💡Okay, so I have this problem where I need to find the sine of angle TPU in a cube labeled PQRSTU. Hmm, cubes can be a bit tricky, but let me try to visualize it. First, I should probably figure out where the points T, P, and U are located in the cube.I know that in a cube, all edges are equal, and each face is a square. Let me try to sketch a cube in my mind. Let's label the cube's vertices as P, Q, R, S, T, U, V, W, but wait, the cube only has eight vertices, so maybe the labeling is different. Wait, the cube is labeled PQRSTU, so that's six letters, but a cube has eight vertices. Maybe some letters are repeated or perhaps it's a different labeling convention. Hmm, maybe I should clarify that.Wait, perhaps each letter corresponds to a different vertex, so P, Q, R, S, T, U, V, W. But the problem mentions PQRSTU, so maybe it's a specific labeling where each face is labeled with these letters. Maybe P is one corner, Q is adjacent to P, R is adjacent to Q, and so on. I think I need to figure out the positions of T, P, and U.Alternatively, maybe it's better to assign coordinates to the cube's vertices to make things clearer. Let me assume that the cube has side length 1 for simplicity, and assign coordinates to each vertex. Let's say point P is at the origin (0,0,0). Then, since it's a cube, the adjacent points would be Q(1,0,0), R(1,1,0), S(0,1,0), and so on for the top layer: T(0,0,1), U(1,0,1), V(1,1,1), W(0,1,1).Wait, let me make sure: if P is (0,0,0), then moving along the x-axis would be Q(1,0,0), then R(1,1,0), S(0,1,0). Then the top layer would be T(0,0,1), U(1,0,1), V(1,1,1), W(0,1,1). Yeah, that seems right.So, point T is (0,0,1), point P is (0,0,0), and point U is (1,0,1). Now, I need to find angle TPU, which is the angle at point P between points T and U. So, in terms of vectors, this angle is between vectors PT and PU.First, let me find the coordinates of vectors PT and PU. Vector PT goes from P(0,0,0) to T(0,0,1), so PT is (0,0,1). Vector PU goes from P(0,0,0) to U(1,0,1), so PU is (1,0,1).Now, to find the angle between two vectors, I can use the dot product formula. The dot product of vectors PT and PU is equal to the product of their magnitudes times the cosine of the angle between them. So, if I can find the dot product and the magnitudes, I can solve for the cosine of the angle, and then take the sine of that angle.First, let's compute the dot product of PT and PU. The dot product is calculated as:PT • PU = (0)(1) + (0)(0) + (1)(1) = 0 + 0 + 1 = 1.Next, let's find the magnitudes of PT and PU. The magnitude of PT is sqrt(0^2 + 0^2 + 1^2) = sqrt(1) = 1. The magnitude of PU is sqrt(1^2 + 0^2 + 1^2) = sqrt(1 + 0 + 1) = sqrt(2).So, according to the dot product formula:PT • PU = |PT| |PU| cos(theta)Plugging in the values:1 = (1)(sqrt(2)) cos(theta)So, cos(theta) = 1 / sqrt(2)Therefore, theta = arccos(1 / sqrt(2)) = 45 degrees.Wait, but the question asks for sin(theta). So, sin(45 degrees) is sqrt(2)/2.But hold on, let me double-check. If the angle between PT and PU is 45 degrees, then sin(theta) is sqrt(2)/2. That seems right.But wait, another thought: in a cube, the angle between two face diagonals that meet at a vertex is actually 60 degrees, not 45 degrees. Hmm, maybe I made a mistake here.Wait, let me think again. If I have two vectors in 3D space, their angle might not be the same as the angle on the face. So, perhaps I need to consider the triangle formed by points T, P, and U.Let me calculate the lengths of the sides of triangle TPU. PT is from P(0,0,0) to T(0,0,1), which is length 1. PU is from P(0,0,0) to U(1,0,1), which is sqrt(2). Now, what about TU? TU is from T(0,0,1) to U(1,0,1). The distance between T and U is sqrt[(1-0)^2 + (0-0)^2 + (1-1)^2] = sqrt[1 + 0 + 0] = 1.So, triangle TPU has sides of length 1, sqrt(2), and 1. Wait, that doesn't make sense because in a triangle, the sum of any two sides must be greater than the third side. Here, 1 + 1 = 2, which is greater than sqrt(2) (~1.414), so that's okay.But if two sides are length 1 and one side is sqrt(2), then it's an isoceles triangle with two sides equal to 1 and base sqrt(2). So, the angle at P is between the two sides of length 1 and sqrt(2). Wait, no, actually, in triangle TPU, the sides are PT = 1, PU = sqrt(2), and TU = 1.Wait, so the sides are 1, 1, sqrt(2). That makes it an isoceles triangle with two sides of length 1 and base sqrt(2). So, the angle at P is between the two sides of length 1 and sqrt(2). Wait, no, actually, in triangle TPU, the sides are PT = 1, PU = sqrt(2), and TU = 1. So, the angle at P is between PT and PU, which are of lengths 1 and sqrt(2), and the opposite side is TU = 1.Wait, so using the Law of Cosines to find the angle at P:TU^2 = PT^2 + PU^2 - 2 * PT * PU * cos(theta)Plugging in the values:1^2 = 1^2 + (sqrt(2))^2 - 2 * 1 * sqrt(2) * cos(theta)Simplify:1 = 1 + 2 - 2 * sqrt(2) * cos(theta)1 = 3 - 2 * sqrt(2) * cos(theta)Subtract 3 from both sides:1 - 3 = -2 * sqrt(2) * cos(theta)-2 = -2 * sqrt(2) * cos(theta)Divide both sides by -2:1 = sqrt(2) * cos(theta)So, cos(theta) = 1 / sqrt(2)Therefore, theta = 45 degrees, as before.So, sin(theta) = sin(45 degrees) = sqrt(2)/2.Wait, but earlier I thought it might be 60 degrees, but that seems incorrect. Maybe I confused it with another angle in the cube.Alternatively, maybe I should consider the vectors again. Vector PT is (0,0,1) and vector PU is (1,0,1). The angle between them is calculated using the dot product, which gave me 45 degrees, so sin(theta) is sqrt(2)/2.But let me also think geometrically. In the cube, point P is connected to T and U. PT is along the z-axis, and PU is along the diagonal of the front face (if we consider P at the origin). So, the angle between the z-axis and the diagonal of the front face.Wait, the