the maximum ( c ) we have is 495 and 492, both less than 500. So, 500 itself is not achieved by either case. So, 500 is not included in the possible ( c ) values.Therefore, our count remains 84.Is there any other edge case I need to consider? For example, when ( x ) is exactly an integer, does that affect the ceiling function? But no, because when ( x ) is an integer, both floor and ceiling functions give ( x ), so the equation simplifies as I did earlier.Another thing: when ( x ) is not an integer, ( lfloor x rfloor ) is the integer part, and ( lceil x rceil ) is the next integer. So, the equation becomes ( 9n + 3(n + 1) = 12n + 3 ), which is correct.So, I think my reasoning is solid here. Therefore, the number of ( c ) values is 84.**Final Answer**The number of values of ( c ) is boxed{84}.

💡Okay, so I'm trying to figure out this probability problem. Let me read it again to make sure I understand it properly."A person is shooting with a total of 5 bullets. The shooting stops either when the target is hit or when all bullets have been fired. If the number of shots fired is denoted by ξ, then the experimental result represented by 'ξ=5' is (　　)"The options are:A: The target was hit on the 5th shotB: The target was not hit on the 5th shotC: The first 4 shots all missed the targetD: The target was hit on the 4th shotAlright, so ξ represents the number of shots fired. The shooting stops when either the target is hit or all bullets are fired. So, if ξ=5, that means all 5 bullets were fired, right? Because if the target was hit before the 5th shot, the shooting would have stopped earlier.So, if ξ=5, that implies that the target was not hit in the first 4 shots. Otherwise, the shooting would have stopped before the 5th shot. Therefore, the 5th shot must have been fired, but whether the target was hit on the 5th shot or not is not specified by ξ=5. ξ=5 just tells us that the shooting went all the way to the 5th bullet.Looking at the options:A says the target was hit on the 5th shot. But ξ=5 doesn't necessarily mean that; it could mean that the target was hit on the 5th shot, but it could also mean that the target wasn't hit at all in all 5 shots. So A is too specific.B says the target was not hit on the 5th shot. But again, ξ=5 doesn't specify whether the 5th shot hit or not. It just means that the shooting went to the 5th shot, which could be because the target was hit on the 5th shot or because all 5 shots missed.C says the first 4 shots all missed the target. That makes sense because if any of the first 4 shots had hit the target, the shooting would have stopped there, and ξ would be less than 5. So, for ξ to be 5, the first 4 shots must have all missed.D says the target was hit on the 4th shot. But if the target was hit on the 4th shot, the shooting would have stopped there, so ξ would be 4, not 5. So D is incorrect.Therefore, the correct answer should be C: The first 4 shots all missed the target.

💡First, let's denote the two numbers as ( x ) and ( y ).We know that the product of two numbers is equal to the product of their Least Common Multiple (LCM) and Highest Common Factor (HCF). This relationship can be expressed as:[x times y = text{LCM}(x, y) times text{HCF}(x, y)]Given that the LCM of the two numbers is 600 and their product is 18,000, we can substitute these values into the equation:[18,000 = 600 times text{HCF}(x, y)]To find the HCF, we can solve for it by dividing both sides of the equation by 600:[text{HCF}(x, y) = frac{18,000}{600} = 30]Therefore, the Highest Common Factor of the two numbers is 30.

