💡Okay, so I need to find a monic quadratic polynomial with real coefficients that has (3 + isqrt{2}) as a root. Hmm, let me think about how to approach this.First, I remember that if a polynomial has real coefficients and a complex root, then its complex conjugate must also be a root. So, if (3 + isqrt{2}) is a root, then (3 - isqrt{2}) must also be a root. That makes sense because the coefficients are real, so the imaginary parts have to cancel out somehow.Alright, so now I know both roots of the quadratic: (3 + isqrt{2}) and (3 - isqrt{2}). Since it's a quadratic, these are the only two roots. Now, to find the polynomial, I can use the fact that if (r_1) and (r_2) are roots of a quadratic, then the polynomial can be written as ((x - r_1)(x - r_2)).Let me write that down:[(x - (3 + isqrt{2}))(x - (3 - isqrt{2}))]Now, I need to multiply these two binomials. I think I can use the formula for the product of conjugates, which is ((a - b)(a + b) = a^2 - b^2). In this case, (a) is (x - 3) and (b) is (isqrt{2}). So, applying the formula:[(x - 3)^2 - (isqrt{2})^2]Let me compute each part separately. First, expanding ((x - 3)^2):[(x - 3)^2 = x^2 - 6x + 9]Okay, that's straightforward. Now, the other part is ((isqrt{2})^2). Let's compute that:[(isqrt{2})^2 = i^2 cdot (sqrt{2})^2 = (-1) cdot 2 = -2]So, putting it all together:[(x - 3)^2 - (isqrt{2})^2 = (x^2 - 6x + 9) - (-2)]Subtracting a negative is the same as adding, so:[x^2 - 6x + 9 + 2 = x^2 - 6x + 11]Therefore, the quadratic polynomial is (x^2 - 6x + 11). Let me double-check to make sure I didn't make any mistakes.First, I started with the roots (3 + isqrt{2}) and (3 - isqrt{2}), which is correct because of the real coefficients. Then, I used the conjugate product formula, which seems right. Expanding ((x - 3)^2) gave me (x^2 - 6x + 9), which is correct. Then, ((isqrt{2})^2) is indeed (-2). Subtracting that from the previous result gives (x^2 - 6x + 11), which looks correct.Wait, let me verify by plugging in (x = 3 + isqrt{2}) into the polynomial to see if it equals zero.Compute (x^2 - 6x + 11) at (x = 3 + isqrt{2}):First, (x^2):[(3 + isqrt{2})^2 = 9 + 6isqrt{2} + (isqrt{2})^2 = 9 + 6isqrt{2} - 2 = 7 + 6isqrt{2}]Then, (-6x):[-6(3 + isqrt{2}) = -18 - 6isqrt{2}]Adding the constant term 11:[7 + 6isqrt{2} - 18 - 6isqrt{2} + 11 = (7 - 18 + 11) + (6isqrt{2} - 6isqrt{2}) = 0 + 0i = 0]Perfect, it does equal zero. So, the polynomial is correct.I think I've got it right. The monic quadratic polynomial with real coefficients that has (3 + isqrt{2}) as a root is (x^2 - 6x + 11).**Final Answer**The monic quadratic polynomial is boxed{x^2 - 6x + 11}.