diagonal of the front face is in the x-z plane, from (0,0,0) to (1,0,1). So, the angle between the z-axis and this diagonal can be found using trigonometry.If I consider the triangle formed by (0,0,0), (1,0,0), and (1,0,1), which is a right triangle with legs of length 1 and 1. The angle at (0,0,0) between the z-axis and the diagonal is 45 degrees, which matches our previous calculation.Therefore, sin(theta) is sqrt(2)/2.Wait, but earlier I thought the angle might be 60 degrees because sometimes in cubes, face diagonals can form equilateral triangles, but in this case, it's not the case. The triangle TPU is not equilateral because two sides are length 1 and one side is sqrt(2). So, it's an isoceles triangle with two sides of length 1 and base sqrt(2), giving an angle of 45 degrees at P.Therefore, sin(angle TPU) is sqrt(2)/2.But wait, let me double-check using another method. Maybe using coordinates to find the vectors and then compute the sine.We have vectors PT = (0,0,1) and PU = (1,0,1). The cross product of PT and PU is |PT||PU|sin(theta) n, where n is the unit vector perpendicular to both.The magnitude of the cross product is |PT||PU|sin(theta).So, let's compute the cross product of PT and PU.PT = (0,0,1)PU = (1,0,1)Cross product PT × PU = |i   j   k|                       0   0   1                       1   0   1= i*(0*1 - 1*0) - j*(0*1 - 1*1) + k*(0*0 - 0*1)= i*(0 - 0) - j*(0 - 1) + k*(0 - 0)= 0i + 1j + 0k= (0,1,0)The magnitude of the cross product is sqrt(0^2 + 1^2 + 0^2) = 1.Now, |PT| = 1, |PU| = sqrt(2), so:|PT × PU| = |PT||PU|sin(theta) => 1 = 1 * sqrt(2) * sin(theta)Therefore, sin(theta) = 1 / sqrt(2) = sqrt(2)/2.Yes, that confirms it. So, sin(angle TPU) is sqrt(2)/2.Wait, but earlier I thought maybe it's 60 degrees, but that seems to be incorrect. So, I think the correct answer is sqrt(2)/2.But just to make sure, let me think about the cube again. If I look at the cube from the top, the diagonal from P to U is in the x-z plane, making a 45-degree angle with the z-axis. So, the angle between PT (along z-axis) and PU (diagonal in x-z plane) is indeed 45 degrees, so sin(theta) is sqrt(2)/2.Therefore, I think the answer is sqrt(2)/2.

💡Okay, so I have this problem here: "A number ( x ) is 3 more than the product of its reciprocal and its additive inverse squared. In which interval does the number lie?" The options are given as intervals from -6 to 9, so I need to figure out where ( x ) falls.First, let me parse the problem sentence by sentence. The number ( x ) is 3 more than something. That something is the product of its reciprocal and its additive inverse squared. Hmm, that sounds a bit complicated, but I can break it down.Let me recall some basic definitions to make sure I understand the terms:1. **Reciprocal of ( x )**: That's ( frac{1}{x} ). So, if ( x ) is a number, its reciprocal is just 1 divided by that number.2. **Additive inverse of ( x )**: That's ( -x ). So, if ( x ) is positive, the additive inverse is negative, and vice versa.3. **Squared**: That means multiplying the number by itself. So, if I have the additive inverse squared, that would be ( (-x)^2 ). Wait, squaring a negative number makes it positive, right? So, ( (-x)^2 = x^2 ).Okay, so putting it all together: the product of the reciprocal of ( x ) and the additive inverse squared of ( x ) is ( frac{1}{x} times (-x)^2 ). But since ( (-x)^2 = x^2 ), this simplifies to ( frac{1}{x} times x^2 ).Let me write that out:[frac{1}{x} times x^2]Simplifying this, the ( x ) in the denominator cancels with one ( x ) in the numerator, leaving me with just ( x ). So, the product of the reciprocal and the additive inverse squared is just ( x ).Now, the problem says that ( x ) is 3 more than this product. So, mathematically, that should be:[x = left( frac{1}{x} times (-x)^2 right) + 3]But as I simplified earlier, that becomes:[x = x + 3]Wait, that can't be right. If I subtract ( x ) from both sides, I get:[0 = 3]Which is obviously not true. Hmm, did I make a mistake somewhere?Let me go back. The problem says "3 more than the product of its reciprocal and its additive inverse squared." So, maybe I misinterpreted the order of operations. Is it the product of the reciprocal and the additive inverse, and then that product is squared? Or is it the reciprocal multiplied by the square of the additive inverse?Looking back at the problem: "the product of its reciprocal and its additive inverse squared." The way it's phrased, "additive inverse squared" is a single term, so it's the square of the additive inverse. So, it's reciprocal multiplied by (additive inverse squared). So, my initial interpretation was correct: ( frac{1}{x} times (-x)^2 ).But that leads to ( x = x + 3 ), which is a contradiction. That suggests there's no solution, but that can't be right because the problem is asking which interval ( x ) lies in, implying there is a solution.Wait, maybe I made a mistake in simplifying ( frac{1}{x} times (-x)^2 ). Let me double-check that.So, ( (-x)^2 = x^2 ), correct. Then, ( frac{1}{x} times x^2 = x ). So, that part is correct. So, the equation becomes ( x = x + 3 ), which is impossible. Hmm.Is there another way to interpret the problem? Maybe the reciprocal of the product of ( x ) and its additive inverse squared? That would be ( frac{1}{x times (-x)^2} ). Let me try that.So, if it's the reciprocal of the product of ( x ) and its additive inverse squared, then:[frac{1}{x times (-x)^2}]Simplify ( (-x)^2 ) to ( x^2 ), so:[frac{1}{x times x^2} = frac{1}{x^3}]Then, the problem says ( x ) is 3 more than this, so:[x = frac{1}{x^3} + 3]That seems more complicated, but at least it's a valid equation. Let me write that down:[x = frac{1}{x^3} + 3]Hmm, this is a nonlinear equation. Maybe I can multiply both sides by ( x^3 ) to eliminate the denominator:[x times x^3 = 1 + 3x^3]Simplify the left side:[x^4 = 1 + 3x^3]Bring all terms to one side:[x^4 - 3x^3 - 1 = 0]This is a quartic equation, which might be difficult to solve. Maybe I can factor it or find rational roots.Using the Rational Root Theorem, possible rational roots are factors of the constant term over factors of the leading coefficient. The constant term is -1, and the leading coefficient is 1, so possible roots are ( pm1 ).Let me test ( x = 1 ):[1^4 - 3(1)^3 - 1 = 1 - 3 - 1 = -3 neq 0]Not a root. Now, ( x = -1 ):[(-1)^4 - 3(-1)^3 - 1 = 1 + 3 - 1 = 3 neq 0]Also not a root. So, there are no rational roots. Maybe I need to use numerical methods or graphing to approximate the roots.Alternatively, perhaps I misinterpreted the problem again. Let me go back to the original statement."A number ( x ) is 3 more than the product of its reciprocal and its additive inverse squared."So, "product of its reciprocal and its additive inverse squared" is ( frac{1}{x} times (-x)^2 ), which is ( x ). So, ( x = x + 3 ), which is impossible. That suggests that either the problem is miswritten, or I'm misinterpreting it.Wait, maybe the problem is saying "the product of its reciprocal and its additive inverse, squared." That would be different. So, the product of reciprocal and additive inverse is ( frac{1}{x} times (-x) = -1 ). Then, squared would be ( (-1)^2 = 1 ). So, the equation would be ( x = 1 + 3 = 4 ).That makes sense. So, maybe the problem was meant to be interpreted as the product of reciprocal and additive inverse, and then that product is squared. So, the equation would be:[x = left( frac{1}{x} times (-x) right)^2 + 3]Simplify inside the parentheses:[frac{1}{x} times (-x) = -1]Then, squared:[(-1)^2 = 1]So, the equation becomes:[x = 1 + 3 = 4]That works! So, ( x = 4 ). Now, looking at the intervals given:(A) -6 ≤ x ≤ -3(B) -3 < x ≤ 0(C) 0 < x ≤ 3(D) 3 < x ≤ 6(E) 6 < x ≤ 9So, 4 falls into interval (D) 3 < x ≤ 6.Wait, but I'm a bit confused because earlier I thought the problem was leading to a contradiction, but interpreting it differently gives a valid solution. Maybe the problem was ambiguous in wording, but the correct interpretation is that the product of reciprocal and additive inverse is taken first, and then that product is squared, making the equation solvable.So, to summarize:1. Reciprocal of ( x ) is ( frac{1}{x} ).2. Additive inverse of ( x ) is ( -x ).3. Product of reciprocal and additive inverse is ( frac{1}{x} times (-x) = -1 ).4. Squared, that becomes ( (-1)^2 = 1 ).5. Then, 3 more than that is ( 1 + 3 = 4 ).Therefore, ( x = 4 ), which is in interval (D).I think that's the correct approach. Initially, I misread the problem, but upon re-examining the wording, interpreting it as the product of reciprocal and additive inverse first, then squaring that product, leads to a valid solution.

💡Alright, so I have this logic problem here, and I need to figure out which statements must be true if the original statement is false. Let me take it step by step.The original statement is: "All books in this bookstore are new editions." And it's given that this statement is false. So, my first thought is, if "all" books are not new editions, then what does that mean? Well, if it's not true that every single book is a new edition, then there must be at least one book that isn't a new edition. That seems straightforward.Now, let's look at the options one by one.**Statement I: All books in this bookstore are old editions.**Hmm, okay. So if the original statement is false, does that mean all books are old? Wait, not necessarily. The original statement being false just means that not every book is a new edition. It doesn't say anything about all of them being old. There could be a mix of new and old editions. So, Statement I might not be true. It's possible that some books are old and some are new, right?**Statement II: There is at least one book in this bookstore that is not a new edition.**Alright, this seems related to the original statement being false. If "all books are new editions" is false, then the opposite must be true. The opposite of "all are new" is "there exists at least one that is not new." So, Statement II must be true.**Statement III: No book in this bookstore is a new edition.**Wait a minute. If the original statement is false, does that mean none are new? That seems too strong. The original statement being false only tells us that not all are new, but it doesn't exclude the possibility that some are new. So, Statement III might not be true either. It could be that some books are new and some are old.**Statement IV: Not all books in this bookstore are new editions.**This one is interesting. It's almost like a direct negation of the original statement. If "all books are new" is false, then "not all books are new" must be true. So, Statement IV must be true.Okay, so summarizing my thoughts:- Statement I: Not necessarily true.- Statement II: Must be true.- Statement III: Not necessarily true.- Statement IV: Must be true.So, the statements that must be true are II and IV. Looking back at the options, that's option D.But wait, let me double-check. Is there any chance that Statement I could be true? If all books are old, then certainly not all are new, which would make the original statement false. But the original statement being false doesn't require all books to be old, just that at least one isn't new. So, Statement I isn't necessarily true.Similarly, Statement III is too strong because it claims none are new, which isn't necessarily the case. The original statement being false only tells us that not all are new, but some could still be new.Therefore, I'm confident that Statements II and IV must be true, which corresponds to option D.