💡Alright, so I have this problem where there's a square inscribed in a right triangle. The square is named BCFE, and it's inside the right triangle AGD. The diagram is the same as the previous one, which I guess means the setup is similar to a problem I've seen before. The given information is that AB is 36 units and CD is 72 units. I need to find the area of the square BCFE.First, let me visualize the problem. There's a right triangle AGD, with the right angle at G, I assume. The square BCFE is inscribed in this triangle, meaning all its sides touch the triangle. Points B and C are on the legs of the triangle, and points E and F are on the hypotenuse. So, the square is kind of snug inside the triangle.Given that AB is 36 units and CD is 72 units, I need to figure out how these lengths relate to the square. AB is probably a segment on one leg of the triangle, and CD is a segment on the other leg. Since the square is inscribed, the sides of the square will be parallel to the legs of the triangle.I remember that when a square is inscribed in a right triangle, it creates smaller similar triangles within the original triangle. So, the triangle above the square and the triangle to the side of the square are similar to the original triangle AGD. This similarity should help me set up proportions to find the side length of the square.Let me denote the side length of the square BCFE as x. Then, the segments AB and CD are parts of the legs of the original triangle AGD. Since AB is 36 units and CD is 72 units, these would be the lengths from the vertices A and D to the points where the square touches the legs.Now, considering the similar triangles, the triangle above the square (let's call it triangle AEB) should be similar to the original triangle AGD. Similarly, the triangle to the side of the square (triangle FDC) should also be similar to triangle AGD.Because of this similarity, the ratios of corresponding sides should be equal. For triangle AEB and triangle AGD, the ratio of AB to AG should be equal to the ratio of BE to GD. Similarly, for triangle FDC and triangle AGD, the ratio of CD to GD should be equal to the ratio of CF to AG.Wait, maybe I need to think about this more carefully. Let me try to write down the proportions.Let’s denote the legs of triangle AGD as AG and GD. Since it's a right triangle, AG and GD are perpendicular. The square BCFE has sides of length x, so BE and CF are both equal to x.Looking at triangle AEB, which is similar to triangle AGD, the sides are proportional. So, AB corresponds to AG, and BE corresponds to GD. Therefore, the ratio of AB to AG should be equal to the ratio of BE to GD.Similarly, looking at triangle FDC, which is also similar to triangle AGD, CD corresponds to GD, and CF corresponds to AG. So, the ratio of CD to GD should be equal to the ratio of CF to AG.But I don't know the lengths of AG and GD. Maybe I can express them in terms of x.Let’s assume that AG is the vertical leg and GD is the horizontal leg of the right triangle AGD. Then, the square BCFE is sitting inside, touching both legs and the hypotenuse.So, from point A, moving down AG, we have segment AB of length 36 units, then the square of height x, and then from the square to point G, which would be another segment. Similarly, from point D, moving along GD, we have segment CD of length 72 units, then the square of width x, and then from the square to point G, another segment.Wait, maybe it's better to think in terms of the entire legs. The total length of AG would be AB plus the height from B to G, which is the same as the side of the square, x. Similarly, the total length of GD would be CD plus the width from C to D, which is also x.But that might not be accurate because the square is inscribed, so the segments AB and CD are not necessarily aligned directly with the legs. Maybe I need to consider the similar triangles more carefully.Let me try to set up the proportions again. Since triangle AEB is similar to triangle AGD, the ratio of AB to AG is equal to the ratio of BE to GD. Similarly, the ratio of CD to GD is equal to the ratio of CF to AG.Let’s denote AG as the vertical leg and GD as the horizontal leg. Then, AB is a segment on AG, and CD is a segment on GD.So, AB = 36 units, which is part of AG, and CD = 72 units, which