💡Okay, so I have this problem with two parts. Let me try to figure out both step by step. First, the problem says that lines m and n intersect at point P. Line m is given by 2x - y - 3 = 0, and line n is x + y - 3 = 0. I need to find the intersection point P. To find P, I can solve these two equations simultaneously. Let me write them down:1. 2x - y - 3 = 02. x + y - 3 = 0Hmm, maybe I can add these two equations together to eliminate y. Let's see:Adding equation 1 and equation 2:(2x - y - 3) + (x + y - 3) = 0 + 0Simplify:2x - y - 3 + x + y - 3 = 0Combine like terms:(2x + x) + (-y + y) + (-3 - 3) = 0So, 3x + 0y - 6 = 0Which simplifies to 3x - 6 = 0So, 3x = 6 => x = 2Wait, hold on, that can't be right because when I plug x=2 into equation 2, which is x + y - 3 = 0, I get 2 + y - 3 = 0 => y = 1. So P is (2,1). Hmm, but in the initial problem, it was stated as P, so maybe that's correct.Wait, but in the solution above, the user wrote that P is (1,2). Did I make a mistake? Let me check again.Equation 1: 2x - y - 3 = 0Equation 2: x + y - 3 = 0Let me solve equation 2 for y: y = 3 - xNow substitute into equation 1:2x - (3 - x) - 3 = 02x - 3 + x - 3 = 03x - 6 = 0 => 3x = 6 => x = 2Then y = 3 - x = 3 - 2 = 1So P is indeed (2,1). So the initial solution had P as (1,2), which is incorrect. So I need to correct that.Now, moving on to part (1). A line l passes through P(2,1) and the distances from points A(1,3) and B(3,2) to line l are equal. I need to find the equation of line l.Okay, so the distances from A and B to line l are equal. That means that line l is either the perpendicular bisector of segment AB or it's parallel to AB.Wait, but line l passes through P, so it's not necessarily the perpendicular bisector, but it must satisfy the condition that the distances from A and B to l are equal.Alternatively, another approach is that the set of all lines for which the distances from A and B are equal is the perpendicular bisector of AB or lines parallel to AB.But since line l passes through P, which is a specific point, it might be one of these lines.Alternatively, maybe line l is such that it makes equal angles with the lines from A and B, but I think the condition is that the distances are equal, which is a different condition.Wait, perhaps another way is to use the formula for the distance from a point to a line.Let me denote the equation of line l as y = mx + c. Since it passes through P(2,1), we can write 1 = 2m + c => c = 1 - 2m.So the equation is y = mx + (1 - 2m).Now, the distance from point A(1,3) to line l is |m*1 - 1 + (1 - 2m)| / sqrt(m^2 + 1). Wait, no, the formula is |Ax + By + C| / sqrt(A^2 + B^2). So, let me write line l in standard form.From y = mx + c, it can be written as mx - y + c = 0. So A = m, B = -1, C = c.So the distance from A(1,3) to l is |m*1 + (-1)*3 + c| / sqrt(m^2 + 1) = |m - 3 + c| / sqrt(m^2 + 1).Similarly, the distance from B(3,2) to l is |m*3 + (-1)*2 + c| / sqrt(m^2 + 1) = |3m - 2 + c| / sqrt(m^2 + 1).Since these distances are equal, we have |m - 3 + c| = |3m - 2 + c|.But since c = 1 - 2m, substitute that in:|m - 3 + (1 - 2m)| = |3m - 2 + (1 - 2m)|Simplify inside the absolute values:Left side: |m - 3 + 1 - 2m| = |-m - 2| = | - (m + 2)| = |m + 2|Right side: |3m - 2 + 1 - 2m| = |m - 1|So we have |m + 2| = |m - 1|This equation can be solved by considering the cases where the expressions inside the absolute values are equal or opposites.Case 1: m + 2 = m - 1This simplifies to 2 = -1, which is impossible. So no solution here.Case 2: m + 2 = -(m - 1)So m + 2 = -m + 1Bring m to left and constants to right:m + m = 1 - 22m = -1 => m = -1/2So the slope m is -1/2.Therefore, the equation of line l is y = (-1/2)x + c, and since it passes through P(2,1):1 = (-1/2)*2 + c => 1 = -1 + c => c = 2So the equation is y = (-1/2)x + 2, which can