💡Okay, so I have this problem here where a number ( k ) is randomly taken from the interval ([-1, 2]). I need to find the probability that the line ( y = k(x - 4) ) intersects with the circle ( x^2 + y^2 = 4 ). The options are given as A through D, and I need to figure out which one is correct.First, let me understand what the problem is asking. I have a circle centered at the origin with a radius of 2 because the equation is ( x^2 + y^2 = 4 ). The line is given in the form ( y = k(x - 4) ), which is a straight line with slope ( k ) and passing through the point ( (4, 0) ). So, this line is going to pass through the point ( (4, 0) ) and have a slope ( k ).I need to find the probability that this line intersects the circle. Since ( k ) is chosen randomly from the interval ([-1, 2]), the probability will be the length of the interval where the line intersects the circle divided by the total length of the interval ([-1, 2]).To find when the line intersects the circle, I can use the condition that the distance from the center of the circle to the line must be less than or equal to the radius of the circle. The center of the circle is at ( (0, 0) ), and the radius is 2.The formula for the distance from a point ( (x_0, y_0) ) to the line ( ax + by + c = 0 ) is ( frac{|ax_0 + by_0 + c|}{sqrt{a^2 + b^2}} ). So, I need to write the equation of the line in the standard form ( ax + by + c = 0 ).Given the line ( y = k(x - 4) ), let me rearrange it:( y = kx - 4k )Subtracting ( y ) from both sides:( kx - y - 4k = 0 )So, in standard form, it's ( kx - y - 4k = 0 ). Here, ( a = k ), ( b = -1 ), and ( c = -4k ).Now, the distance from the center ( (0, 0) ) to this line is:( frac{|k cdot 0 + (-1) cdot 0 - 4k|}{sqrt{k^2 + (-1)^2}} = frac{| -4k |}{sqrt{k^2 + 1}} = frac{4|k|}{sqrt{k^2 + 1}} )This distance must be less than or equal to the radius of the circle, which is 2, for the line to intersect the circle.So, the inequality is:( frac{4|k|}{sqrt{k^2 + 1}} leq 2 )Let me solve this inequality step by step.First, multiply both sides by ( sqrt{k^2 + 1} ):( 4|k| leq 2sqrt{k^2 + 1} )Divide both sides by 2:( 2|k| leq sqrt{k^2 + 1} )Now, square both sides to eliminate the square root. Since both sides are non-negative, squaring is a valid operation here.( (2|k|)^2 leq (sqrt{k^2 + 1})^2 )Simplify both sides:( 4k^2 leq k^2 + 1 )Subtract ( k^2 ) from both sides:( 3k^2 leq 1 )Divide both sides by 3:( k^2 leq frac{1}{3} )Take square roots:( |k| leq frac{sqrt{3}}{3} )So, ( k ) must satisfy:( -frac{sqrt{3}}{3} leq k leq frac{sqrt{3}}{3} )Now, I need to find the probability that ( k ) falls within this interval when it's randomly selected from ([-1, 2]).First, let's find the length of the interval where the line intersects the circle. The interval is from ( -frac{sqrt{3}}{3} ) to ( frac{sqrt{3}}{3} ). The length is:( frac{sqrt{3}}{3} - left(-frac{sqrt{3}}{3}right) = frac{2sqrt{3}}{3} )Next, the total length of the interval from which ( k ) is chosen is from (-1) to (2), which is:( 2 - (-1) = 3 )Therefore, the probability ( P ) is the ratio of these two lengths:( P = frac{frac{2sqrt{3}}{3}}{3} = frac{2sqrt{3}}{9} )So, the probability is ( frac{2sqrt{3}}{9} ), which corresponds to option C.Wait, let me double-check my steps to make sure I didn't make a mistake.1. I converted the line equation to standard form correctly.2. Calculated the distance from the center to the line correctly.3. Set up the inequality correctly with the radius.4. Solved the inequality step by step, squaring both sides, which should be valid since both sides are non-negative.5. Found the interval for ( k ) correctly.6. Calculated the length of the interval and the total interval correctly.7. Divided them to get the probability.Everything seems to check out. I think the answer is indeed option C: ( frac{2sqrt{3}}{9} ).**Final Answer**boxed{C}

💡Okay, so I need to figure out how many of the first 20 rows in Pascal's triangle, starting from row 2, consist entirely of odd numbers except for the 1s at each end. Hmm, let's see. I remember that Pascal's triangle is built from binomial coefficients, and each row corresponds to the coefficients of (a + b)^n. The entries in each row are calculated by adding the two numbers directly above them.First, I should recall that the entries in Pascal's triangle can be even or odd. I think there's a pattern to the parity of these numbers. Maybe if I look at the first few rows, I can spot a pattern.Let me write out the first few rows:Row 0: 1Row 1: 1 1Row 2: 1 2 1Row 3: 1 3 3 1Row 4: 1 4 6 4 1Row 5: 1 5 10 10 5 1Row 6: 1 6 15 20 15 6 1Row 7: 1 7 21 35 35 21 7 1Row 8: 1 8 28 56 70 56 28 8 1Okay, so looking at these, I can see that the numbers in the middle can be even or odd. The question is asking about rows where all the numbers, except for the 1s at the ends, are odd. So, for example, row 3 is 1 3 3 1, and both 3s are odd, so that fits. Row 2 is 1 2 1, which has an even number in the middle, so that doesn't fit. Similarly, row 4 has even numbers in the middle.So, from rows 0 to 8, only row 3 fits the criteria. Let me check a few more rows to see if there's a pattern.Row 9: 1 9 36 84 126 126 84 36 9 1Row 10: 1 10 45 120 210 252 210 120 45 