is part of GD.The square has side length x, so BE = x and CF = x.From triangle AEB ~ triangle AGD, we have:AB / AG = BE / GDSimilarly, from triangle FDC ~ triangle AGD, we have:CD / GD = CF / AGSo, substituting the known values:36 / AG = x / GDand72 / GD = x / AGNow, we have two equations:1) 36 / AG = x / GD2) 72 / GD = x / AGLet me write these as:36 / AG = x / GD --> Equation 172 / GD = x / AG --> Equation 2If I solve Equation 1 for x, I get x = (36 * GD) / AGSimilarly, from Equation 2, x = (72 * AG) / GDSo, setting these equal:(36 * GD) / AG = (72 * AG) / GDCross-multiplying:36 * GD^2 = 72 * AG^2Divide both sides by 36:GD^2 = 2 * AG^2So, GD = AG * sqrt(2)Now, let's go back to Equation 1:36 / AG = x / GDBut GD = AG * sqrt(2), so:36 / AG = x / (AG * sqrt(2))Simplify:36 / AG = x / (AG * sqrt(2))Multiply both sides by AG:36 = x / sqrt(2)So, x = 36 * sqrt(2)But wait, that would make the side length x = 36√2, and the area would be x^2 = (36√2)^2 = 36^2 * 2 = 1296 * 2 = 2592.Hmm, that seems familiar. Let me check if this makes sense.Alternatively, maybe I made a mistake in setting up the proportions. Let me try another approach.Let’s denote AG as a and GD as b. Then, the area of triangle AGD is (a*b)/2.The square BCFE has side length x, so its area is x^2.The square divides the triangle into two smaller similar triangles and the square. The areas of the smaller triangles can be expressed in terms of x.From the similarity, the ratio of the sides of the smaller triangles to the original triangle is the same.So, for triangle AEB, the sides are proportional to the original triangle. The base of triangle AEB is AB = 36, and the height is x. Similarly, the base of triangle FDC is CD = 72, and the height is x.But I think the proportions should relate the sides of the smaller triangles to the original triangle.Wait, maybe I should use the fact that the ratios of the areas of similar triangles are the squares of the ratios of their corresponding sides.But I'm not sure if that's necessary here. Let me try to express a and b in terms of x.From triangle AEB ~ triangle AGD, the ratio of similarity is AB / AG = 36 / a.Similarly, the ratio of BE / GD = x / b.Since the triangles are similar, these ratios are equal:36 / a = x / b --> Equation 1Similarly, from triangle FDC ~ triangle AGD, the ratio of CD / GD = 72 / b.And the ratio of CF / AG = x / a.So, 72 / b = x / a --> Equation 2Now, from Equation 1: 36 / a = x / b --> 36b = a xFrom Equation 2: 72 / b = x / a --> 72a = b xSo, we have:36b = a x --> Equation 172a = b x --> Equation 2Let me solve these equations simultaneously.From Equation 1: a = (36b) / xSubstitute a into Equation 2:72 * (36b / x) = b xSimplify:(72 * 36b) / x = b xMultiply both sides by x:72 * 36b = b x^2Divide both sides by b (assuming b ≠ 0):72 * 36 = x^2Calculate 72 * 36:72 * 36 = (70*36) + (2*36) = 2520 + 72 = 2592So, x^2 = 2592Therefore, the area of the square BCFE is 2592 square units.Wait, that's the same result as before. So, even though I approached it differently, I still got x^2 = 2592.I think this makes sense. The key was setting up the proportions based on similar triangles and solving for x^2 directly, which gives the area of the square.I don't see any mistakes in this reasoning, so I'm confident that the area of square BCFE is 2592 square units.

💡First, I need to understand the domains and ranges of the functions ( r(x) ) and ( s(x) ). The function ( r(x) ) has a domain of ({ -2, -1, 0, 1 }) and a range of ({ 1, 3, 5, 7 }). This means ( r(x) ) can output the values 1, 3, 5, or 7. On the other hand, ( s(x) ) has a domain of ({ 0, 1, 2, 3, 4, 5 }) and is defined by ( s(x) = 2x + 1 ).To find the possible values of ( s(r(x)) ), I need to ensure that the outputs of ( r(x) ) fall within the domain of ( s(x) ). From ( r(x) )'s range ({ 1, 3, 5, 7 }), the values 1, 3, and 5 are within the domain of ( s(x) ), while 7 is not. Therefore, the valid inputs for ( s(x) ) are 1, 3, and 5.Next, I calculate the corresponding outputs of ( s(x) ) for these valid inputs:- ( s(1) = 2 times 1 + 1 = 3 )- ( s(3) = 2 times 3 + 1 = 7 )- ( s(5) = 2 times 5 + 1 = 11 )Finally, I sum these outputs to find the total sum of all possible values of ( s(r(x)) ):- ( 3 + 7 + 11 = 21 )Therefore, the sum of all possible values of ( s(r(x)) ) is 21.