be written as x + 2y - 4 = 0.Alternatively, another possibility is that line l is vertical, but since the distances are equal, maybe another line exists. Wait, let me check.If line l is vertical, its equation would be x = 2, since it passes through P(2,1). Let's check the distances from A(1,3) and B(3,2) to x=2.Distance from A(1,3) to x=2 is |1 - 2| = 1.Distance from B(3,2) to x=2 is |3 - 2| = 1.So both distances are equal, so x=2 is another possible line.Therefore, there are two possible lines: x + 2y - 4 = 0 and x = 2.Wait, but in the initial solution, the user wrote x + 2y - 5 = 0 and x=2, but that seems incorrect because when I solved, I got x + 2y - 4 = 0.Wait, let me check again.From m = -1/2, c = 2, so equation is y = (-1/2)x + 2, which is equivalent to x + 2y = 4, so x + 2y - 4 = 0.Yes, that's correct. So the two possible lines are x + 2y - 4 = 0 and x = 2.Now, moving on to part (2). A line l1 passes through P(2,1) and intersects the positive x-axis at A and positive y-axis at B. The area of triangle ABO is 4. Find the equation of l1.Okay, so line l1 passes through (2,1) and intersects x-axis at (a,0) and y-axis at (0,b). The area of triangle ABO is (1/2)*a*b = 4, so a*b = 8.Also, since the line passes through (2,1), we can write the equation of the line in intercept form: x/a + y/b = 1.Substituting (2,1):2/a + 1/b = 1We also have a*b = 8.So we have two equations:1. 2/a + 1/b = 12. a*b = 8Let me solve these equations.From equation 2, b = 8/a.Substitute into equation 1:2/a + 1/(8/a) = 1 => 2/a + a/8 = 1Multiply both sides by 8a to eliminate denominators:16 + a^2 = 8aBring all terms to one side:a^2 - 8a + 16 = 0This is a quadratic equation: a^2 - 8a + 16 = 0Solving for a:a = [8 ± sqrt(64 - 64)] / 2 = [8 ± 0]/2 = 4So a = 4, then b = 8/a = 2.Therefore, the intercepts are at (4,0) and (0,2).So the equation of the line is x/4 + y/2 = 1, which can be rewritten as x + 2y = 4, or y = (-1/2)x + 2.Wait, but in the initial solution, the user wrote the equation as y = (-1/2)x + 2, which is correct.Alternatively, another approach is to consider the slope-intercept form.Let me denote the slope as m. Since the line passes through (2,1), the equation is y - 1 = m(x - 2).This line intersects the x-axis at y=0:0 - 1 = m(x - 2) => -1 = m(x - 2) => x = 2 - 1/mSimilarly, it intersects the y-axis at x=0:y - 1 = m(0 - 2) => y = 1 - 2mSo the intercepts are at (2 - 1/m, 0) and (0, 1 - 2m).The area of triangle ABO is (1/2)*|2 - 1/m|*|1 - 2m| = 4Since the intercepts are on positive axes, 2 - 1/m > 0 and 1 - 2m > 0.So 2 - 1/m > 0 => 1/m < 2 => m > 1/2 (if m positive) or m < 0.But 1 - 2m > 0 => 2m < 1 => m < 1/2.So combining, m must be negative.So m < 0.Now, the area equation:(1/2)*(2 - 1/m)*(1 - 2m) = 4Multiply both sides by 2:(2 - 1/m)*(1 - 2m) = 8Expand the left side:2*(1 - 2m) - (1/m)*(1 - 2m) = 8Simplify:2 - 4m - (1/m - 2) = 8Combine terms:2 - 4m - 1/m + 2 = 8Wait, that seems incorrect. Let me expand correctly:(2 - 1/m)(1 - 2m) = 2*1 + 2*(-2m) - (1/m)*1 + (1/m)*(2m)= 2 - 4m - 1/m + 2Wait, that's 2 + 2 = 4, so 4 - 4m - 1/m = 8So 4 - 4m - 1/m = 8Bring 8 to left:4 - 4m - 1/m - 8 = 0 => -4 - 4m - 1/m = 0Multiply both sides by m to eliminate denominator:-4m - 4m^2 - 1 = 0Rearrange:4m^2 + 4m + 1 = 0This is a quadratic equation: 4m^2 + 4m + 1 = 0Solving for m:m = [-4 ± sqrt(16 - 16)] / 8 = [-4 ± 0]/8 = -4/8 = -1/2So m = -1/2Therefore, the equation of the line is y - 1 = (-1/2)(x - 2)Simplify:y - 1 = (-1/2)x + 1y = (-1/2)x + 2Which is the same as before.So the equation is y = (-1/2)x + 2, or x + 2y = 4.Therefore, the equation of l1 is x + 2y = 4.Wait, but in the initial solution, the user wrote the equation as y = (-1/2)x + 2, which is correct.So, summarizing:Part (1): The equations are x + 2y - 4 = 0 and x = 2.Part (2): The equation is y = (-1/2)x + 2.I think that's it.

