answer：Okay, so I have this problem to prove using mathematical induction: [(n+1) times (n+2) times ldots times (n+n) = 2^n times 1 times 3 times ldots times (2n-1)]Hmm, induction. Alright, I remember that induction involves two main steps: the base case and the inductive step. Let me try to recall how that works.First, the base case. I need to show that the statement holds true for the initial value of ( n ). Usually, that's ( n = 1 ). Let me plug in ( n = 1 ) into both sides of the equation.Left side: When ( n = 1 ), it's just ( (1+1) ) which is ( 2 ).Right side: ( 2^1 times 1 times 3 times ldots times (2 times 1 - 1) ). Wait, ( 2 times 1 - 1 = 1 ), so it's ( 2 times 1 = 2 ).Okay, both sides are equal when ( n = 1 ). So the base case holds. That wasn't too bad.Now, the inductive step. I need to assume that the statement is true for some arbitrary positive integer ( k ), and then show that it must also be true for ( k + 1 ).So, let's assume that for ( n = k ), the equation holds:[(k+1) times (k+2) times ldots times (k+k) = 2^k times 1 times 3 times ldots times (2k - 1)]This is my induction hypothesis. Now, I need to show that:[(k+2) times (k+3) times ldots times (k+1 + (k+1)) = 2^{k+1} times 1 times 3 times ldots times (2(k+1) - 1)]Wait, let me make sure I understand the left side correctly. For ( n = k + 1 ), the product starts at ( (k+1) + 1 = k + 2 ) and goes up to ( (k+1) + (k+1) = 2k + 2 ). So, the left side is:[(k+2) times (k+3) times ldots times (2k + 2)]And the right side is:[2^{k+1} times 1 times 3 times ldots times (2(k+1) - 1) = 2^{k+1} times 1 times 3 times ldots times (2k + 1)]Alright, so I need to show that the left side equals the right side. Let me see how I can relate this to the induction hypothesis.Looking at the left side for ( n = k + 1 ), it's the product from ( k + 2 ) to ( 2k + 2 ). I can think of this as the product from ( k + 1 ) to ( 2k + 2 ) divided by ( (k + 1) ). But wait, that might complicate things.Alternatively, maybe I can express the left side for ( k + 1 ) in terms of the left side for ( k ). Let's see:The left side for ( k ) is:[(k+1) times (k+2) times ldots times (2k)]And the left side for ( k + 1 ) is:[(k+2) times (k+3) times ldots times (2k + 2)]So, the left side for ( k + 1 ) is the left side for ( k ) multiplied by ( (2k + 1) times (2k + 2) ) divided by ( (k + 1) ). Wait, that might not be straightforward.Hold on, perhaps another approach. Let me write out the left side for ( k + 1 ):[(k+2)(k+3)ldots(2k+2)]This can be written as:[(k+1 + 1)(k+1 + 2)ldots(2(k+1))]Which is similar to the original product but shifted by 1. Hmm, maybe I can factor out something.Alternatively, perhaps I can express the left side for ( k + 1 ) as the left side for ( k ) multiplied by ( (2k + 1)(2k + 2) ) divided by ( (k + 1) ). Let me check:If I take the left side for ( k ):[(k+1)(k+2)ldots(2k)]And multiply it by ( (2k + 1)(2k + 2) ), I get:[(k+1)(k+2)ldots(2k)(2k + 1)(2k + 2)]But that's actually the product up to ( 2k + 2 ), which is the left side for ( k + 1 ) multiplied by ( (k + 1) ). Wait, no:Wait, the left side for ( k + 1 ) is from ( k + 2 ) to ( 2k + 2 ), which is the same as the left side for ( k ) multiplied by ( (2k + 1)(2k + 2) ) divided by ( (k + 1) ). Let me verify:Left side for ( k + 1 ):[(k+2)(k+3)ldots(2k+2)]This can be written as:[frac{(k+1)(k+2)ldots(2k+2)}{(k+1)}]Which is:[frac{text{Left side for } k times (2k + 1)(2k + 2)}{(k + 1)}]So, yes, that's correct. Therefore, the left side for ( k + 1 ) is:[frac{text{Left side for } k times (2k + 1)(2k + 2)}{(k + 1)}]Now, from the induction hypothesis, the left side for ( k ) is equal to ( 2^k times 1 times 3 times ldots times (2k - 1) ). So, substituting that in:Left side for ( k + 1 ):[frac{2^k times 1 times 3 times ldots times (2k - 1) times (2k + 1)(2k + 2)}{(k + 1)}]Simplify the fraction:[frac{2k + 2}{k + 1} = 2]So, the expression becomes:[2^k times 1 times 3 times ldots times (2k - 1) times (2k + 1) times 2]Which is:[2^{k + 1} times 1 times 3 times ldots times (2k - 1) times (2k + 1)]But the right side for ( k + 1 ) is:[2^{k + 1} times 1 times 3 times ldots times (2(k + 1) - 1) = 2^{k + 1} times 1 times 3 times ldots times (2k + 1)]So, both sides are equal. Therefore, if the statement holds for ( n = k ), it also holds for ( n = k + 1 ).Since we've verified the base case and the inductive step, by mathematical induction, the given equation holds for all positive integers ( n ).

