answer：Okay, so I need to find the equation of the directrix of the parabola given by the equation ( y = frac{x^2 - 4x + 4}{8} ). Hmm, I remember that the directrix is a line that's used in the definition of a parabola. It's kind of like the opposite of the focus; for any point on the parabola, the distance to the focus equals the distance to the directrix. But I'm a bit fuzzy on the exact steps to find it, so I need to work through this carefully.First, I think it would help to rewrite the given equation in a more standard form. The equation is ( y = frac{x^2 - 4x + 4}{8} ). I notice that the numerator is a quadratic expression, so maybe I can complete the square to make it easier to identify the vertex and other features of the parabola.Let me try completing the square for the quadratic in the numerator. The expression is ( x^2 - 4x + 4 ). To complete the square, I take the coefficient of x, which is -4, divide it by 2 to get -2, and then square it to get 4. So, ( x^2 - 4x + 4 ) can be written as ( (x - 2)^2 ). That simplifies the equation to ( y = frac{(x - 2)^2}{8} ).Now, this looks more like the standard form of a parabola. The standard form for a parabola that opens upward or downward is ( y = a(x - h)^2 + k ), where (h, k) is the vertex of the parabola. In this case, comparing ( y = frac{(x - 2)^2}{8} ) to the standard form, I can see that h is 2 and k is 0. So, the vertex of the parabola is at (2, 0).Next, I need to recall how to find the directrix. I remember that for a parabola in the form ( y = a(x - h)^2 + k ), the focus is located at ( (h, k + frac{1}{4a}) ) and the directrix is the line ( y = k - frac{1}{4a} ). Let me verify that. If the parabola opens upward, the focus is above the vertex, and the directrix is below the vertex. Since our equation is ( y = frac{1}{8}(x - 2)^2 ), the value of a is ( frac{1}{8} ). So, plugging into the formula for the focus, it should be at ( (2, 0 + frac{1}{4 times frac{1}{8}}) ). Calculating ( frac{1}{4a} ), where ( a = frac{1}{8} ), we get ( frac{1}{4 times frac{1}{8}} = frac{1}{frac{1}{2}} = 2 ). So, the focus is at (2, 0 + 2) = (2, 2). Wait, if the focus is at (2, 2), then the directrix should be a line that's the same distance below the vertex as the focus is above the vertex. Since the vertex is at (2, 0) and the focus is 2 units above, the directrix should be 2 units below the vertex. That would make the directrix the line ( y = 0 - 2 = -2 ).But hold on, I think I might have made a mistake here. Let me double-check the formula for the directrix. The formula I used was ( y = k - frac{1}{4a} ). Plugging in the values, ( k = 0 ) and ( a = frac{1}{8} ), so ( frac{1}{4a} = frac{1}{4 times frac{1}{8}} = frac{1}{frac{1}{2}} = 2 ). Therefore, the directrix should be ( y = 0 - 2 = -2 ). Wait, but earlier I thought the focus was at (2, 2), which seems correct because ( frac{1}{4a} = 2 ). So, the focus is 2 units above the vertex, and the directrix is 2 units below the vertex. That makes sense because the parabola opens upward, so the focus is above and the directrix is below.But let me think again. The standard form is ( y = a(x - h)^2 + k ), and the distance from the vertex to the focus (and also to the directrix) is ( frac{1}{4a} ). In this case, ( a = frac{1}{8} ), so ( frac{1}{4a} = frac{1}{4 times frac{1}{8}} = frac{1}{frac{1}{2}} = 2 ). So, the focus is at (h, k + ( frac{1}{4a} )) = (2, 0 + 2) = (2, 2), and the directrix is the line ( y = k - frac{1}{4a} ) = 0 - 2 = -2.Wait, but in the initial problem, the equation was ( y = frac{x^2 - 4x + 4}{8} ), which simplifies to ( y = frac{(x - 2)^2}{8} ). So, the vertex is at (2, 0), and the parabola opens upward because the coefficient is positive.Therefore, the focus is 2 units above the vertex at (2, 2), and the directrix is 2 units below the vertex at y = -2.But wait, in the initial problem, the equation was ( y = frac{x^2 - 4x + 4}{8} ). Let me confirm that I didn't make a mistake in completing the square. The numerator is ( x^2 - 4x + 4 ), which factors to ( (x - 2)^2 ), so that part is correct.So, the equation becomes ( y = frac{(x - 2)^2}{8} ), which is indeed in the standard form ( y = a(x - h)^2 + k ) with h = 2, k = 0, and a = 1/8.Therefore, the distance from the vertex to the focus is ( frac{1}{4a} = frac{1}{4 times frac{1}{8}} = 2 ). So, the focus is at (2, 0 + 2) = (2, 2), and the directrix is the line y = 0 - 2 = -2.Wait, but