💡Okay, so I have this problem here: I need to find the real number ( a ) such that the function ( f(x) = frac{sin x}{(x - a)(x + 1)} ) is an odd function. Hmm, okay. Let me think about what it means for a function to be odd.From what I remember, an odd function satisfies the condition ( f(-x) = -f(x) ) for all ( x ) in its domain. So, if I plug in ( -x ) into the function, it should equal the negative of the original function. That seems straightforward. Let me write that down:( f(-x) = -f(x) )So, substituting ( f(x) ) into this equation, I get:( frac{sin(-x)}{(-x - a)(-x + 1)} = -frac{sin x}{(x - a)(x + 1)} )Wait, okay, let me make sure I substituted correctly. The numerator becomes ( sin(-x) ), which I know is equal to ( -sin x ). So, replacing ( sin(-x) ) with ( -sin x ), the equation becomes:( frac{-sin x}{(-x - a)(-x + 1)} = -frac{sin x}{(x - a)(x + 1)} )Hmm, now I can factor out the negative sign in the numerator on the left side. Let me do that:( frac{-sin x}{(-x - a)(-x + 1)} = -frac{sin x}{(x - a)(x + 1)} )So, if I multiply both sides by ( -1 ), I can get rid of the negative signs. Let's see:Left side: ( frac{-sin x}{(-x - a)(-x + 1)} times (-1) = frac{sin x}{(-x - a)(-x + 1)} )Right side: ( -frac{sin x}{(x - a)(x + 1)} times (-1) = frac{sin x}{(x - a)(x + 1)} )So now, the equation simplifies to:( frac{sin x}{(-x - a)(-x + 1)} = frac{sin x}{(x - a)(x + 1)} )Since ( sin x ) is not zero for all ( x ), I can divide both sides by ( sin x ), assuming ( sin x neq 0 ). So, that gives me:( frac{1}{(-x - a)(-x + 1)} = frac{1}{(x - a)(x + 1)} )Alright, so now I have an equation without the sine term. Let me write that as:( frac{1}{(-x - a)(-x + 1)} = frac{1}{(x - a)(x + 1)} )To make this easier, maybe I can cross-multiply to eliminate the denominators. Let's try that:( (x - a)(x + 1) = (-x - a)(-x + 1) )Okay, so now I have an equation where both sides are products of two binomials. Let me expand both sides to see if I can find ( a ).Starting with the left side:( (x - a)(x + 1) )Using the distributive property (FOIL):First: ( x times x = x^2 )Outer: ( x times 1 = x )Inner: ( -a times x = -a x )Last: ( -a times 1 = -a )So, combining these terms:( x^2 + x - a x - a )Combine like terms:( x^2 + (1 - a)x - a )Okay, that's the left side. Now, let's expand the right side:( (-x - a)(-x + 1) )Again, using FOIL:First: ( -x times -x = x^2 )Outer: ( -x times 1 = -x )Inner: ( -a times -x = a x )Last: ( -a times 1 = -a )So, combining these terms:( x^2 - x + a x - a )Combine like terms:( x^2 + (-1 + a)x - a )So, now we have both sides expanded:Left side: ( x^2 + (1 - a)x - a )Right side: ( x^2 + (a - 1)x - a )Now, since these two expressions are equal for all ( x ), their corresponding coefficients must be equal. Let's set the coefficients equal to each other.First, the ( x^2 ) terms:Left: ( 1 )Right: ( 1 )They are equal, so nothing to do here.Next, the ( x ) terms:Left: ( (1 - a) )Right: ( (a - 1) )So, setting them equal:( 1 - a = a - 1 )Hmm, let me solve this equation for ( a ).Bring ( a ) to the left and constants to the right:( 1 + 1 = a + a )( 2 = 2a )Divide both sides by 2:( a = 1 )Okay, so ( a = 1 ). Let me check the constant terms just to make sure.Left side constant term: ( -a )Right side constant term: ( -a )So, they are equal regardless of ( a ). So, that doesn't give us any new information.Therefore, the only condition we get is ( a = 1 ).Let me double-check my steps to make sure I didn't make a mistake.1. I started by using the definition