answer：Okay, so I have this problem about two right circular cylinders that have the same volume. The radius of the second cylinder is 20% more than the radius of the first. I need to figure out the relationship between their heights. Hmm, let me think about how to approach this.First, I remember that the volume of a right circular cylinder is given by the formula:[ V = pi r^2 h ]where ( V ) is the volume, ( r ) is the radius, and ( h ) is the height. Since both cylinders have the same volume, I can set their volume equations equal to each other.Let me denote the radius and height of the first cylinder as ( r_1 ) and ( h_1 ), respectively. For the second cylinder, the radius is 20% more than the first, so I can express that as:[ r_2 = r_1 + 0.2r_1 = 1.2r_1 ]Okay, so the radius of the second cylinder is 1.2 times the radius of the first. Now, since the volumes are equal, I can write:[ pi r_1^2 h_1 = pi r_2^2 h_2 ]I can cancel out the ( pi ) from both sides since it's a common factor:[ r_1^2 h_1 = r_2^2 h_2 ]Now, substituting ( r_2 = 1.2r_1 ) into the equation:[ r_1^2 h_1 = (1.2r_1)^2 h_2 ]Let me compute ( (1.2r_1)^2 ):[ (1.2r_1)^2 = 1.44r_1^2 ]So, substituting back in:[ r_1^2 h_1 = 1.44r_1^2 h_2 ]Now, I can divide both sides by ( r_1^2 ) to simplify:[ h_1 = 1.44h_2 ]Hmm, so this means that ( h_1 ) is 1.44 times ( h_2 ). To express this as a percentage, 1.44 is the same as 144%, which means ( h_1 ) is 44% more than ( h_2 ).Let me double-check that. If ( h_1 = 1.44h_2 ), then the increase from ( h_2 ) to ( h_1 ) is 0.44 times ( h_2 ), which is 44%. So yes, that makes sense.Looking back at the options given:A) The second height is 44% less than the first.B) The first height is 44% more than the second.C) The second height is 80% of the first.D) The first height is 80% of the second.E) Neither height is 44% more or less than the other.From my calculation, ( h_1 = 1.44h_2 ), which means the first height is 44% more than the second. So option B seems correct.But just to be thorough, let me consider the other options:A) If the second height were 44% less than the first, that would mean ( h_2 = h_1 - 0.44h_1 = 0.56h_1 ). But from my equation, ( h_2 = frac{h_1}{1.44} approx 0.694h_1 ), which is about a 30.6% decrease, not 44%. So A is incorrect.C) The second height is 80% of the first. That would mean ( h_2 = 0.8h_1 ). But from my calculation, ( h_2 approx 0.694h_1 ), which is about 69.4%, not 80%. So C is incorrect.D) The first height is 80% of the second. That would mean ( h_1 = 0.8h_2 ). But my equation shows ( h_1 = 1.44h_2 ), which is 144%, not 80%. So D is incorrect.E) Neither height is 44% more or less than the other. But from my calculation, ( h_1 ) is indeed 44% more than ( h_2 ), so E is incorrect.Therefore, the correct answer is B: The first height is 44% more than the second.I think that makes sense. The radius increase leads to a squared increase in the volume term, so the height has to compensate by decreasing, but since the volume is the same, the height actually increases by a factor that's the inverse of the squared radius factor. So, since the radius increased by 1.2, the height increases by ( 1/(1.2)^2 = 1/1.44 approx 0.694 ), but wait, that seems contradictory. Wait, no, actually, let me think again.Wait, if the radius increases by 20%, the cross-sectional area increases by ( (1.2)^2 = 1.44 ), so to keep the volume the same, the height must decrease by the same factor, right? So if the cross-sectional area is 1.44 times larger, the height should be ( 1/1.44 ) times the original height to keep the volume the same.But in the problem, the first cylinder has radius ( r_1 ) and the second has ( r_2 = 1.2r_1 ). So if the second cylinder has a larger radius, its cross-sectional area is larger, so to have the same volume, its height must be smaller. So actually, ( h_2 ) should be smaller than ( h_1 ).Wait, but in my earlier calculation, I had ( h_1 = 1.44h_2 ), which implies ( h_2 = h_1 / 1.44 approx 0.694h_1 ). So ( h_2 ) is about 69.4% of ( h_1 ), meaning it's decreased by about 30.6%, not 44%. So that would mean the second height is approximately 30.6% less than the first, not 44%.But according to the options, A says