10 1Row 11: 1 11 55 165 330 462 462 330 165 55 11 1Row 12: 1 12 66 220 495 792 924 792 495 220 66 12 1Row 13: 1 13 78 286 715 1287 1716 1716 1287 715 286 78 13 1Row 14: 1 14 91 364 1001 2002 3003 3432 3003 2002 1001 364 91 14 1Row 15: 1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 1Row 16: 1 16 120 560 1820 4368 8008 11440 12870 11440 8008 4368 1820 560 120 16 1Row 17: 1 17 136 680 2380 6188 12376 19448 24310 24310 19448 12376 6188 2380 680 136 17 1Row 18: 1 18 153 816 3060 8568 18564 31824 43758 48620 43758 31824 18564 8568 3060 816 153 18 1Row 19: 1 19 171 969 3876 11628 27132 50388 75582 92378 75582 50388 27132 11628 3876 969 171 19 1Looking at these, it seems that only row 3 meets the criteria. Rows 1, 2, 4, 5, etc., all have even numbers in the middle. So, is row 3 the only one? But wait, I remember hearing something about powers of 2 in relation to Pascal's triangle. Maybe rows that are powers of 2 have all odd numbers except for the 1s.Let me check row 1, which is 1 1. That's just two 1s, so technically, there are no numbers in the middle, but since the question starts from row 2, row 1 isn't considered. Row 2 is 1 2 1, which has an even number. Row 4 is 1 4 6 4 1, which also has even numbers. Row 8 is 1 8 28 56 70 56 28 8 1, which definitely has even numbers. So, maybe my initial thought was wrong.Wait, maybe it's not the powers of 2, but something else. I think it might be related to the binary representation of the row number. If the row number is one less than a power of 2, then all the numbers in that row are odd. Let me see.Row 3 is 1 3 3 1, which fits. 3 is one less than 4, which is 2^2. Let's check row 7, which is 1 7 21 35 35 21 7 1. All the middle numbers are odd. 7 is one less than 8, which is 2^3. So, row 7 also fits. Similarly, row 15 is one less than 16, which is 2^4. Let me check row 15.Row 15: 1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 1Yes, all the middle numbers are odd. So, rows that are one less than a power of 2 have all odd numbers except for the 1s at the ends.So, in the first 20 rows, starting from row 2, the rows that fit the criteria are rows 3, 7, 15. Let me count them: 3, 7, 15. That's three rows.Wait, but earlier when I listed the rows, row 15 is within the first 20 rows. So, that would make three rows: 3, 7, and 15.But earlier, I thought only row 3 fit. Maybe I need to verify.Let me check row 7 again:Row 7: 1 7 21 35 35 21 7 1Yes, all middle numbers are odd.Row 15: 1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 1All middle numbers are odd.So, within the first 20 rows, starting from row 2, the rows that fit are 3, 7, and 15. That's three rows.Wait, but the user asked starting from row 2, so row 2 is excluded. So, rows 3, 7, 15 are within rows 2 to 19? Wait, row 15 is row 15, which is less than 20, so yes.But earlier, I thought only row 3 fit. Maybe I was mistaken.Wait, let me think again. The key is that a row n in Pascal's triangle has all odd numbers except for the 1s if and only if n is of the form 2^k - 1. So, n = 1, 3, 7, 15, 31, etc.But since we're starting from row 2, n >= 2, so the first such row is n=3, then n=7, n=15. Next would be n=31, which is beyond 20.So, in the first 20 rows, starting from row 2, the rows that fit are 3, 7, and 15. That's three rows.But earlier, I thought only row 3 fit. Maybe I need to verify.Wait, let me check row 7 and row 15 again.Row 7: 1 7 21 35 35 21 7 1All middle numbers are odd.Row 15: 1 15 105 455 1365 3003 5005 6435 6435 5005 3003 1365 455 105 15 1All middle numbers are odd.So, yes, rows 3, 7, and 15 fit the criteria.Therefore, the answer should be 3.Wait, but in the initial thought, I thought only row 3 fit. Maybe I was confused earlier.So, to summarize, rows that are one less than a power of 2 have all odd numbers except for the 1s. These are rows 3 (2^2 -1), 7 (2^3 -1), 15 (2^4 -1), etc.Within the first 20 rows, starting from row 2, the rows that fit are 3, 7, and 15. So, three rows.Therefore, the answer is 3.

💡Okay, so I have this problem about two circles intersecting at points P and Q. The coordinates of point P are (3, 2), and the product of the radii of the two circles is 13/2. Both circles are tangent to the line l: y = kx and the x-axis. I need to find the equation of the line l.Hmm, let me start by visualizing the problem. We have two circles that intersect at points P and Q. Both circles are tangent to the x-axis and another line l, which is given by y = kx. Since both circles are tangent to the x-axis, their centers must be at a distance equal to their radii from the x-axis. So, if the radius of a circle is r, its center must be at (h, r) for some h.Also, since both circles are tangent to the line l: y = kx, the distance from the center of each circle to this line must also be equal to their radii. The distance from a point (h, r) to the line y = kx is given by the formula |kh - r| / sqrt(k^2 + 1). Since this distance must equal the radius r, we have |kh - r| / sqrt(k^2 + 1) = r.Let me write that down for both circles. Let’s denote the centers of the two circles as (h1, r1) and (h2, r2), with radii r1 and r2 respectively. Then, for each circle, we have:|k*h1 - r1| / sqrt(k^2 + 