💡Okay, so I have this quadratic equation, right? It's x squared plus p times x plus q equals zero. And I need to find the values of p and q such that the equation has two distinct solutions, which are 2p and p plus q. Hmm, that sounds a bit tricky, but let me try to break it down.First, I remember that for a quadratic equation of the form ax squared plus bx plus c equals zero, the solutions can be found using the quadratic formula. But in this case, I know the solutions already: they are 2p and p plus q. So maybe I can use Vieta's formulas here. Vieta's formulas relate the coefficients of the polynomial to the sum and product of its roots. Yeah, that sounds right.So, for a quadratic equation ax squared plus bx plus c equals zero, the sum of the roots is negative b over a, and the product of the roots is c over a. In my equation, a is 1 because the coefficient of x squared is 1. So the sum of the roots should be negative p, and the product should be q.Let me write that down:Sum of roots: 2p + (p + q) = -pProduct of roots: 2p * (p + q) = qOkay, so starting with the sum of the roots. Let's compute that:2p + p + q = -pCombine like terms:3p + q = -pHmm, so if I subtract 3p from both sides, I get:q = -p - 3pWhich simplifies to:q = -4pAlright, so that's one equation: q equals negative four p. Now, let's move on to the product of the roots.Product of roots: 2p times (p + q) equals q.So, let's substitute q with -4p from the previous equation:2p * (p + (-4p)) = -4pSimplify inside the parentheses:2p * (-3p) = -4pMultiply:-6p squared equals -4pHmm, let's write that as:-6p² = -4pI can multiply both sides by -1 to make it positive:6p² = 4pNow, let's bring all terms to one side to form a quadratic equation:6p² - 4p = 0Factor out a common term, which is 2p:2p(3p - 2) = 0So, either 2p equals zero or 3p minus 2 equals zero.Case 1: 2p = 0Then p = 0.But wait, if p is zero, then the roots would be 2p = 0 and p + q = 0 + q = q. So both roots would be zero and q. But for the roots to be distinct, q must not be equal to zero. However, if p is zero, from our earlier equation q = -4p, q would be zero as well. So both roots would be zero, which are not distinct. Therefore, p = 0 is not a valid solution because it doesn't give us two distinct roots.Case 2: 3p - 2 = 0Then 3p = 2So p = 2/3Alright, so p is 2/3. Now, let's find q using q = -4p.q = -4*(2/3) = -8/3So q is -8/3.Let me double-check to make sure these values satisfy the original conditions.First, the quadratic equation becomes:x² + (2/3)x - 8/3 = 0Let's see if 2p and p + q are indeed the roots.2p = 2*(2/3) = 4/3p + q = (2/3) + (-8/3) = (2 - 8)/3 = -6/3 = -2So the roots should be 4/3 and -2.Let's plug x = 4/3 into the equation:(4/3)² + (2/3)*(4/3) - 8/3= 16/9 + 8/9 - 8/3Convert 8/3 to 24/9:16/9 + 8/9 - 24/9 = (16 + 8 - 24)/9 = 0/9 = 0Good, that works.Now, plug x = -2 into the equation:(-2)² + (2/3)*(-2) - 8/3= 4 - 4/3 - 8/3Convert 4 to 12/3:12/3 - 4/3 - 8/3 = (12 - 4 - 8)/3 = 0/3 = 0Perfect, that also works.Additionally, since the discriminant of the quadratic equation must be positive for the roots to be distinct, let's check that.The discriminant D is b² - 4ac.Here, a = 1, b = 2/3, c = -8/3.So D = (2/3)² - 4*(1)*(-8/3)= 4/9 + 32/3Convert 32/3 to 96/9:4/9 + 96/9 = 100/9Which is positive, so the roots are indeed distinct.Therefore, the values p = 2/3 and q = -8/3 satisfy the given conditions.