the second height is 44% less, which is not what I got. So maybe I made a mistake in interpreting which height is which.Wait, let me clarify:- First cylinder: radius ( r_1 ), height ( h_1 )- Second cylinder: radius ( r_2 = 1.2r_1 ), height ( h_2 )Volumes are equal:[ pi r_1^2 h_1 = pi r_2^2 h_2 ]Substituting ( r_2 = 1.2r_1 ):[ r_1^2 h_1 = (1.2r_1)^2 h_2 ][ r_1^2 h_1 = 1.44r_1^2 h_2 ][ h_1 = 1.44h_2 ]So ( h_1 = 1.44h_2 ), meaning ( h_1 ) is 44% more than ( h_2 ). So the first height is 44% more than the second. Therefore, option B is correct.But earlier, I thought that since the radius increased, the height should decrease. But in reality, since the second cylinder has a larger radius, its height must be smaller to keep the volume the same. So ( h_2 ) is smaller than ( h_1 ), which aligns with ( h_1 = 1.44h_2 ). So the first height is larger, specifically 44% larger than the second.So, to avoid confusion, it's important to note that the height of the first cylinder is 44% more than the second, not the other way around. So option B is correct.I think I initially got confused when thinking about the percentage decrease, but since the question asks for the relationship between the heights, and we found that ( h_1 ) is 44% more than ( h_2 ), that's the correct answer.Another way to think about it is:If ( h_1 = 1.44h_2 ), then ( h_2 = h_1 / 1.44 approx 0.694h_1 ), which is a decrease of about 30.6% from ( h_1 ). But the question is asking for the relationship between the heights, not necessarily the percentage decrease or increase from one to the other. Since ( h_1 ) is 44% more than ( h_2 ), that's the direct relationship.So, to summarize:- The radius of the second cylinder is 20% larger, so its cross-sectional area is ( 1.2^2 = 1.44 ) times larger.- To maintain the same volume, the height must be ( 1/1.44 approx 0.694 ) times the original height.- Therefore, ( h_2 approx 0.694h_1 ), meaning ( h_1 ) is approximately 44% larger than ( h_2 ).Thus, the correct answer is B: The first height is 44% more than the second.

answer：Okay, so I have this problem where there's an ellipse given by the equation (frac{x^2}{16} + frac{y^2}{9} = 1). A point (P) is moving on this ellipse in the first quadrant. From this point (P), two tangents (PA) and (PB) are drawn to the circle (x^2 + y^2 = 9). The points where these tangents touch the circle are (A) and (B). Then, the line connecting (A) and (B) intersects the x-axis at point (M) and the y-axis at point (N). I need to find the minimum area of the triangle (MON), where (O) is the origin.Alright, let's break this down step by step.First, I need to understand the setup. There's an ellipse and a circle. The ellipse has semi-major axis 4 and semi-minor axis 3. The circle has radius 3, centered at the origin. Point (P) is on the ellipse in the first quadrant, so both its x and y coordinates are positive.From point (P), we draw two tangents to the circle. The points where these tangents touch the circle are (A) and (B). Then, the line (AB) is drawn, and it intersects the x-axis at (M) and the y-axis at (N). The triangle (MON) is formed by the origin and these two intercepts. We need to find the minimum area of this triangle as point (P) moves along the ellipse.I think the first step is to parameterize point (P) on the ellipse. For an ellipse, a common parameterization is using an angle (theta). So, for the ellipse (frac{x^2}{16} + frac{y^2}{9} = 1), the parameterization would be (P(4costheta, 3sintheta)), where (theta) is the parameter varying from 0 to (2pi). Since (P) is in the first quadrant, (theta) will be between 0 and (frac{pi}{2}).Next, I need to find the equations of the tangents from (P) to the circle (x^2 + y^2 = 9). The circle is centered at the origin with radius 3. The tangents from an external point to a circle can be found using the formula for the tangent lines.The general equation for a tangent to the circle (x^2 + y^2 = r^2) from an external point ((x_1, y_1)) is given by:[xx_1 + yy_1 = r^2]But wait, is that correct? Hmm, no, actually, that's the equation of the polar line of the point ((x_1, y_1)) with respect to the circle. If the point is outside the circle, this polar line is