💡First, I need to determine the minimum number of games the team must win to accumulate at least 48 points.Let ( x ) represent the number of games the team wins. Since there are 32 games in total, the team will lose ( 32 - x ) games.For each win, the team earns 3 points, and for each loss, they lose 1 point. Therefore, the total points can be expressed as:[3x - (32 - x)]To find the minimum number of wins required to reach at least 48 points, I set up the inequality:[3x - (32 - x) geq 48]Simplifying the inequality:[3x - 32 + x geq 48][4x - 32 geq 48][4x geq 80][x geq 20]Thus, the team must win at least 20 games to have a chance of advancing to the playoffs.

💡Okay, so I have this problem where a real number ( x ) satisfies ( x^3 + frac{1}{x^3} = 18 ). I need to find ( {x} + left{frac{1}{x}right} ), where ( { cdot } ) denotes the fractional part. The options given are (A) ( frac{1}{2} ), (B) ( 3 - sqrt{5} ), (C) ( frac{3 - sqrt{5}}{2} ), and (D) 1.Hmm, fractional parts. I remember that the fractional part of a number is the part after the decimal point. For example, if ( x = 2.5 ), then ( {x} = 0.5 ). So, I need to find the sum of the fractional parts of ( x ) and ( frac{1}{x} ).First, let me try to find ( x ) from the given equation. The equation is ( x^3 + frac{1}{x^3} = 18 ). I think I can use some algebraic identities to simplify this. Maybe if I let ( y = x + frac{1}{x} ), then I can express ( x^3 + frac{1}{x^3} ) in terms of ( y ).I recall that ( (x + frac{1}{x})^3 = x^3 + 3x + frac{3}{x} + frac{1}{x^3} ). So, that simplifies to ( x^3 + frac{1}{x^3} + 3(x + frac{1}{x}) ). Therefore, ( x^3 + frac{1}{x^3} = (x + frac{1}{x})^3 - 3(x + frac{1}{x}) ).Let me write that down:[x^3 + frac{1}{x^3} = left( x + frac{1}{x} right)^3 - 3left( x + frac{1}{x} right)]So, substituting ( y = x + frac{1}{x} ), the equation becomes:[y^3 - 3y = 18]Which simplifies to:[y^3 - 3y - 18 = 0]Now, I need to solve this cubic equation for ( y ). Let me try to find rational roots using the Rational Root Theorem. The possible rational roots are factors of 18 divided by factors of 1, so possible roots are ( pm1, pm2, pm3, pm6, pm9, pm18 ).Let me test ( y = 3 ):[3^3 - 3*3 - 18 = 27 - 9 - 18 = 0]Oh, that works! So, ( y = 3 ) is a root. Therefore, I can factor the cubic equation as:[(y - 3)(y^2 + 3y + 6) = 0]The quadratic factor ( y^2 + 3y + 6 ) doesn't have real roots because the discriminant ( 9 - 24 = -15 ) is negative. So, the only real solution is ( y = 3 ).Therefore, ( x + frac{1}{x} = 3 ). Now, let me solve for ( x ). Let me set ( x + frac{1}{x} = 3 ), which can be rewritten as:[x^2 - 3x + 1 = 0]Using the quadratic formula:[x = frac{3 pm sqrt{9 - 4}}{2} = frac{3 pm sqrt{5}}{2}]So, the solutions are ( x = frac{3 + sqrt{5}}{2} ) and ( x = frac{3 - sqrt{5}}{2} ).Now, I need to find the fractional parts of ( x ) and ( frac{1}{x} ). Let me compute the numerical values of these solutions to understand their integer and fractional parts.First, ( sqrt{5} ) is approximately 2.236. So,1. ( x_1 = frac{3 + sqrt{5}}{2} approx frac{3 + 2.236}{2} = frac{5.236}{2} = 2.618 )2. ( x_2 = frac{3 - sqrt{5}}{2} approx frac{3 - 2.236}{2} = frac{0.764}{2} = 0.382 )So, ( x_1 approx 2.618 ) and ( x_2 approx 0.382 ).Now, let's find ( frac{1}{x_1} ) and ( frac{1}{x_2} ):1. ( frac{1}{x_1} approx frac{1}{2.618} approx 0.382 )2. ( frac{1}{x_2} approx frac{1}{0.382} approx 2.618 )Interesting, so ( x_1 ) and ( x_2 ) are reciprocals of each other. That makes sense because if ( x ) satisfies ( x + frac{1}{x} = 3 ), then ( frac{1}{x} ) will also satisfy the same equation.Now, let's find the fractional parts.For ( x_1 approx 2.618 ):- The integer part is 2, so the fractional part ( {x_1} = 2.618 - 2 = 0.618 )For ( frac{1}{x_1} approx 0.382 ):- The integer part is 0, so the fractional part ( left{ frac{1}{x_1} right} = 0.382 )Similarly, for ( x_2 approx 0.382 ):- The integer part is 0, so the fractional part ( {x_2} = 0.382 )For ( frac{1}{x_2} approx 2.618 ):- The integer part is 2, so the fractional part ( left{ frac{1}{x_2} right} = 2.618 - 2 = 0.618 )So, depending on which ( x ) we take, the fractional parts are either ( 0.618 ) and ( 0.382 ), or ( 0.382 ) and ( 0.618 ). In either case, their sum is:[0.618 + 0.382 = 1.0]Wait, that's exactly 1. So, regardless of which ( x ) we take, the sum of the fractional parts is 1.But let me check if this is always the case. Suppose ( x ) is a number greater than 1, so ( {x} = x - lfloor x rfloor ). And ( frac{1}{x} ) is less than 1, so ( left{ frac{1}{x} right} = frac{1}{x} ).If ( x ) is between 1 and 2, then ( lfloor x rfloor = 1 ), so ( {x} = x - 1 ). And ( frac{1}{x} ) is between 0.5 and 1, so ( left{ frac{1}{x} right} = frac{1}{x} ).Wait, but in our case, ( x_1 approx 2.618 ), which is between 2 and 3, so ( lfloor x_1 rfloor = 2 ), so ( {x_1} = x_1 - 2 approx 0.618 ). And ( frac{1}{x_1} approx 0.382 ), which is less than 1, so ( left{ frac{1}{x_1} right} = 0.382 ).Similarly, ( x_2 approx 0.382 ), which is less than 1, so ( {x_2} = 0.382 ). And ( frac{1}{x_2} approx 2.618 ), which is between 2 and 3, so ( left{ frac{1}{x_2} right} = 2.618 - 2 = 0.618 ).So, in both cases, the sum of the fractional parts is ( 0.618 + 0.382 = 1.0 ).But wait, 0.618 is approximately ( frac{sqrt{5} - 1}{2} ), which is about 0.618, and 0.382 is approximately ( frac{3 - sqrt{5}}{2} ), which is about 0.382.Wait, let me compute ( frac{sqrt{5} - 1}{2} ):[frac{sqrt{5} - 1}{2} approx frac{2.236 - 1}{2} = frac{1.236}{2} = 0.618]And ( frac{3 - sqrt{5}}{2} approx frac{3 - 2.236}{2} = frac{0.764}{2} = 0.382 )So, ( {x} ) and ( left{ frac{1}{x} right} ) are ( frac{sqrt{5} - 1}{2} ) and ( frac{3 - sqrt{5}}{2} ), or vice versa, depending on which ( x ) we take.But in either case, their sum is:[frac{sqrt{5} - 1}{2} + frac{3 - sqrt{5}}{2} = frac{(sqrt{5} - 1) + (3 - sqrt{5})}{2} = frac{2}{2} = 1]So, regardless of the specific ( x ), the sum of the fractional parts is 1.Therefore, the answer should be 1, which is option (D).Wait, but let me double-check if there's any case where this might not hold. Suppose ( x ) is an integer. But in our case, ( x ) is ( frac{3 pm sqrt{5}}{2} ), which are irrational numbers, so they are not integers. Therefore, their fractional parts are well-defined and non-zero.Also, since ( x ) and ( frac{1}{x} ) are reciprocals, one is greater than 1 and the other is less than 1. So, one has a fractional part, and the other is less than 1, so its fractional part is itself.Therefore, their sum is always 1.Hence, the answer is 1.**Final Answer**boxed{D}

💡First, I need to expand the given complex expression ((a + i)(1 - ai)). I'll use the distributive property to multiply each term in the first parentheses by each term in the second parentheses.After expanding, I'll simplify the expression by combining like terms and remembering that (i^2 = -1). This will give me a complex number in the form of (x + yi), where (x) is the real part and (y) is the imaginary part.Since the product is equal to 2, which is a real number, the imaginary part of the expanded expression must be zero. This gives me two separate equations: one for the real part and one for the imaginary part.I'll solve the equation from the real part first to find the value of (a). Then, I'll substitute this value back into the equation from the imaginary part to ensure consistency.Finally, I'll verify that the solution satisfies both equations, confirming the correct value of (a).