I'm a bit confused because sometimes I see the standard form written as ( (x - h)^2 = 4p(y - k) ), where p is the distance from the vertex to the focus. In that case, comparing ( y = frac{(x - 2)^2}{8} ) to ( (x - h)^2 = 4p(y - k) ), we can rewrite it as ( (x - 2)^2 = 8(y - 0) ), so 4p = 8, which means p = 2. Therefore, the focus is at (h, k + p) = (2, 0 + 2) = (2, 2), and the directrix is the line y = k - p = 0 - 2 = -2.Yes, that confirms it. So, the directrix is y = -2.Wait, but in the initial problem, the equation was ( y = frac{x^2 - 4x + 4}{8} ). Let me just plot a point to make sure. If x = 2, then y = 0, which is the vertex. If x = 0, then y = (0 - 0 + 4)/8 = 4/8 = 0.5. So, the point (0, 0.5) is on the parabola. The focus is at (2, 2), so the distance from (0, 0.5) to (2, 2) should equal the distance from (0, 0.5) to the directrix y = -2.Calculating the distance from (0, 0.5) to (2, 2): distance = sqrt[(2 - 0)^2 + (2 - 0.5)^2] = sqrt[4 + 2.25] = sqrt[6.25] = 2.5.Distance from (0, 0.5) to the directrix y = -2: since the directrix is a horizontal line, the distance is the vertical distance from (0, 0.5) to y = -2, which is |0.5 - (-2)| = |2.5| = 2.5.So, both distances are equal, which confirms that the directrix is indeed y = -2.Wait, but in the initial problem, the equation was ( y = frac{x^2 - 4x + 4}{8} ). Let me make sure I didn't make any calculation errors. The numerator is ( x^2 - 4x + 4 ), which is ( (x - 2)^2 ), so the equation is ( y = frac{(x - 2)^2}{8} ). That's correct.So, in standard form, it's ( y = a(x - h)^2 + k ), where a = 1/8, h = 2, k = 0. The distance from the vertex to the focus is ( frac{1}{4a} = frac{1}{4 times frac{1}{8}} = 2 ). Therefore, the focus is at (2, 0 + 2) = (2, 2), and the directrix is y = 0 - 2 = -2.Yes, that all checks out. So, the equation of the directrix is y = -2.

answer：Alright, so I have this expression: ( x^2 - y^2 - z^2 + 3yz + x + 2y - z ). The question is asking if it has any linear factor with integer coefficients. The options given are A through D, with A being "None" and the others being specific linear expressions. First, I need to recall what a linear factor is. A linear factor would be something like ( ax + by + cz + d ), where ( a ), ( b ), ( c ), and ( d ) are integers. So, I need to check if the given quadratic expression can be factored into two such linear expressions.Looking at the expression, it has terms with ( x^2 ), ( y^2 ), ( z^2 ), and cross terms like ( yz ). It also has linear terms in ( x ), ( y ), and ( z ). Since it's a quadratic expression in three variables, factoring it might be a bit tricky, but let's see.I remember that sometimes quadratic expressions can be factored by grouping or by completing the square. Maybe I can try to rearrange the terms to see if any patterns emerge.Let me write down the expression again:( x^2 - y^2 - z^2 + 3yz + x + 2y - z )I notice that ( -y^2 - z^2 + 3yz ) can be rewritten as ( -(y^2 + z^2 - 3yz) ). Hmm, that looks like it could be part of a perfect square. Let me check:( (y - frac{3}{2}z)^2 = y^2 - 3yz + frac{9}{4}z^2 )But in our expression, we have ( y^2 + z^2 - 3yz ), which is different. So, maybe that's not the right approach.Alternatively, maybe I can group the quadratic terms and the linear terms separately:Quadratic terms: ( x^2 - y^2 - z^2 + 3yz )Linear terms: ( x + 2y - z )If I can factor the quadratic part, maybe I can then see if the linear terms fit into the factorization.Looking at the quadratic part: ( x^2 - y^2 - z^2 + 3yz )This resembles a quadratic form. Maybe I can factor it as a product of two linear terms. Let's assume it factors as ( (x + a y + b z)(x + c y + d z) ). Then, expanding this would give:( x^2 + (a + c)xy + (b + d)xz + (ac)y^2 + (ad + bc)yz + (bd)z^2 )Comparing this with our quadratic part ( x^2 - y^2 - z^2 + 3yz ), we can set up equations:1. Coefficient of ( x^2 ): 1 = 1 (which is satisfied)2. Coefficient of ( xy ): 0 = a + c3. Coefficient of ( xz ): 0 = b + d4. Coefficient of ( y^2 ): -1 = ac5. Coefficient of ( yz ): 3 = ad + bc6. Coefficient of ( z^2 ): -1 = bdFrom equation 2: ( c = -a )From equation 3: ( d = -b )Substituting into equation 4: ( -1 = a(-a) = -a^2 ) => ( a^2 = 1 ) => ( a = 1 ) or ( a = -1 )Similarly, substituting into equation 6: ( -1 = b(-b) = -b^2 ) => ( b^2 = 1 ) => ( b = 1 ) or ( b = -1 )Let's consider ( a = 1 ) and ( b = 1 ):Then, ( c = -1 ) and ( d = -1 )Now, check