the tangent from that point to the circle. So, in this case, since (P) is on the ellipse, which is outside the circle (since the ellipse has a larger semi-major axis), the polar line of (P) with respect to the circle will be the tangent line (AB).So, the equation of the tangent line (AB) from point (P(4costheta, 3sintheta)) to the circle (x^2 + y^2 = 9) is:[x cdot 4costheta + y cdot 3sintheta = 9]Simplifying, we can write this as:[4xcostheta + 3ysintheta = 9]Wait, is that correct? Let me check. The standard equation for the polar line is (xx_1 + yy_1 = r^2), so substituting (x_1 = 4costheta) and (y_1 = 3sintheta), we get:[x cdot 4costheta + y cdot 3sintheta = 9]Yes, that seems right.So, the equation of line (AB) is (4xcostheta + 3ysintheta = 9).Now, I need to find the points where this line intersects the x-axis and y-axis. The x-intercept (M) is found by setting (y = 0), and the y-intercept (N) is found by setting (x = 0).Calculating the x-intercept (M):[4xcostheta = 9 implies x = frac{9}{4costheta}]So, the coordinates of (M) are (left(frac{9}{4costheta}, 0right)).Calculating the y-intercept (N):[3ysintheta = 9 implies y = frac{9}{3sintheta} = frac{3}{sintheta}]So, the coordinates of (N) are (left(0, frac{3}{sintheta}right)).Now, the triangle (MON) is a right triangle with legs along the x-axis and y-axis. The lengths of these legs are (|OM| = frac{9}{4costheta}) and (|ON| = frac{3}{sintheta}).The area (A) of triangle (MON) is given by:[A = frac{1}{2} times |OM| times |ON| = frac{1}{2} times frac{9}{4costheta} times frac{3}{sintheta}]Simplifying this:[A = frac{1}{2} times frac{27}{4costheta sintheta} = frac{27}{8costheta sintheta}]I can use the double-angle identity for sine here, which is (sin 2theta = 2sintheta costheta). So, (costheta sintheta = frac{1}{2}sin 2theta).Substituting this into the area formula:[A = frac{27}{8 times frac{1}{2}sin 2theta} = frac{27}{4sin 2theta}]So, the area is (frac{27}{4sin 2theta}).Now, I need to find the minimum value of this area as (theta) varies. Since (theta) is in the first quadrant, (2theta) will vary from 0 to (pi). The sine function (sin 2theta) reaches its maximum value of 1 when (2theta = frac{pi}{2}), which implies (theta = frac{pi}{4}).Therefore, the maximum value of (sin 2theta) is 1, and substituting this into the area formula gives:[A_{text{min}} = frac{27}{4 times 1} = frac{27}{4}]So, the minimum area of triangle (MON) is (frac{27}{4}).Wait, let me double-check if this makes sense. When (theta = frac{pi}{4}), point (P) is at ((4cosfrac{pi}{4}, 3sinfrac{pi}{4}) = left(4 times frac{sqrt{2}}{2}, 3 times frac{sqrt{2}}{2}right) = (2sqrt{2}, frac{3sqrt{2}}{2})). Then, the tangent line equation becomes:[4x times frac{sqrt{2}}{2} + 3y times frac{sqrt{2}}{2} = 9]Simplifying:[2sqrt{2}x + frac{3sqrt{2}}{2}y = 9]Dividing both sides by (sqrt{2}):[2x + frac{3}{2}y = frac{9}{sqrt{2}}]Wait, that seems a bit messy. Maybe I made a mistake in simplifying.Wait, no, actually, the equation was already simplified to (4xcostheta + 3ysintheta = 9). Plugging in (theta = frac{pi}{4}), we get:[4x times frac{sqrt{2}}{2} + 3y times frac{sqrt{2}}{2} = 9]Which simplifies to:[2sqrt{2}x + frac{3sqrt{2}}{2}y = 9]To find the intercepts, set (y = 0):[2sqrt{2}x = 9 implies x = frac{9}{2sqrt{2}} = frac{9sqrt{2}}{4}]Similarly, set (x = 0):[frac{3sqrt{2}}{2}y = 9 implies y = frac{9 times 2}{3sqrt{2}} = frac{6}{sqrt{2}} = 3sqrt{2}]So, the intercepts are (Mleft(frac{9sqrt{2}}{4}, 0right)) and (Nleft(0, 3sqrt{2}right)).Calculating the area:[A = frac{1}{2} times frac{9sqrt{2}}{4} times 3sqrt{2} = frac{1}{2} times frac{27 times 2}{4} = frac{1}{2} times frac{54}{4} = frac{27}{4}]Yes, that checks out. So, the minimum area is indeed (frac{27}{4}).I think this makes sense. As point (P) moves around the ellipse, the area of triangle (MON) changes, and the minimum occurs when the sine term in the area formula is maximized, which happens at (theta = frac{pi}{4}).I don't see any mistakes in the reasoning. The key steps were parameterizing point (P), finding the equation of the tangent line, determining the intercepts, expressing the area in terms of (theta), and then using trigonometric identities to find the minimum area.