💡Okay, so Johann has 64 fair coins, and he's going to flip all of them. Now, any coin that lands on tails is going to be tossed again, and this process repeats up to three more times for each coin. So, in total, each coin can be tossed up to four times. The question is asking for the expected number of coins that show heads after these tosses. The options are 60, 56, 52, 48, or 64.Alright, let's break this down. First, I know that each coin is fair, so the probability of getting heads on any single toss is 1/2, and the same for tails. Now, Johann is flipping all the coins, and if a coin lands on tails, he flips it again, up to three more times. So, each coin can be flipped a maximum of four times.I think the key here is to figure out the probability that a single coin ends up showing heads after these multiple tosses. Once I have that probability, I can multiply it by the total number of coins, which is 64, to get the expected number of heads.So, let's focus on one coin. What's the probability that this coin shows heads after up to four tosses? Well, the coin will keep getting tossed until it lands on heads or until it's been tossed four times. So, the coin could land on heads on the first toss, or if it's tails, it gets tossed again, and so on.Let me think about the different scenarios:1. The coin lands on heads on the first toss. The probability of this is 1/2.2. If it lands on tails the first time (probability 1/2), then it gets tossed again. The probability of getting heads on the second toss is then 1/2, so the combined probability for this scenario is (1/2) * (1/2) = 1/4.3. If it's tails again on the second toss (probability 1/2), it gets tossed a third time. The probability of getting heads on the third toss is 1/2, so the combined probability is (1/2)^3 = 1/8.4. If it's tails on the third toss as well (probability 1/2), it gets tossed a fourth time. The probability of getting heads on the fourth toss is 1/2, so the combined probability is (1/2)^4 = 1/16.Now, if the coin is tails on all four tosses, it will still show tails, but we're only interested in the probability of it showing heads. So, we can add up the probabilities of it showing heads on each of the four tosses.So, the total probability P of a coin showing heads after up to four tosses is:P = 1/2 + 1/4 + 1/8 + 1/16Let me compute that:1/2 is 8/16,1/4 is 4/16,1/8 is 2/16,1/16 is 1/16.Adding them up: 8/16 + 4/16 + 2/16 + 1/16 = 15/16.So, the probability that a single coin shows heads after up to four tosses is 15/16.Now, since Johann has 64 coins, the expected number of heads is 64 multiplied by 15/16.Let me calculate that:64 * (15/16) = (64/16) * 15 = 4 * 15 = 60.So, the expected number of coins showing heads after these tosses is 60.Wait, but let me double-check my reasoning. Is there another way to think about this?Alternatively, instead of calculating the probability of getting heads on each toss and adding them up, I could consider the probability of the coin never showing heads in all four tosses and subtract that from 1.The probability of getting tails on all four tosses is (1/2)^4 = 1/16. So, the probability of getting at least one head in four tosses is 1 - 1/16 = 15/16, which matches what I calculated earlier.Therefore, the expected number of heads is indeed 64 * 15/16 = 60.So, the answer should be 60, which is option A.**Final Answer**boxed{A}

💡Alright, so I've got this geometry problem here about a cone and some reflections. Let me try to visualize it first. There's a cone with its vertex at the origin, and a plane Π that passes through this vertex and is perpendicular to the cone's axis. Point A is somewhere on this plane Π. Then, there's a point M on the cone such that a light ray from A to M reflects off the cone's surface and becomes parallel to plane Π. I need to find the locus of the projections of these points M onto plane Π.Hmm, okay. Let me break this down. First, the cone. Since the plane Π passes through the vertex and is perpendicular to the axis, it must be a plane that slices the cone along its axis. So, if I imagine the cone standing upright, Π is like a horizontal plane cutting through the tip. Point A is somewhere on this horizontal plane.Now, point M is on the cone. A light ray goes from A to M, reflects off the cone, and then becomes parallel to Π. Since Π is perpendicular to the cone's axis, being parallel to