equation 5: ( ad + bc = (1)(-1) + (1)(-1) = -1 -1 = -2 ), which is not equal to 3. So this doesn't work.Next, try ( a = 1 ) and ( b = -1 ):Then, ( c = -1 ) and ( d = 1 )Check equation 5: ( ad + bc = (1)(1) + (1)(-1) = 1 -1 = 0 ), which is not 3.Next, ( a = -1 ) and ( b = 1 ):Then, ( c = 1 ) and ( d = -1 )Check equation 5: ( ad + bc = (-1)(-1) + (-1)(1) = 1 -1 = 0 ), not 3.Finally, ( a = -1 ) and ( b = -1 ):Then, ( c = 1 ) and ( d = 1 )Check equation 5: ( ad + bc = (-1)(1) + (-1)(1) = -1 -1 = -2 ), not 3.So, none of these combinations work. That suggests that the quadratic part ( x^2 - y^2 - z^2 + 3yz ) doesn't factor into linear terms with integer coefficients. But wait, maybe I missed something. Let me double-check the calculations.When ( a = 1 ) and ( b = 1 ):( ad + bc = (1)(-1) + (1)(-1) = -1 -1 = -2 )Nope, still -2.When ( a = 1 ) and ( b = -1 ):( ad + bc = (1)(1) + (1)(-1) = 1 -1 = 0 )Still not 3.Same with the others.Hmm, maybe the quadratic part doesn't factor nicely, which would mean the entire expression doesn't have a linear factor with integer coefficients. But let's not give up yet.Maybe I should consider the entire expression, including the linear terms, when trying to factor it. Perhaps the linear terms can help in completing the square or something.Let me try to rearrange the expression:( x^2 + x - y^2 + 2y - z^2 - z + 3yz )Now, group similar terms:( x^2 + x - y^2 + 2y - z^2 - z + 3yz )Maybe I can complete the square for each variable.Starting with ( x ):( x^2 + x = x^2 + x + frac{1}{4} - frac{1}{4} = (x + frac{1}{2})^2 - frac{1}{4} )For ( y ):( -y^2 + 2y = -(y^2 - 2y) = -(y^2 - 2y + 1 - 1) = -(y - 1)^2 + 1 )For ( z ):( -z^2 - z = -(z^2 + z) = -(z^2 + z + frac{1}{4} - frac{1}{4}) = -(z + frac{1}{2})^2 + frac{1}{4} )So, substituting these back into the expression:( (x + frac{1}{2})^2 - frac{1}{4} - (y - 1)^2 + 1 - (z + frac{1}{2})^2 + frac{1}{4} + 3yz )Simplify the constants:( -frac{1}{4} + 1 + frac{1}{4} = 1 )So, the expression becomes:( (x + frac{1}{2})^2 - (y - 1)^2 - (z + frac{1}{2})^2 + 3yz + 1 )Hmm, not sure if this helps. The cross term ( 3yz ) is still there, and the squares are not combining nicely.Maybe another approach. Let's consider that if the expression factors into two linear terms, say ( (ax + by + cz + d)(ex + fy + gz + h) ), then the product should match the original expression.Given that the leading term is ( x^2 ), likely ( a = e = 1 ) or ( a = -1 ) and ( e = -1 ). Let's assume ( a = 1 ) and ( e = 1 ) for simplicity.So, let's suppose:( (x + m y + n z + p)(x + q y + r z + s) )Expanding this:( x^2 + (m + q)xy + (n + r)xz + (mq)y^2 + (mr + nq)yz + (nr)z^2 + (p + s)x + (ps)y + (pr)z + (pqrs) )Wait, actually, the constant term would be ( p cdot s ), but in our original expression, there is no constant term. So, ( p cdot s = 0 ). That means either ( p = 0 ) or ( s = 0 ).But in our original expression, we have linear terms ( x + 2y - z ). So, the coefficients of ( x ), ( y ), and ( z ) in the expanded form must match.Let me write down the coefficients:1. ( x^2 ): 1 (matches)2. ( xy ): ( m + q ) (should be 0)3. ( xz ): ( n + r ) (should be 0)4. ( y^2 ): ( mq ) (should be -1)5. ( yz ): ( mr + nq ) (should be 3)6. ( z^2 ): ( nr ) (should be -1)7. ( x ): ( p + s ) (should be 1)8. ( y ): ( ps ) (should be 2)9. ( z ): ( pr ) (should be -1)10. Constant term: ( pqrs ) (should be 0)From equation 2: ( m + q = 0 ) => ( q = -m )From equation 3: ( n + r = 0 ) => ( r = -n )From equation 4: ( mq = -1 ). Since ( q = -m ), this becomes ( m(-m) = -m^2 = -1 ) => ( m^2 = 1 ) => ( m = 1 ) or ( m = -1 )Similarly, from equation 6: ( nr = -1 ). Since ( r = -n ), this becomes ( n(-n) = -n^2 = -1 ) => ( n^2 = 1 ) => ( n = 1 ) or ( n = -1 )Let's consider ( m = 1 ) and ( n = 1 ):Then, ( q = -1 ) and ( r = -1 )From equation 5: ( mr + nq = (1)(-1) + (1)(-1) = -1 -1 = -2 ), which should be 3. Not matching.Next, ( m = 1 ) and ( n = -1 ):Then, ( q = -1 ) and ( r = 1 )From equation 5: ( mr + nq = (1)(1) + (-1)(-1) = 1 + 1 = 2 ), which is not 3.Next, ( m = -1 ) and ( n = 1 ):Then, ( q = 1 ) and ( r = -1 )From equation 5: ( mr + nq = (-1)(-1) + (1)(1) = 1 + 1 = 2 ), not 3.Finally, ( m = -1 ) and ( n = -1 ):Then, ( q = 1 ) and ( r = 1 )From equation 5: ( mr + nq = (-1)(1) + (-1)(1) = -1 -1 = -2 ), not 3.So, none of these combinations work. That means the quadratic part doesn't factor into linear terms with integer coefficients, which