1) = r1and|k*h2 - r2| / sqrt(k^2 + 1) = r2.Since both circles are above the x-axis and tangent to it, their centers are above the x-axis, so r1 and r2 are positive. Also, since the line l is y = kx, which passes through the origin, and both circles are tangent to it, the centers must lie on the same side of the line l. Therefore, the expressions inside the absolute value should have the same sign, so we can drop the absolute value:(k*h1 - r1) / sqrt(k^2 + 1) = r1and(k*h2 - r2) / sqrt(k^2 + 1) = r2.Wait, but actually, the sign depends on the position relative to the line. If the center is above the line l, then (k*h - r) would be positive or negative? Let me think. The line l is y = kx, so for a point (h, r), if r > k*h, then it's above the line, otherwise below. Since the circles are tangent to the line l, and also tangent to the x-axis, which is below the line l if k > 0. Wait, actually, the x-axis is y=0, and the line l is y=kx. So depending on the slope k, the line l could be above or below the x-axis. But since both circles are tangent to both the x-axis and the line l, and intersect at point P(3,2), which is above the x-axis, the line l must also be above the x-axis. So, the centers of the circles are above the x-axis and above the line l. Therefore, r > k*h for both circles. So, (k*h - r) would be negative, hence |k*h - r| = r - k*h. So, the distance formula becomes (r - k*h) / sqrt(k^2 + 1) = r.So, for each circle:(r - k*h) / sqrt(k^2 + 1) = rMultiply both sides by sqrt(k^2 + 1):r - k*h = r*sqrt(k^2 + 1)Then, rearranged:r - r*sqrt(k^2 + 1) = k*hSo,h = [r - r*sqrt(k^2 + 1)] / kSimplify:h = r [1 - sqrt(k^2 + 1)] / kHmm, that's for each circle. So, for circle C1, h1 = r1 [1 - sqrt(k^2 + 1)] / k, and for circle C2, h2 = r2 [1 - sqrt(k^2 + 1)] / k.Now, since both circles pass through point P(3, 2), we can write the equations of the circles and substitute (3, 2) into them.The equation of circle C1 is:(x - h1)^2 + (y - r1)^2 = r1^2Similarly, for circle C2:(x - h2)^2 + (y - r2)^2 = r2^2Substituting (3, 2) into C1's equation:(3 - h1)^2 + (2 - r1)^2 = r1^2Similarly, for C2:(3 - h2)^2 + (2 - r2)^2 = r2^2Let me expand these equations.Starting with C1:(3 - h1)^2 + (2 - r1)^2 = r1^2Expanding:(9 - 6h1 + h1^2) + (4 - 4r1 + r1^2) = r1^2Combine like terms:9 - 6h1 + h1^2 + 4 - 4r1 + r1^2 = r1^2Simplify:13 - 6h1 + h1^2 - 4r1 + r1^2 = r1^2Subtract r1^2 from both sides:13 - 6h1 + h1^2 - 4r1 = 0Similarly, for C2:13 - 6h2 + h2^2 - 4r2 = 0So, both equations reduce to:h^2 - 6h - 4r + 13 = 0Where h is h1 or h2, and r is r1 or r2.But we already have expressions for h1 and h2 in terms of r1 and r2. Recall that:h = r [1 - sqrt(k^2 + 1)] / kSo, substituting h into the equation h^2 - 6h - 4r + 13 = 0:[r (1 - sqrt(k^2 + 1))/k]^2 - 6 [r (1 - sqrt(k^2 + 1))/k] - 4r + 13 = 0Let me denote sqrt(k^2 + 1) as s for simplicity. So, s = sqrt(k^2 + 1). Then, 1 - s is 1 - sqrt(k^2 + 1).So, substituting:[r (1 - s)/k]^2 - 6 [r (1 - s)/k] - 4r + 13 = 0Let me compute each term:First term: [r (1 - s)/k]^2 = r^2 (1 - s)^2 / k^2Second term: -6 [r (1 - s)/k] = -6r (1 - s)/kThird term: -4rFourth term: +13So, putting it all together:r^2 (1 - s)^2 / k^2 - 6r (1 - s)/k - 4r + 13 = 0This is a quadratic equation in terms of r. Let me write it as:A r^2 + B r + C = 0Where:A = (1 - s)^2 / k^2B = -6 (1 - s)/k - 4C = 13So, quadratic equation is:[(1 - s)^2 / k^2] r^2 + [ -6 (1 - s)/k - 4 ] r + 13 = 0Now, since both circles C1 and C2 satisfy this equation, their radii r1 and r2 are the roots of this quadratic equation. Therefore, by Vieta's formulas, the product of the roots r1 * r2 = C / A.Given that the product of the radii is 13/2, we have:r1 * r2 = 13 / 2 = C / A = 13 / [ (1 - s)^2 / k^2 ]So,13 / 2 = 13 / [ (1 - s)^2 / k^2 ]Simplify:1/2 = 1 / [ (1 - s)^2 / k^2 ]Which implies:(1 - s)^2 / k^2 = 2Therefore,(1 - s)^2 = 2 k^2Take square roots:1 - s = sqrt(2) kBut s = sqrt(k^2 + 1), so:1 - sqrt(k^2 + 1) = sqrt(2) kLet me write that:sqrt(k^2 + 1) = 1 - sqrt(2) kWait, but sqrt(k^2 + 1) is always positive, and 1 - sqrt(2) k must also be positive because 1 - s = sqrt(2) k, and s = sqrt(k^2 + 1) > |k|, so 1 - s is negative if k is positive. Wait, hold on, maybe I made a mistake in the sign.Earlier, when we had |k*h - r| = r - k*h because r > k*h, so we had:(r - k*h)/sqrt(k^2 + 1) = rWhich led to:r - k*h = r sqrt(k^2 + 1)Then,r - r sqrt(k^2 + 1) = k*hSo,h = [r (1 - sqrt(k^2 + 1))]/kBut sqrt(k^2 + 1) > |k|, so 1 - sqrt(k^2 + 1) is negative, so h is negative if k is positive. But h is the x-coordinate of the center, which can be positive or negative. However, since the circles pass through (3, 2), which is in the first quadrant, the centers are likely in the first quadrant as well, so h should be positive. Therefore, [1 - sqrt(k^2 + 1)] is negative, so k must be negative to make h positive. Therefore, k is negative.So, in the equation sqrt(k^2 + 1) = 1 - sqrt(2) k, since k is negative, let me denote k = -m, where m > 0.Then, sqrt(m^2 + 1) = 1 + sqrt(2) mLet me square both sides:m^2 + 1 = (1 + sqrt(2) m)^2Expand the right side:1 + 2 sqrt(2) m + 2 m^2So,m^2 + 1 = 1 + 2 sqrt(2) m + 2 m^2Subtract 1 from both sides:m^2 = 2 sqrt(2) m + 2 m^2Bring all terms to one side:0 = 2 sqrt(2) m + 