answer：Alright, so I've got this inequality to solve: (a{x}^{2}-(a+2)x+2 < 0), where (a) is a real number. Hmm, okay. I remember that quadratic inequalities can be a bit tricky, especially when the coefficient of (x^2) isn't fixed—it depends on (a). So, I guess I need to consider different cases based on the value of (a).First off, let's think about what happens when (a = 0). If (a) is zero, the equation simplifies a lot. Plugging in (a = 0), the inequality becomes (-2x + 2 < 0). That simplifies to (-2x < -2), and dividing both sides by (-2) (remembering to flip the inequality sign because I'm dividing by a negative) gives (x > 1). So, when (a = 0), the solution is all real numbers greater than 1.Okay, that was straightforward. Now, what if (a) isn't zero? Well, if (a) isn't zero, we're dealing with a quadratic equation. The general form is (ax^2 + bx + c), and in this case, it's (a{x}^{2}-(a+2)x+2). To solve the inequality (a{x}^{2}-(a+2)x+2 < 0), I need to find the roots of the quadratic equation (a{x}^{2}-(a+2)x+2 = 0) and then determine where the quadratic expression is negative.So, let's find the roots using the quadratic formula. The quadratic formula is (x = frac{-b pm sqrt{b^2 - 4ac}}{2a}). Here, (a) is the coefficient of (x^2), which is (a), (b) is the coefficient of (x), which is (-(a + 2)), and (c) is the constant term, which is 2.Plugging these into the quadratic formula:(x = frac{-(-(a + 2)) pm sqrt{(-(a + 2))^2 - 4 cdot a cdot 2}}{2 cdot a})Simplifying that:(x = frac{a + 2 pm sqrt{(a + 2)^2 - 8a}}{2a})Let's compute the discriminant inside the square root:((a + 2)^2 - 8a = a^2 + 4a + 4 - 8a = a^2 - 4a + 4)Hmm, that's a perfect square: (a^2 - 4a + 4 = (a - 2)^2)So, the roots simplify to:(x = frac{a + 2 pm |a - 2|}{2a})Now, the absolute value (|a - 2|) can be split into two cases: when (a - 2) is non-negative and when it's negative.**Case 1: (a geq 2)**In this case, (|a - 2| = a - 2), so the roots become:(x = frac{a + 2 pm (a - 2)}{2a})Let's compute both roots:First root with the plus sign:(x = frac{a + 2 + a - 2}{2a} = frac{2a}{2a} = 1)Second root with the minus sign:(x = frac{a + 2 - (a - 2)}{2a} = frac{a + 2 - a + 2}{2a} = frac{4}{2a} = frac{2}{a})So, the roots are (x = 1) and (x = frac{2}{a}).Now, since (a geq 2), (frac{2}{a}) is less than or equal to 1. So, the quadratic opens upwards (since (a > 0)) and the expression is negative between the roots. Therefore, the solution is ( frac{2}{a} < x < 1 ).**Case 2: (a < 2)**Here, (|a - 2| = 2 - a), so the roots become:(x = frac{a + 2 pm (2 - a)}{2a})Again, computing both roots:First root with the plus sign:(x = frac{a + 2 + 2 - a}{2a} = frac{4}{2a} = frac{2}{a})Second root with the minus sign:(x = frac{a + 2 - (2 - a)}{2a} = frac{a + 2 - 2 + a}{2a} = frac{2a}{2a} = 1)So, the roots are still (x = 1) and (x = frac{2}{a}). However, since (a < 2), (frac{2}{a}) is greater than 1 (assuming (a > 0)). But wait, we need to consider the sign of (a) as well. If (a) is positive, the parabola opens upwards, and the expression is negative between the roots. So, the solution is (1 < x < frac{2}{a}).If (a) is negative, the parabola opens downwards. In this case, the expression is negative outside the interval between the roots. So, the solution would be (x < 1) or (x > frac{2}{a}). But since (a) is negative, (frac{2}{a}) is negative, which complicates things a bit. Let me think about this.If (a < 0), then (frac{2}{a}) is negative, and 1 is positive. So, the quadratic expression (a{x}^{2}-(a+2)x+2) will be negative when (x < frac{2}{a}) or (x > 1). But since (frac{2}{a}) is negative and 1 is positive, these intervals don't overlap, so the solution is (x < frac{2}{a}) or (x > 1).Wait, but I need to make sure about the direction of the inequality. Since (a < 0), the parabola opens downward, so the expression is negative outside the interval between the roots. So, yes, (x < frac{2}{a}) or (x > 1).Let me summarize:- If (a = 0): (x > 1)- If (a > 0): - If (0 < a < 2): (1 < x < frac{2}{a}) - If (a = 2): The quadratic becomes (2x^2 - 4x + 2 = 0), which simplifies to (x^2 - 2x + 1 = 0), so (x = 1) is a repeated root. Since the parabola opens upwards, the expression is never negative, so no solution. - If (a > 2): (frac{2}{a} < x < 1)- If (a < 0): (x < frac{2}{a}) or (x > 1)I think that covers all the cases. Let me double-check.When (a = 0), we already saw it simplifies to a linear inequality, giving (x > 1). That seems right.For (a > 0), depending on whether (a) is less than or greater than 2, the order of the roots changes. Since the parabola opens upwards, the expression is negative between the roots when (a > 0). So, if (a < 2), (frac{2}{a} > 1), so the solution is between 1 and (frac{2}{a}). If (a > 2), (frac{2}{a} < 1), so the solution is between (frac{2}{a}) and 1. If (a = 2), the quadratic becomes a perfect square, and there's no solution because the expression is never negative.For (a < 0), the parabola opens downward, so the expression is negative outside the interval between the roots. Since (frac{2}{a}) is negative and 1 is positive, the solution is (x < frac{2}{a}) or (x > 1).I think that makes sense. I don't see any mistakes in this reasoning. So, the final answer should cover all these cases.