suggests the entire expression doesn't have a linear factor with integer coefficients.But let's not stop here. Maybe I made a mistake in assuming ( a = 1 ) and ( e = 1 ). What if one of them is -1?Let's try ( a = 1 ) and ( e = -1 ):So, ( (x + m y + n z + p)(-x + q y + r z + s) )Expanding this:( -x^2 + (q - m)xy + (r - n)xz + (-mq)y^2 + (mr - nq)yz + (-nr)z^2 + (-p + s)x + (ps)y + (pr)z + (pqrs) )Comparing with the original expression:1. ( x^2 ): -1 (should be 1) => Not matching. So, this doesn't work.Similarly, if ( a = -1 ) and ( e = 1 ), it would also lead to a negative ( x^2 ) term, which doesn't match.What if both ( a = -1 ) and ( e = -1 ):( (-x + m y + n z + p)(-x + q y + r z + s) )Expanding:( x^2 + (-q - m)xy + (-r - n)xz + (mq)y^2 + (mr + nq)yz + (nr)z^2 + (-p - s)x + (ps)y + (pr)z + (pqrs) )Now, comparing:1. ( x^2 ): 1 (matches)2. ( xy ): ( -q - m ) (should be 0)3. ( xz ): ( -r - n ) (should be 0)4. ( y^2 ): ( mq ) (should be -1)5. ( yz ): ( mr + nq ) (should be 3)6. ( z^2 ): ( nr ) (should be -1)7. ( x ): ( -p - s ) (should be 1)8. ( y ): ( ps ) (should be 2)9. ( z ): ( pr ) (should be -1)10. Constant term: ( pqrs ) (should be 0)From equation 2: ( -q - m = 0 ) => ( q = -m )From equation 3: ( -r - n = 0 ) => ( r = -n )From equation 4: ( mq = -1 ). Since ( q = -m ), this becomes ( m(-m) = -m^2 = -1 ) => ( m^2 = 1 ) => ( m = 1 ) or ( m = -1 )From equation 6: ( nr = -1 ). Since ( r = -n ), this becomes ( n(-n) = -n^2 = -1 ) => ( n^2 = 1 ) => ( n = 1 ) or ( n = -1 )Let's try ( m = 1 ) and ( n = 1 ):Then, ( q = -1 ) and ( r = -1 )From equation 5: ( mr + nq = (1)(-1) + (1)(-1) = -1 -1 = -2 ), which should be 3. Not matching.Next, ( m = 1 ) and ( n = -1 ):Then, ( q = -1 ) and ( r = 1 )From equation 5: ( mr + nq = (1)(1) + (-1)(-1) = 1 + 1 = 2 ), not 3.Next, ( m = -1 ) and ( n = 1 ):Then, ( q = 1 ) and ( r = -1 )From equation 5: ( mr + nq = (-1)(-1) + (1)(1) = 1 + 1 = 2 ), not 3.Finally, ( m = -1 ) and ( n = -1 ):Then, ( q = 1 ) and ( r = 1 )From equation 5: ( mr + nq = (-1)(1) + (-1)(1) = -1 -1 = -2 ), not 3.Again, none of these work. So, even if I consider negative coefficients for ( x ), it doesn't help.Maybe I should try a different approach. Perhaps the expression can be factored as ( (x + a y + b z + c)(d y + e z + f) ), but that would make it a product of a linear and a quadratic term, which isn't what we're looking for. We need both factors to be linear.Alternatively, maybe the expression can be factored by grouping. Let's see.Looking at the original expression:( x^2 - y^2 - z^2 + 3yz + x + 2y - z )Let me try to group the quadratic terms and the linear terms:Quadratic terms: ( x^2 - y^2 - z^2 + 3yz )Linear terms: ( x + 2y - z )If I can factor the quadratic part, maybe I can then see if the linear terms fit into the factorization.Wait, earlier I tried factoring the quadratic part and it didn't work. Maybe I need to consider that the entire expression might have a linear factor, not just the quadratic part.Let me assume that the expression factors as ( (x + a y + b z + c)(d x + e y + f z + g) ). Since the leading term is ( x^2 ), likely ( d = 1 ) or ( d = -1 ). Let's assume ( d = 1 ).So, ( (x + a y + b z + c)(x + e y + f z + g) )Expanding this:( x^2 + (a + e)xy + (b + f)xz + (ae)y^2 + (af + be)yz + (bf)z^2 + (c + g)x + (ce + bg)y + (cf)z + (cg) )Comparing with the original expression:1. ( x^2 ): 1 (matches)2. ( xy ): ( a + e ) (should be 0)3. ( xz ): ( b + f ) (should be 0)4. ( y^2 ): ( ae ) (should be -1)5. ( yz ): ( af + be ) (should be 3)6. ( z^2 ): ( bf ) (should be -1)7. ( x ): ( c + g ) (should be 1)8. ( y ): ( ce + bg ) (should be 2)9. ( z ): ( cf ) (should be -1)10. Constant term: ( cg ) (should be 0)From equation 2: ( a + e = 0 ) => ( e = -a )From equation 3: ( b + f = 0 ) => ( f = -b )From equation 4: ( ae = -1 ). Since ( e = -a ), this becomes ( a(-a) = -a^2 = -1 ) => ( a^2 = 1 ) => ( a = 1 ) or ( a = -1 )From equation 6: ( bf = -1 ). Since ( f = -b ), this becomes ( b(-b) = -b^2 = -1 ) => ( b^2 = 1 ) => ( b = 1 ) or ( b = -1 )Let's consider ( a = 1 ) and ( b = 1 ):Then, ( e = -1 ) and ( f = -1 )From equation 5: ( af + be = (1)(-1) + (1)(-1) = -1 -1 = -2 ), which should be 3. Not matching.Next, ( a = 1 ) and ( b = -1 ):Then, ( e = -1 ) and ( f = 1 )From equation 5: ( af + be = (1)(1) + (1)(-1) = 1 -1 = 0 ), not 3.Next, ( a = -1 ) and ( b = 1 ):Then, ( e = 1 ) and ( f = -1 )From equation 5: ( af + be = (-1)(-1) + (-1)(1) = 