of an odd function: ( f(-x) = -f(x) ).2. Substituted ( f(x) ) into this equation, correctly replacing ( x ) with ( -x ) and recognizing that ( sin(-x) = -sin x ).3. Simplified the equation by canceling out ( -sin x ) on both sides, leading to an equation without the sine term.4. Cross-multiplied to eliminate denominators, resulting in a polynomial equation.5. Expanded both sides, carefully applying the distributive property.6. Compared coefficients of like terms, leading to an equation for ( a ).7. Solved for ( a ), getting ( a = 1 ).8. Verified that the constant terms are equal regardless of ( a ), so no additional constraints.Everything seems to check out. I don't see any mistakes in my reasoning. So, I think ( a = 1 ) is the correct answer.**Final Answer**The real number ( a ) is boxed{1}.

💡Okay, so I need to find the inverse of this matrix: (begin{pmatrix} 4 & -3 5 & -2 end{pmatrix}). Hmm, I remember that not all matrices have inverses, so first I should check if this one does. I think the way to do that is by calculating the determinant. Alright, the determinant of a 2x2 matrix (begin{pmatrix} a & b c & d end{pmatrix}) is (ad - bc). Let me apply that here. So for this matrix, (a = 4), (b = -3), (c = 5), and (d = -2). Plugging those into the formula, the determinant should be (4*(-2) - (-3)*5). Let me compute that: (4*(-2)) is (-8), and (-3*5) is (-15), but since it's subtracting a negative, it becomes positive. So (-8 + 15) equals (7). Okay, so the determinant is 7. Since 7 is not zero, that means the matrix does have an inverse. Good, I don't have to give the zero matrix as an answer. Now, how do I find the inverse? I think there's a formula for the inverse of a 2x2 matrix. Let me recall it. I believe the inverse of a matrix (begin{pmatrix} a & b c & d end{pmatrix}) is (frac{1}{ad - bc}) times (begin{pmatrix} d & -b -c & a end{pmatrix}). So, applying that to our matrix, which has (a = 4), (b = -3), (c = 5), and (d = -2), the inverse should be (frac{1}{7}) times (begin{pmatrix} -2 & 3 -5 & 4 end{pmatrix}). Let me write that out step by step. First, the determinant is 7, so the scalar factor is (frac{1}{7}). Then, swapping (a) and (d), we get (-2) and (4). Then, changing the signs of (b) and (c), we get (3) and (-5). So putting it all together, the inverse matrix is (begin{pmatrix} -2/7 & 3/7 -5/7 & 4/7 end{pmatrix}). Wait, let me double-check that. If I multiply the original matrix by this inverse, I should get the identity matrix. Let me do a quick multiplication to verify. Multiplying (begin{pmatrix} 4 & -3 5 & -2 end{pmatrix}) by (begin{pmatrix} -2/7 & 3/7 -5/7 & 4/7 end{pmatrix}):First element: (4*(-2/7) + (-3)*(-5/7)) equals (-8/7 + 15/7) which is (7/7 = 1).Second element: (4*(3/7) + (-3)*(4/7)) equals (12/7 - 12/7 = 0).Third element: (5*(-2/7) + (-2)*(-5/7)) equals (-10/7 + 10/7 = 0).Fourth element: (5*(3/7) + (-2)*(4/7)) equals (15/7 - 8/7 = 7/7 = 1).So, the product is indeed the identity matrix. That means my inverse is correct. I think I've got it. The inverse is (begin{pmatrix} -2/7 & 3/7 -5/7 & 4/7 end{pmatrix}). Just to make sure I didn't make any calculation mistakes, let me go through the steps again. Determinant: (4*(-2) - (-3)*5 = -8 + 15 = 7). Correct. Inverse formula: (frac{1}{7}) times (begin{pmatrix} -2 & 3 -5 & 4 end{pmatrix}). Yep, that looks right. Dividing each element by 7 gives the fractions I have. And the multiplication check worked out perfectly. So, I'm confident that's the correct inverse.