answer：Alright, so I'm trying to figure out this game between Alice and Barbara. It's a bit tricky, but let me break it down step by step.First, in part (a), the game starts with zero on the board. Alice goes first, and each player can replace the current number ( x ) with a number ( y ) such that ( 0 < y - x < 1 ). The first player to write a number greater than or equal to 2010 wins. Hmm, okay, so each turn, a player can increase the number by any amount between just above 0 and just below 1.I wonder if there's a strategy where one player can force the other into a losing position. Maybe something like mirroring the opponent's moves? Let me think. If Barbara can always respond to Alice's move in a way that keeps the game under control, maybe she can force Alice into a position where Alice has to make the final move to reach 2010.Wait, so if Barbara mirrors Alice's moves by adding ( 1 - z ) whenever Alice adds ( z ), then after each pair of turns, the total increase would be 1. So, starting from 0, after Alice's first move, it's ( z ), then Barbara adds ( 1 - z ) to make it 1. Then Alice adds ( z' ) to make it ( 1 + z' ), and Barbara adds ( 1 - z' ) to make it 2. This pattern continues, and after each of Barbara's turns, the number is an integer.So, if Barbara keeps doing this, after her 2009th turn, the number would be 2009. Then it's Alice's turn, and she has to add some ( z ) such that ( 0 < z < 1 ), making the number between 2009 and 2010. Then Barbara can add the remaining amount to reach exactly 2010 and win. That makes sense. So Barbara can force a win by mirroring Alice's moves.Now, moving on to part (b). This part is a bit different. The first player to write a number greater than or equal to 2010 on her 2011th turn or later wins. But if a player writes a number greater than or equal to 2010 on her 2010th turn or earlier, she loses immediately. So, the goal is to reach 2010 on or after the 2011th turn.Hmm, this seems like a timing game. The players have to be careful not to reach 2010 too early, but also want to be the one to reach it on the 2011th turn or later. Let me think about how the turns work.Since Alice goes first, she has the 1st, 3rd, 5th, etc., turns. Barbara has the 2nd, 4th, 6th, etc. So, the 2010th turn would be Barbara's 1005th turn, and the 2011th turn would be Alice's 1006th turn.Wait, so if Alice can force the game to reach 2010 on her 1006th turn, she wins. But if Barbara can force it to reach 2010 on her 1005th turn, she would lose because it's before the 2011th turn. So, Barbara needs to avoid reaching 2010 on her 1005th turn, and Alice wants to reach it on her 1006th turn.How can Alice ensure that? Maybe she can control the increments in such a way that Barbara is forced to make the last move before 2010, but then Alice can finish it on her next turn.Alternatively, maybe Alice can use a similar mirroring strategy but adjust it to account for the timing. If Alice starts by making a small move, and then mirrors Barbara's moves, she can control the progression of the number.Wait, let's think about the total number of turns. If the game goes up to 2010, that's 2010 increments of 1, but since each turn can add less than 1, it might take more turns. But in this case, the constraint is on the turn number, not the number itself.So, the key is to reach 2010 on the 2011th turn. Since Alice goes first, she can plan her moves to ensure that on her 1006th turn (which is the 2011th turn overall), she can make the final increment to reach 2010.But how? Maybe by controlling the increments so that after Barbara's 1005th turn, the number is just below 2010, and then Alice can add the remaining amount on her 1006th turn.But Barbara might try to prevent that. Maybe Barbara can also use a strategy to reach 2010 earlier, but she has to avoid it because it would cause her to lose.Wait, so if Alice can force the game to reach 2010 on her 1006th turn, she wins. Barbara, on the other hand, wants to avoid reaching 2010 on her 1005th turn, which would make her lose. So, Barbara might try to