1 -1 = 0 ), not 3.Finally, ( a = -1 ) and ( b = -1 ):Then, ( e = 1 ) and ( f = 1 )From equation 5: ( af + be = (-1)(1) + (-1)(1) = -1 -1 = -2 ), not 3.So, again, none of these combinations work. This suggests that the expression doesn't factor into two linear terms with integer coefficients.But wait, maybe I missed something in the linear terms. Let's look at equations 7, 8, 9, and 10.From equation 7: ( c + g = 1 )From equation 8: ( ce + bg = 2 )From equation 9: ( cf = -1 )From equation 10: ( cg = 0 )From equation 10: ( cg = 0 ) => Either ( c = 0 ) or ( g = 0 )If ( c = 0 ), then from equation 7: ( g = 1 )From equation 9: ( cf = -1 ) => ( 0 cdot f = -1 ) => 0 = -1, which is impossible.If ( g = 0 ), then from equation 7: ( c = 1 )From equation 9: ( cf = -1 ) => ( 1 cdot f = -1 ) => ( f = -1 )From equation 8: ( ce + bg = 2 ) => ( 1 cdot e + b cdot 0 = e = 2 )But earlier, we have ( e = -a ). So, ( e = 2 ) => ( a = -2 )But earlier, we had ( a^2 = 1 ) => ( a = 1 ) or ( a = -1 ). So, ( a = -2 ) contradicts that.Therefore, this approach also doesn't work.Given all these attempts, it seems that the expression cannot be factored into two linear terms with integer coefficients. Therefore, the answer is likely A. None.But just to be thorough, let me check the options given:B. ( -x + y + frac{3}{2}z )C. ( x - y - frac{3}{2}z + 1 )D. ( x + y - frac{3}{2}z )These options have fractions, which suggests that even if the expression could be factored, it would require non-integer coefficients, which contradicts the question's requirement for integer coefficients.Therefore, the correct answer is A. None.

answer：Alright, so I have this problem about a square table where each cell has either a plus or a minus sign, and the total number of pluses and minuses are equal. I need to prove that there are either two rows or two columns with the same number of plus signs. Hmm, okay, let me try to wrap my head around this.First, since it's a square table, let's say it's an ( n times n ) table. The total number of cells is ( n^2 ). Since the number of pluses and minuses are equal, that means there are ( frac{n^2}{2} ) pluses and ( frac{n^2}{2} ) minuses. So, ( n^2 ) must be even, which implies that ( n ) is even as well. Wait, is that necessarily true? Let me think. If ( n ) is odd, ( n^2 ) would still be odd, right? So, if ( n ) is odd, ( n^2 ) is odd, and you can't have an equal number of pluses and minuses because you can't split an odd number into two equal integers. So, actually, ( n ) must be even for this to hold. Okay, so ( n ) is even, say ( n = 2k ) for some integer ( k ).So, the table is ( 2k times 2k ), and there are ( 2k^2 ) pluses and ( 2k^2 ) minuses. Now, I need to show that either two rows have the same number of pluses or two columns have the same number of pluses.Let me think about the possible number of pluses in a row. In any row, the number of pluses can range from 0 to ( 2k ). So, there are ( 2k + 1 ) possible values for the number of pluses in a row. But there are ( 2k ) rows. If all the rows had a distinct number of pluses, then the number of pluses in each row would have to be a permutation of ( 0, 1, 2, ldots, 2k ). But wait, that's ( 2k + 1 ) different numbers, and we only have ( 2k ) rows. So, by the pigeonhole principle, at least two rows must have the same number of pluses. Is that right?Wait, hold on. If we have ( 2k ) rows and ( 2k + 1 ) possible values for the number of pluses, then it's possible that each row has a unique number of pluses, except one value is missing. So, maybe not necessarily two rows have the same number. Hmm, so perhaps my initial thought was incorrect.Let me think again. Suppose that all rows have distinct numbers of pluses. Then, the total number of pluses would be the sum of ( 0 + 1 + 2 + ldots + 2k ), minus the missing number. The sum of ( 0 + 1 + 2 + ldots + 2k ) is ( frac{2k(2k + 1)}{2} = k(2k + 1) ). So, the total number of pluses would be ( k(2k + 1) - m ), where ( m ) is the missing number.But we know that the total number of pluses is ( 2k^2 ). So, ( k(2k + 1) - m = 2k^2 ). Simplifying, ( 2k^2 + k - m = 2k^2 ), which implies ( k - m = 0 ), so ( m = k ). Therefore, the missing number is ( k ). So, if all rows have distinct numbers of pluses, then the number ( k ) must be missing.So, in this case, the number of pluses in the rows would be ( 0, 1, 2, ldots, k-1, k+1, ldots, 2k ). So, one