2 m^2 - m^2Simplify:0 = 2 sqrt(2) m + m^2Factor:m (2 sqrt(2) + m) = 0So, solutions are m = 0 or m = -2 sqrt(2). But m > 0, so m = -2 sqrt(2) is invalid. Wait, that can't be. Did I make a mistake?Wait, when I squared both sides, I might have introduced an extraneous solution. Let me check.Original equation after substitution:sqrt(m^2 + 1) = 1 + sqrt(2) mBut sqrt(m^2 + 1) is always positive, and 1 + sqrt(2) m is also positive since m > 0. So, squaring should be okay.But when I squared, I got:m^2 + 1 = 1 + 2 sqrt(2) m + 2 m^2Which simplifies to:0 = 2 sqrt(2) m + m^2So, m (2 sqrt(2) + m) = 0Solutions are m = 0 or m = -2 sqrt(2). But m > 0, so no solution? That can't be.Wait, maybe my substitution was wrong. Let me go back.We had:1 - sqrt(k^2 + 1) = sqrt(2) kBut I set k = -m, so:1 - sqrt(m^2 + 1) = -sqrt(2) mMultiply both sides by -1:sqrt(m^2 + 1) - 1 = sqrt(2) mSo,sqrt(m^2 + 1) = 1 + sqrt(2) mWhich is the same as before. So, when I square, I get:m^2 + 1 = 1 + 2 sqrt(2) m + 2 m^2Simplify:0 = 2 sqrt(2) m + m^2So, m (2 sqrt(2) + m) = 0Again, m = 0 or m = -2 sqrt(2). But m > 0, so no solution. That suggests that there is no solution, which contradicts the problem statement.Wait, maybe I made a mistake earlier in the sign. Let me go back to the distance formula.We had:The distance from (h, r) to y = kx is |k h - r| / sqrt(k^2 + 1) = rSince the center is above the line l, which is y = kx, and the line l is passing through the origin, if k is positive, the line l is in the first quadrant, and the center is above it, so r > k h. Therefore, |k h - r| = r - k h.But if k is negative, then the line l is in the second and fourth quadrants. Since the center is in the first quadrant, and the line l is y = kx with k negative, the center is above the line l if r > k h, but since k is negative, k h is negative, so r > k h is always true because r is positive. So, in that case, |k h - r| = r - k h regardless.Wait, but earlier, when I set k = -m, I got an inconsistency. Maybe I should not have substituted k = -m but instead kept k as negative.Let me try again without substitution.We have:sqrt(k^2 + 1) = 1 - sqrt(2) kBut since k is negative, let me denote k = -m where m > 0.Then:sqrt(m^2 + 1) = 1 + sqrt(2) mWhich is the same as before. So, squaring:m^2 + 1 = 1 + 2 sqrt(2) m + 2 m^2Simplify:0 = 2 sqrt(2) m + m^2Which gives m = 0 or m = -2 sqrt(2). But m > 0, so no solution. Hmm, that's a problem.Wait, maybe I made a mistake in the earlier steps. Let me go back.We had:(r - k h) / sqrt(k^2 + 1) = rWhich led to:r - k h = r sqrt(k^2 + 1)Then,r - r sqrt(k^2 + 1) = k hSo,h = [r (1 - sqrt(k^2 + 1))]/kBut if k is negative, then h is positive because 1 - sqrt(k^2 + 1) is negative, and k is negative, so negative divided by negative is positive.But when we plugged into the quadratic equation, we ended up with no solution. Maybe I need to re-express the quadratic equation correctly.Let me re-express the quadratic equation step.We had:[(1 - s)^2 / k^2] r^2 + [ -6 (1 - s)/k - 4 ] r + 13 = 0Where s = sqrt(k^2 + 1)Given that r1 * r2 = 13/2, and from Vieta's formula, r1 * r2 = C / A = 13 / [ (1 - s)^2 / k^2 ] = 13 k^2 / (1 - s)^2So,13 k^2 / (1 - s)^2 = 13 / 2Simplify:k^2 / (1 - s)^2 = 1 / 2So,(1 - s)^2 = 2 k^2Which is the same as before.So,1 - s = sqrt(2) kBut s = sqrt(k^2 + 1), so:1 - sqrt(k^2 + 1) = sqrt(2) kNow, since k is negative, let me write k = -m where m > 0.Then,1 - sqrt(m^2 + 1) = -sqrt(2) mMultiply both sides by -1:sqrt(m^2 + 1) - 1 = sqrt(2) mNow, square both sides:(m^2 + 1) - 2 sqrt(m^2 + 1) + 1 = 2 m^2Simplify:m^2 + 2 - 2 sqrt(m^2 + 1) = 2 m^2Bring all terms to one side:- m^2 + 2 - 2 sqrt(m^2 + 1) = 0Multiply both sides by -1:m^2 - 2 + 2 sqrt(m^2 + 1) = 0Rearrange:2 sqrt(m^2 + 1) = 2 - m^2Divide both sides by 2:sqrt(m^2 + 1) = 1 - (m^2)/2Now, square both sides again:m^2 + 1 = 1 - m^2 + (m^4)/4Simplify:m^2 + 1 = 1 - m^2 + (m^4)/4Subtract 1 from both sides:m^2 = -m^2 + (m^4)/4Bring all terms to one side:2 m^2 - (m^4)/4 = 0Factor:m^2 (2 - m^2 /4) = 0So,m^2 = 0 or 2 - m^2 /4 = 0m^2 = 0 gives m = 0, which is invalid since m > 0.2 - m^2 /4 = 0 => m^2 = 8 => m = 2 sqrt(2)So, m = 2 sqrt(2). Therefore, k = -m = -2 sqrt(2)Wait, but earlier, when I set k = -m, I got m = 2 sqrt(2), so k = -2 sqrt(2). But let me check if this satisfies the original equation.Let me compute sqrt(m^2 + 1) = sqrt(8 + 1) = 3And 1 - (m^2)/2 = 1 - 8/2 = 1 - 4 = -3But sqrt(m^2 + 1) = 3, and 1 - (m^2)/2 = -3, so 3 = -3? That's not possible.Wait, that suggests that even after squaring, we introduced an extraneous solution. So, m = 2 sqrt(2) is not a valid solution.Hmm, this is confusing. Maybe I made a mistake in the algebra.Let me go back to the equation after substitution:sqrt(m^2 + 1) = 1 - (m^2)/2But sqrt(m^2 + 1) is positive, and 1 - (m^2)/2 must also be positive.So,1 - (m^2)/2 > 0 => m^2 < 2 => m < sqrt(2)But we found m = 2 sqrt(2) which is greater than sqrt(2), so it's invalid.Therefore, there is no solution? But the problem states that such circles exist. So, maybe I made a mistake in the earlier steps.Wait, perhaps I should consider that the centers are below the line l, which would change the sign in the distance formula. Let me think again.If the center is