keep the number below 2010 until after her 1005th turn.But how can Alice ensure that? Maybe by starting with a small increment and then mirroring Barbara's moves to control the progression.Let me try to outline a possible strategy:1. Alice starts by writing a number ( a ) such that ( 0 < a < 1 ). Let's say she writes ( a = 0.5 ).2. Then Barbara has to write a number ( b ) such that ( 0.5 < b < 1.5 ). Suppose Barbara writes ( b = 0.5 + z ), where ( 0 < z < 1 ).3. Alice then writes ( b + (1 - z) ), which would be ( 0.5 + z + (1 - z) = 1.5 ).4. This pattern continues, with Alice mirroring Barbara's moves to ensure that after each pair of turns, the number increases by 1.5. After 2009 such pairs, the number would be ( 2009 ).6. Then it's Alice's turn, and she can write ( 2009 + z ), where ( 0 < z < 1 ), making the number between 2009 and 2010.7. Barbara then has to write a number ( y ) such that ( 2009 + z < y < 2010 + z ). But since ( z < 1 ), ( y ) can be at most just below 2010.8. Then it's Alice's turn again, and she can write ( y + (1 - (y - (2009 + z))) ), which would be exactly 2010.Wait, that might not be precise. Let me think again.Actually, after Barbara's 1005th turn, the number would be just below 2010. Then it's Alice's 1006th turn, and she can add the remaining amount to reach 2010.But Barbara might try to reach 2010 on her 1005th turn, but she can't because she would lose. So she has to keep it below 2010 until after her 1005th turn.Therefore, Alice can force the game to reach 2010 on her 1006th turn, making her the winner.So, in part (a), Barbara can win by mirroring Alice's moves, but in part (b), Alice can win by controlling the timing of the increments to reach 2010 on her 1006th turn.I think that's the gist of it. Let me try to formalize it a bit more.In part (a), Barbara can always respond to Alice's move by adding ( 1 - z ), ensuring that after each of her turns, the number is an integer. This forces Alice to make the final increment to reach 2010.In part (b), Alice can start with a small increment and then mirror Barbara's moves, ensuring that after Barbara's 1005th turn, the number is just below 2010. Then Alice can make the final increment on her 1006th turn to reach 2010, winning the game.Yeah, that makes sense. So, in part (a), Barbara has the winning strategy, and in part (b), Alice does.

answer：Okay, so I have this sequence defined by y₁, y₂, y₃, and so on. The first term is y₁ equals the fourth root of 4, which is √[4]{4}. Then y₂ is that fourth root of 4 raised to the cube root of 4, so (√[4]{4})^{∛4}. And in general, each term is the previous term raised to the cube root of 4. So, yₙ = (y_{n-1})^{∛4} for n > 1. The question is asking for the smallest value of n where yₙ is an integer.Alright, let me try to break this down. Maybe if I express everything in terms of exponents with base 2, it will be easier to handle. Since 4 is 2 squared, so 4 = 2². That might help simplify things.Starting with y₁: √[4]{4} is the same as 4^(1/4). Since 4 is 2², this becomes (2²)^(1/4). When you raise a power to a power, you multiply the exponents. So, 2^(2*(1/4)) = 2^(1/2). So, y₁ is 2^(1/2). Got that.Now, y₂ is y₁ raised to the cube root of 4. So, that's (2^(1/2))^{∛4}. Again, 4 is 2², so the cube root of 4 is 4^(1/3) = (2²)^(1/3) = 2^(2/3). So, y₂ is (2^(1/2))^{2^(2/3)}. Hmm, that seems a bit complicated. Wait, actually, when you raise a power to another power, you multiply the exponents. So, it's 2^{(1/2) * 2^(2/3)}. Let me compute that exponent.First, 2^(2/3) is the cube root of 4, which is approximately 1.5874, but maybe I can keep it as 2^(2/3) for exactness. So, (1/2) * 2^(2/3) = (2^(-1)) * 2^(2/3) = 2^(-1 + 2/3) = 2^(-1/3). Wait, that doesn't seem right. Let me check that again.Wait, no, when you multiply exponents with the same base, you add the exponents. But in this case, it's 2^(1/2) raised to 2^(2/3). So, the exponent is (1/2) * 2^(2/3). Let me compute that.