row has 0 pluses, one row has 1 plus, and so on, up to one row having ( 2k ) pluses, but skipping ( k ).Now, let's think about the columns. Each column also has a certain number of pluses. The total number of pluses is ( 2k^2 ), so the average number of pluses per column is ( frac{2k^2}{2k} = k ). So, on average, each column has ( k ) pluses.But if all the rows have distinct numbers of pluses, and the number ( k ) is missing, then the number of pluses in each row is either less than ( k ) or greater than ( k ). So, in the rows, we have ( k ) rows with fewer than ( k ) pluses and ( k ) rows with more than ( k ) pluses.Now, let's consider the columns. Each column has some number of pluses. If we sum the number of pluses in all columns, we get ( 2k^2 ). If all columns had distinct numbers of pluses, then similar to the rows, the total number of pluses would be ( 0 + 1 + 2 + ldots + 2k - m' ), where ( m' ) is the missing number. But the total is ( 2k^2 ), so ( k(2k + 1) - m' = 2k^2 ), which again gives ( m' = k ). So, the missing number for the columns would also be ( k ).But wait, if both rows and columns are missing ( k ) pluses, does that lead to a contradiction? Let me see.If all rows have distinct numbers of pluses, missing ( k ), then each row has either fewer than ( k ) or more than ( k ) pluses. Similarly, if all columns have distinct numbers of pluses, missing ( k ), then each column has either fewer than ( k ) or more than ( k ) pluses.But let's think about the total number of pluses. If all rows have distinct numbers of pluses, missing ( k ), then the total number of pluses is ( k(2k + 1) - k = 2k^2 + k - k = 2k^2 ), which matches. Similarly for columns.But now, if both rows and columns are arranged such that they miss ( k ), is that possible? Let me try to visualize a small case. Let's take ( k = 1 ), so ( n = 2 ). Then, the table is ( 2 times 2 ), with 2 pluses and 2 minuses.If all rows have distinct numbers of pluses, missing ( k = 1 ), then the rows would have 0 and 2 pluses. Similarly, the columns would have 0 and 2 pluses. But in a ( 2 times 2 ) table, if one row has 0 pluses and the other has 2 pluses, then the columns would each have 1 plus, which contradicts the idea that columns are missing ( k = 1 ). So, in this case, it's impossible for both rows and columns to have distinct numbers of pluses missing ( k ).Therefore, in the ( 2 times 2 ) case, it's impossible to have all rows and all columns with distinct numbers of pluses. Hence, there must be either two rows or two columns with the same number of pluses.Let me try to generalize this. Suppose we have a ( 2k times 2k ) table. If all rows have distinct numbers of pluses, missing ( k ), then the total number of pluses is ( 2k^2 ). Similarly, if all columns have distinct numbers of pluses, missing ( k ), the total number of pluses is also ( 2k^2 ). However, arranging both rows and columns to miss ( k ) leads to a contradiction, as seen in the ( 2 times 2 ) case.Therefore, it's impossible for both rows and columns to have all distinct numbers of pluses. Hence, there must be either two rows or two columns with the same number of pluses.Wait, but in the ( 2 times 2 ) case, we saw that if rows have distinct numbers of pluses, columns cannot have distinct numbers, and vice versa. So, in general, for any ( 2k times 2k ) table, it's impossible for both rows and columns to have all distinct numbers of pluses. Therefore, at least one of them must have two rows or two columns with the same number of pluses.So, I think this reasoning holds. By assuming that both rows and columns have distinct numbers of pluses, we arrive at a contradiction, implying that at least two rows or two columns must have the same number of pluses.Another way to think about it is using the pigeonhole principle. For the rows, there are ( 2k ) rows and ( 2k + 1 ) possible numbers of pluses (from 0 to ( 2k )). So, if all rows had distinct numbers, one number would be missing. Similarly for columns. But as we've seen, missing the same number in both rows and columns leads to a contradiction, so it's impossible for both to have all distinct counts. Therefore, at least one of them must have a repeated count.I think that solidifies the proof. So, in conclusion, in any such square table with an equal number of pluses and minuses, there must be either two rows or two columns with the same number of pluses.