below the line l, then r < k h, so |k h - r| = k h - r.But since the circles are tangent to the x-axis and the line l, and intersect at (3, 2), which is above the x-axis, the centers must be above the x-axis. If the line l is above the x-axis, then the centers could be above or below the line l. But since the circles are tangent to both the x-axis and the line l, and intersect above the x-axis, it's more likely that the centers are above the line l.Wait, but if the line l is above the x-axis, then the centers being above the line l would mean that the circles are above the line l, but they also have to intersect at (3, 2). Maybe the line l is below the x-axis? But the problem says both circles are tangent to the x-axis and the line l. If the line l is below the x-axis, then the centers would be above the x-axis but below the line l, which would complicate things.Wait, no, the line l is given as y = kx, which passes through the origin. If k is positive, it's in the first and third quadrants; if k is negative, it's in the second and fourth quadrants. Since the circles are tangent to the x-axis and intersect at (3, 2), which is in the first quadrant, the line l must be in the first quadrant, so k is positive.But earlier, when I assumed k is positive, I ended up with no solution. Maybe I need to re-examine the distance formula.Wait, if k is positive, and the center is (h, r), then the distance from (h, r) to y = kx is |k h - r| / sqrt(k^2 + 1). Since the center is above the line l, which is y = kx, then r > k h, so |k h - r| = r - k h.Therefore, the distance is (r - k h)/sqrt(k^2 + 1) = rSo,r - k h = r sqrt(k^2 + 1)Then,r - r sqrt(k^2 + 1) = k hSo,h = [r (1 - sqrt(k^2 + 1))]/kBut since k is positive, and 1 - sqrt(k^2 + 1) is negative, h is negative. But the center is in the first quadrant, so h must be positive. Therefore, this suggests that our assumption that the center is above the line l is incorrect. Therefore, the center must be below the line l, so r < k h, hence |k h - r| = k h - r.So, the distance is (k h - r)/sqrt(k^2 + 1) = rTherefore,k h - r = r sqrt(k^2 + 1)So,k h = r (1 + sqrt(k^2 + 1))Thus,h = [r (1 + sqrt(k^2 + 1))]/kNow, h is positive, as expected, since k is positive and r is positive.Okay, so this changes things. Let me redo the earlier steps with this correct expression.So, h = [r (1 + sqrt(k^2 + 1))]/kNow, substituting h into the equation from the circle passing through (3, 2):(3 - h)^2 + (2 - r)^2 = r^2Expanding:9 - 6 h + h^2 + 4 - 4 r + r^2 = r^2Simplify:13 - 6 h + h^2 - 4 r = 0So,h^2 - 6 h - 4 r + 13 = 0Now, substituting h = [r (1 + sqrt(k^2 + 1))]/k into this equation:[r (1 + sqrt(k^2 + 1))/k]^2 - 6 [r (1 + sqrt(k^2 + 1))/k] - 4 r + 13 = 0Let me denote s = sqrt(k^2 + 1) again.So,[r (1 + s)/k]^2 - 6 [r (1 + s)/k] - 4 r + 13 = 0Expanding:r^2 (1 + s)^2 / k^2 - 6 r (1 + s)/k - 4 r + 13 = 0This is a quadratic in r:A r^2 + B r + C = 0Where:A = (1 + s)^2 / k^2B = -6 (1 + s)/k - 4C = 13So, the quadratic equation is:[(1 + s)^2 / k^2] r^2 + [ -6 (1 + s)/k - 4 ] r + 13 = 0Given that the product of the roots r1 * r2 = 13/2, and from Vieta's formula, r1 * r2 = C / A = 13 / [ (1 + s)^2 / k^2 ] = 13 k^2 / (1 + s)^2So,13 k^2 / (1 + s)^2 = 13 / 2Simplify:k^2 / (1 + s)^2 = 1 / 2Therefore,(1 + s)^2 = 2 k^2Taking square roots:1 + s = sqrt(2) kBut s = sqrt(k^2 + 1), so:1 + sqrt(k^2 + 1) = sqrt(2) kNow, let me solve for k.Let me denote t = sqrt(k^2 + 1). Then, t = sqrt(k^2 + 1), so t^2 = k^2 + 1.The equation becomes:1 + t = sqrt(2) kBut t = sqrt(k^2 + 1), so:1 + sqrt(k^2 + 1) = sqrt(2) kLet me isolate sqrt(k^2 + 1):sqrt(k^2 + 1) = sqrt(2) k - 1Now, square both sides:k^2 + 1 = (sqrt(2) k - 1)^2Expand the right side:2 k^2 - 2 sqrt(2) k + 1So,k^2 + 1 = 2 k^2 - 2 sqrt(2) k + 1Subtract k^2 + 1 from both sides:0 = k^2 - 2 sqrt(2) kFactor:k (k - 2 sqrt(2)) = 0So, k = 0 or k = 2 sqrt(2)But k = 0 would make the line l horizontal, which is the x-axis, but the circles are already tangent to the x-axis, so they can't be tangent to another horizontal line unless they are the same circle, which they are not. Therefore, k = 2 sqrt(2)But wait, let me check if this satisfies the original equation.Compute sqrt(k^2 + 1) = sqrt( (8) + 1 ) = 3Then, 1 + sqrt(k^2 + 1) = 1 + 3 = 4And sqrt(2) k = sqrt(2) * 2 sqrt(2) = 4So, 1 + sqrt(k^2 + 1) = sqrt(2) k, which is 4 = 4. Correct.Therefore, k = 2 sqrt(2)So, the equation of the line l is y = 2 sqrt(2) xWait, but earlier, when I assumed k was negative, I got no solution, but when I corrected the distance formula to have the center below the line l, I got k positive, and it worked out.So, the line l is y = 2 sqrt(2) xBut let me just verify with the quadratic equation.Given k = 2 sqrt(2), s = sqrt(k^2 + 1) = sqrt(8 + 1) = 3So, A = (1 + s)^2 / k^2 = (4)^2 / 8 = 16 / 8 = 2B = -6 (1 + s)/k - 4 = -6 * 4 / (2 sqrt(2)) - 4 = -24 / (2 sqrt(2)) - 4 = -12 / sqrt(2) - 4 = -6 sqrt(2) - 4C = 13So, quadratic equation is:2 r^2 + (-6 sqrt(2) - 4) r + 13 = 0Let me compute the product of the roots:r1 * r2 = C / A = 13 / 2, which matches the given condition.Also, the sum of the roots:r1 + r2 = (6 sqrt(2) + 4)/2 = 3 sqrt(2) + 2So, that seems consistent.Therefore, the equation of the line l is y = 2 sqrt(2) x