💡Okay, so I have this problem where there's an international society with members from six countries, and there are a total of 1978 members. Each member has a unique identification number from 1 to 1978. I need to prove that there's at least one member whose ID number is either the sum of two other members from the same country or twice the number of another member from the same country.Hmm, let me try to break this down. First, since there are six countries and 1978 members, by the pigeonhole principle, at least one country must have a significant number of members. Let me calculate that. If I divide 1978 by 6, that's approximately 329.67. So, at least one country must have at least 330 members. Let's call this country A.Now, in country A, there are at least 330 members. Let me denote the largest member number in country A as m₁. Since there are 330 members, the smallest number in country A must be less than or equal to m₁ - 329. That's because if you have 330 consecutive numbers, the smallest would be m₁ - 329.So, if I consider the pairs (m₁ - a_i) where a_i are the other member numbers in country A, there are at least 329 such differences. Now, if none of these differences correspond to any member number in country A, then they must belong to the other five countries. Applying the pigeonhole principle again, if these 329 differences are distributed among the remaining five countries, at least one of these countries, say country B, must contain at least 66 of these differences. So, country B has at least 66 members.Now, let's look at country B. Let m₂ be the largest member number in country B. Similarly, the smallest number in country B is less than or equal to m₂ - 65. So, considering the differences m₂ - b_i, where b_i are the other member numbers in country B, we get at least 65 new differences. If these differences are not in country B, they must be in the remaining four countries.Continuing this process, if I take one of the remaining four countries, say country C, and apply the same logic, I'll find that country C must have at least 17 of these differences. Then, moving on to country D, E, and F, each time reducing the number of differences and the number of countries.Eventually, I'll end up with a country where the differences can't be distributed anymore without overlapping with the existing member numbers. This means that at some point, one of these differences must correspond to a member number in the same country, which contradicts our initial assumption.Therefore, this contradiction implies that our assumption was wrong, and there must exist at least one member whose identification number is either the sum of two other members from the same country or twice the number of another member from the same country.I think this makes sense. By repeatedly applying the pigeonhole principle and considering the differences, we force a contradiction, which proves the original statement.**Final Answer**boxed{text{The original statement holds}}

💡First, I recognize that the car's speed increases by 2 km/h every hour. This means that each subsequent hour, the car travels 2 km more than the previous hour.The distance traveled in the first hour is 55 km. For the second hour, the distance increases to 57 km, then to 59 km in the third hour, and so on.This pattern forms an arithmetic sequence where:- The first term (a₁) is 55 km.- The common difference (d) is 2 km.- The number of terms (n) is 12.To find the total distance traveled in 12 hours, I use the formula for the sum of an arithmetic sequence:Sₙ = n/2 * (2a₁ + (n - 1)d)Plugging in the values:S₁₂ = 12/2 * (2*55 + (12 - 1)*2)S₁₂ = 6 * (110 + 22)S₁₂ = 6 * 132S₁₂ = 792 kmTherefore, the total distance traveled in 12 hours is 792 km.