answer：First, I need to understand what it means for two numbers to be 35% and 42% less than a third number. Let's denote the third number as ( X ).If the first number is 35% less than ( X ), it means the first number is 65% of ( X ) because ( 100% - 35% = 65% ).Similarly, if the second number is 42% less than ( X ), it means the second number is 58% of ( X ) because ( 100% - 42% = 58% ).Now, to find out how much percent the second number is less than the first number, I need to calculate the difference between the first and second numbers relative to the first number.The difference between the first and second numbers is ( 65% ) of ( X ) minus ( 58% ) of ( X ), which equals ( 7% ) of ( X ).To find the percentage difference relative to the first number, I divide the difference by the first number and multiply by 100:[frac{7% text{ of } X}{65% text{ of } X} times 100 = frac{7}{65} times 100 approx 10.77%]Therefore, the second number is approximately 10.77% less than the first number.

answer：Okay, so I have these three vector problems to solve. Let me take them one by one. First, the vectors given are a = (1, 2), b = (-2, m), c = a + (t² + 1)b, and d = -k a + (1/t) b. m is a real number, and k and t are positive real numbers. Problem 1: If vector a is parallel to vector b, find the value of m. Alright, so if two vectors are parallel, one must be a scalar multiple of the other. That means there exists some scalar λ such that a = λ b. So, writing this out component-wise, we have:1 = λ*(-2) 2 = λ*mFrom the first equation, 1 = -2λ, so λ = -1/2. Plugging this into the second equation, 2 = (-1/2)*m. Solving for m, we get m = 2*(-2) = -4. So, m is -4.Wait, let me double-check that. If a is parallel to b, then their components must be proportional. So, the ratio of the x-components should equal the ratio of the y-components. So, 1/(-2) should equal 2/m. So, 1/(-2) = 2/m. Cross-multiplying, 1*m = (-2)*2, so m = -4. Yep, that seems right.Problem 2: If vector a is perpendicular to vector b, find the value of m.Alright, if two vectors are perpendicular, their dot product is zero. So, a · b = 0. Calculating the dot product:(1)(-2) + (2)(m) = 0 -2 + 2m = 0 2m = 2 m = 1So, m is 1. Let me verify that. If m is 1, then vector b is (-2, 1). The dot product of a and b is (1)(-2) + (2)(1) = -2 + 2 = 0. Yep, that works.Problem 3: When m = 1, if vector x is perpendicular to vector y, find the minimum value of k.Wait, hold on. The problem mentions vectors x and y, but in the original problem statement, we have vectors c and d defined in terms of a, b, t, and k. So, I think maybe there was a typo, and x and y are supposed to be c and d? Because x and y weren't defined before. Let me check the original problem again.Ah, yes, the original problem says: When m=1, if vector x is perpendicular to vector y, find the minimum value of k. But in the given vectors, we have c and d defined. So, perhaps x and y are c and d? Or maybe x and y are other vectors? Hmm. Wait, the original problem says:"Given vectors a=(1,2), b=(-2,m), c = a + (t² +1)b, d = -k a + (1/t) b, where m∈ℝ, and k,t are positive real numbers.1. If a // b, find m.2. If a ⊥ b, find m.3. When m=1, if x ⊥ y, find the minimum value of k."So, x and y are not defined in the problem. Hmm, that's confusing. Maybe it's a translation issue or a typo. Perhaps x and y are c and d? Because c and d are defined in terms of a, b, t, and k. So, maybe the problem meant c ⊥ d? That would make sense because c and d are defined, and m=1 is given, so we can substitute m=1 into c and d.Alternatively, maybe x and y are other vectors, but since they aren't defined, perhaps it's a mistake. I think it's more likely that x and y are c and d. So, I'll proceed under that assumption.So, when m=1, c = a + (t² +1)b, and d = -k a + (1/t) b. So, let's write out c and d with m=1.First, vector a is (1,2), vector b is (-2,1) because m=1.So, c = a + (t² +1)b = (1,2) + (t² +1)*(-2,1).Calculating c:x-component: 1 + (t² +1)*(-2) = 1 - 2(t² +1) = 1 - 2t² - 2 = -2t² -1y-component: 2 + (t² +1)*1 = 2 + t² +1 = t² +3So, c = (-2t² -1, t² +3)Similarly, d = -k a + (1/t) b = -k*(1,2) + (1/t)*(-2,1)Calculating d:x-component: -k*1 + (1/t)*(-2) = -k - 2/ty-component: -k*2 + (1/t)*1 = -2k + 1/tSo, d = (-k - 2/t, -2k + 1/t)Now, since c is perpendicular to d, their dot product must be zero.So, c · d = 0.Calculating the dot product:(-2t² -1)*(-k - 2/t) + (t² +3)*(-2k + 1/t) = 0Let me compute each part step by step.First term: (-2t² -1)*(-k - 2/t)Let me expand this:= (-2t²)*(-k) + (-2t²)*(-2/t) + (-1)*(-k) + (-1)*(-2/t)= 2k t² + (4t²)/t + k + 2/tSimplify:= 2k t² + 4t + k + 2/tSecond term: (t² +3)*(-2k + 1/t)Expanding:= t²*(-2k) + t²*(1/t) + 3*(-2k) + 3*(1/t)= -2k t² + t + (-6k) + 3/tSimplify:= -2k t² + t -6k + 3/tNow, adding the two terms together:First term + Second term:(2k t² + 4t + k + 2/t) + (-2k t² + t -6k + 3/t) = 