(1/2) * 2^(2/3) = (2^(-1)) * 2^(2/3) = 2^(-1 + 2/3) = 2^(-1/3). So, y₂ is 2^(-1/3). Wait, that would make y₂ less than 1, but y₁ is 2^(1/2) which is about 1.414, and raising that to a positive exponent should give a number greater than 1, right? So, maybe I made a mistake in the exponent calculation.Wait, no, actually, 2^(2/3) is approximately 1.5874, so (1/2) * 1.5874 is approximately 0.7937, which is positive, so 2^0.7937 is about 1.732, which is greater than 1. So, maybe I should keep it as 2^{(1/2)*2^(2/3)} instead of trying to simplify it further. Maybe I can express it differently.Alternatively, let's think about the exponents in terms of logarithms. Maybe taking the logarithm of yₙ will help me see the pattern better.Let me define zₙ = log₂(yₙ). Then, since yₙ = (y_{n-1})^{∛4}, taking log base 2 of both sides gives zₙ = log₂(y_{n-1}^{∛4}) = ∛4 * log₂(y_{n-1}) = ∛4 * z_{n-1}.So, zₙ = ∛4 * z_{n-1}. That's a recursive relation for zₙ. Let's compute the first few terms.We know y₁ = 2^(1/2), so z₁ = log₂(y₁) = log₂(2^(1/2)) = 1/2.Then, z₂ = ∛4 * z₁ = ∛4 * (1/2). Since ∛4 is 2^(2/3), so z₂ = 2^(2/3) * (1/2) = 2^(2/3 - 1) = 2^(-1/3).Wait, that's the same as before. So, z₂ = 2^(-1/3). Then, z₃ = ∛4 * z₂ = 2^(2/3) * 2^(-1/3) = 2^(2/3 - 1/3) = 2^(1/3).Similarly, z₄ = ∛4 * z₃ = 2^(2/3) * 2^(1/3) = 2^(2/3 + 1/3) = 2^1 = 2.So, z₄ = 2, which means y₄ = 2^{z₄} = 2^2 = 4, which is an integer.Wait, so that seems to suggest that y₄ is 4, which is an integer. So, the smallest n is 4.But let me double-check my calculations to make sure I didn't make any mistakes.Starting with z₁ = 1/2.z₂ = ∛4 * z₁ = 2^(2/3) * 2^(-1) = 2^(2/3 - 1) = 2^(-1/3).z₃ = ∛4 * z₂ = 2^(2/3) * 2^(-1/3) = 2^(2/3 - 1/3) = 2^(1/3).z₄ = ∛4 * z₃ = 2^(2/3) * 2^(1/3) = 2^(2/3 + 1/3) = 2^1 = 2.So, z₄ = 2, which means y₄ = 2^2 = 4. Yep, that checks out.So, the sequence of zₙ is: 1/2, 2^(-1/3), 2^(1/3), 2, and so on. Each time, we're multiplying by 2^(2/3), which is the cube root of 4.So, the exponents are: 1/2, (1/2)*(2/3) = 1/3, then (1/3)*(2/3) = 2/9, wait, no, that doesn't match. Wait, no, actually, in the zₙ sequence, each term is multiplied by 2^(2/3), not by 2/3.Wait, no, zₙ = 2^(2/3) * z_{n-1}, so the exponents are being added by 2/3 each time in the logarithmic scale. Wait, no, actually, zₙ is being multiplied by 2^(2/3), so in terms of exponents, it's additive.Wait, maybe I should think of zₙ as a geometric sequence where each term is multiplied by 2^(2/3). So, z₁ = 1/2, z₂ = (1/2)*2^(2/3), z₃ = (1/2)*(2^(2/3))^2, z₄ = (1/2)*(2^(2/3))^3, and so on.So, zₙ = (1/2)*(2^(2/3))^{n-1} = (1/2)*2^{(2/3)(n-1)}.We want yₙ to be an integer, which means zₙ must be an integer because yₙ = 2^{zₙ}. So, zₙ must be an integer.So, zₙ = (1/2)*2^{(2/3)(n-1)} = 2^{-1} * 2^{(2/3)(n-1)} = 2^{(2/3)(n-1) - 1}.We need (2/3)(n-1) - 1 to be an integer because 2 raised to an integer power is an integer.Let me set (2/3)(n-1) - 1 = k, where k is an integer.So, (2/3)(n-1) = k + 1.Multiply both sides by 3: 2(n-1) = 3(k + 1).So, 2(n-1) must be divisible by 3, meaning that n-1 must be divisible by 3/2. But n-1 has to be an integer, so 3 must divide 2(n-1). Since 2 and 3 are coprime, 3 must divide (n-1). So, n-1 is a multiple of 3, say n-1 = 3m, where m is a positive integer.Then, 2(n-1) = 6m, so 6m = 3(k + 1) => 2m = k + 1 => k = 2m - 1.So, k must be an odd integer.Now, let's express zₙ in terms of m:zₙ = 2^{(2/3)(n-1) - 1} = 2^{(2/3)(3m) - 1} = 2^{2m - 1}.So, zₙ = 2^{2m - 1}, which is an integer as long as 2m - 1 is an integer, which it is since m is an integer.So, the smallest n occurs when m is the smallest positive integer such that n = 3m + 1 is minimized.So, m = 1: n = 3*1 + 1 = 4.Therefore, the smallest n is 4.Let me verify this again with the original sequence:y₁ = 2^(1/2)y₂ = (2^(1/2))^{2^(2/3)} = 2^{(1/2)*2^(2/3)} = 2^{2^(2/3 - 1)} = 2^{2^(-1/3)}? Wait, no, that's not correct. Wait, actually, when you raise 2^(1/2) to the power of 2^(2/3), you multiply the exponents: (1/2)*2^(2/3). Hmm, that's not straightforward. Maybe it's better to stick with the zₙ approach.From the zₙ approach, we saw that z₄ = 2, so y₄ = 2^2 = 4, which is indeed an integer.So, yes, n=4 is the smallest such n.