0Combine like terms:2k t² -2k t² = 04t + t = 5tk -6k = -5k2/t + 3/t = 5/tSo, overall:0 + 5t -5k + 5/t = 0Simplify:5t -5k + 5/t = 0Divide both sides by 5:t - k + 1/t = 0So, t + 1/t = kSo, k = t + 1/tNow, we need to find the minimum value of k, given that t is a positive real number.So, k is expressed in terms of t as k = t + 1/t.We need to find the minimum value of k.This is a standard optimization problem. The function f(t) = t + 1/t for t > 0 has its minimum at t = 1, and the minimum value is 2.We can confirm this using calculus or the AM-GM inequality.Using AM-GM inequality: For positive real numbers t and 1/t, the arithmetic mean is (t + 1/t)/2, and the geometric mean is sqrt(t*(1/t)) = 1. By AM-GM, (t + 1/t)/2 ≥ 1, so t + 1/t ≥ 2, with equality when t = 1/t, i.e., t =1.Therefore, the minimum value of k is 2.So, summarizing:1. When a is parallel to b, m = -4.2. When a is perpendicular to b, m =1.3. When m=1, and c is perpendicular to d, the minimum value of k is 2.I think that makes sense. Let me just double-check the calculations for problem 3.Starting from c · d =0:(-2t² -1)(-k - 2/t) + (t² +3)(-2k +1/t) =0Expanding:First term:(-2t²)(-k) = 2k t²(-2t²)(-2/t) = 4t(-1)(-k) = k(-1)(-2/t) = 2/tSecond term:(t²)(-2k) = -2k t²(t²)(1/t) = t(3)(-2k) = -6k(3)(1/t) = 3/tAdding all together:2k t² +4t +k +2/t -2k t² +t -6k +3/tSimplify:2k t² -2k t² =04t + t =5tk -6k =-5k2/t +3/t=5/tSo, 5t -5k +5/t=0Divide by 5: t -k +1/t=0 => k = t +1/tYes, that's correct. So, k = t +1/t, which has a minimum at t=1, k=2. So, the minimum value is 2.I think that's solid.

answer：Okay, so I have this problem where I need to figure out how many ways I can arrange the numbers from 1 to 100 in a circle. The condition is that the sum of every two numbers placed opposite each other has to be the same. Also, arrangements that are rotations of each other count as the same. The answer needs to be in the form (a! cdot b^c). Hmm, let me try to break this down.First, arranging numbers in a circle where opposite pairs have the same sum. That makes me think of pairs of numbers that add up to the same total. Since we're dealing with numbers from 1 to 100, the total sum of all numbers is ( frac{100 times 101}{2} = 5050 ). But wait, that's the sum of all numbers, not the sum of each pair. Each pair of opposite numbers should have the same sum. Let's call this common sum ( S ).If each pair adds up to ( S ), and there are 50 such pairs (since 100 numbers arranged in a circle, each number has one opposite), then the total sum of all numbers should be ( 50 times S ). So, ( 50 times S = 5050 ), which means ( S = frac{5050}{50} = 101 ). Okay, so each pair of opposite numbers must add up to 101.That means the pairs are (1,100), (2,99), (3,98), ..., (50,51). So, each number from 1 to 50 is paired with a number from 51 to 100 such that their sum is 101. So, I have 50 such pairs.Now, arranging these pairs in a circle. Since it's a circle, rotations are considered the same. So, fixing one position might help avoid counting rotations multiple times. Let me think about how to arrange these pairs.If I fix one number, say 1, in a position, then its opposite must be 100. Then, I need to arrange the remaining 49 pairs around the circle. Each pair can be placed in two different ways: either the smaller number first or the larger number first. So, for each pair, there are 2 choices.But wait, since it's a circle, fixing one position already accounts for rotational symmetry. So, once I fix 1 and 100 opposite each other, I just need to arrange the remaining 49 pairs around the circle. Each of these pairs can be arranged in 2 ways, so that would be ( 2^{49} ) ways.But also, the order of the pairs themselves can be permuted. Since there are 49 pairs left, they can be arranged in ( 49! ) ways around the circle. So, putting it all together, the total number of arrangements would be ( 49! times 2^{49} ).Wait, let me double-check. If I fix one pair, say (1,100), then the remaining 49 pairs can be arranged in the circle. Each pair can be placed in two different orientations, so that's ( 2^{49} ). And the number of ways to arrange the 49 pairs around the circle is ( 49! ). So, yes, multiplying those together gives the total number of distinct arrangements, considering rotations as the same.So, the final answer should be ( 49! times 2^{49} ). Let me see if that makes sense. If I didn't fix a position, I would have to divide by the number of rotations, which is 100, but since I fixed one position, I don't need to do that anymore. So, yeah, ( 49! times 2^{49} ) should be the correct count.I think that's it. I don't see any mistakes in this reasoning. Each step seems to follow logically, and I accounted for the rotational symmetry by fixing one position. The number of ways to arrange the remaining pairs and their orientations gives the total number of valid arrangements.