Π would mean the reflected ray is horizontal, right? So, after reflecting, the light ray doesn't go up or down along the cone's axis but moves sideways.I think I need to use some properties of reflection here. When a light ray reflects off a surface, the angle of incidence equals the angle of reflection. But since the cone is a curved surface, the reflection properties might be a bit more complex. Maybe I can use the concept of the normal vector at point M on the cone to determine the direction of the reflected ray.Let me set up a coordinate system. Let's place the vertex of the cone at the origin (0,0,0). The axis of the cone can be along the z-axis. Then, plane Π is the xy-plane because it's perpendicular to the z-axis and passes through the origin. Point A is somewhere in the xy-plane, say at (a, 0, 0) for simplicity.Point M is on the cone. The equation of the cone can be written as x² + y² = (k z)², where k is the slope of the cone's generatrix. So, k = tan(α), where α is the angle between the axis and the generatrix.Now, the light ray from A to M reflects off the cone. The reflected ray is parallel to Π, which is the xy-plane. So, the reflected ray must have a direction vector with zero z-component. That means after reflection, the light ray is moving horizontally.To find the reflection, I need the normal vector at point M on the cone. The gradient of the cone's equation gives the normal vector. The gradient of x² + y² - (k z)² is (2x, 2y, -2k² z). So, the normal vector at M is (x, y, -k² z).The reflection of the incoming ray AM over the normal vector should give the direction of the reflected ray. The formula for reflecting a vector over another vector involves some dot product calculations. Let me recall the formula: if I have a vector v and I want to reflect it over a vector n, the reflected vector v' is given by v' = v - 2(v·n)/(n·n) * n.So, the incoming vector is from A to M, which is (x - a, y, z). The normal vector is (x, y, -k² z). Let me compute the dot product:(v·n) = (x - a)x + y * y + z*(-k² z) = x² - a x + y² - k² z².But since M is on the cone, x² + y² = k² z², so this simplifies to (k² z² - a x) - k² z² = -a x.So, (v·n) = -a x.Now, the magnitude squared of the normal vector n is x² + y² + (k² z)². Again, since x² + y² = k² z², this becomes k² z² + k^4 z² = k² z² (1 + k²).So, (n·n) = k² z² (1 + k²).Putting it into the reflection formula:v' = (x - a, y, z) - 2*(-a x)/(k² z² (1 + k²)) * (x, y, -k² z).Simplify the scalar factor: 2*(-a x)/(k² z² (1 + k²)) = -2 a x / (k² z² (1 + k²)).Wait, no, actually, the formula is v' = v - 2(v·n)/(n·n) * n.So, it's:v' = (x - a, y, z) - 2*(-a x)/(k² z² (1 + k²)) * (x, y, -k² z).So, the scalar is 2*(-a x)/(k² z² (1 + k²)) = -2 a x / (k² z² (1 + k²)).But since we have a negative sign in front, it becomes positive:v' = (x - a, y, z) + 2 a x / (k² z² (1 + k²)) * (x, y, -k² z).Now, let's compute each component:First component:(x - a) + 2 a x / (k² z² (1 + k²)) * x = x - a + 2 a x² / (k² z² (1 + k²)).Second component:y + 2 a x / (k² z² (1 + k²)) * y = y + 2 a x y / (k² z² (1 + k²)).Third component:z + 2 a x / (k² z² (1 + k²)) * (-k² z) = z - 2 a x / (z (1 + k²)).But we know that after reflection, the ray is parallel to Π, which is the xy-plane. So, the z-component of the reflected vector must be zero. Therefore, the third component of v' must be zero:z - 2 a x / (z (1 + k²)) = 0.So, z = 2 a x / (z (1 + k²)).Multiply both sides by z (1 + k²):z² (1 + k²) = 2 a x.But from the cone equation, x² + y² = k² z². So, z² = (x² + y²)/k².Substitute into the equation:(x² + y²)/k² * (1 + k²) = 2 a x.Multiply both sides by k²:(x² + y²)(1 + k²) = 2 a x k².So, x² + y² = (2 a x k²)/(1 + k²).This is the equation of a circle in the xy-plane. Let me rearrange it:x² - (2 a k²)/(1 + k²) x + y² = 0.Completing the square for x:x² - (2 a k²)/(1 + k²) x + (a² k^4)/(1 + k²)^2 + y² = (a² k^4)/(1 + k²)^2.So, (x - (a k²)/(1 + k²))² + y² = (a k²/(1 + k²))².This is a circle centered at (a k²/(1 + k²), 0) with radius a k²/(1 + k²).Therefore, the locus of the projections of points M onto plane Π is a circle with center at (a k²/(1 + k²), 0) and radius a k²/(1 + k²).Wait, but in the problem statement, point A is in plane Π, which is the xy-plane, and we assumed A is at (a, 0, 0). So, the projection of M onto Π is (x, y, 0). So, the locus is indeed this circle in the xy-plane.I think that makes sense. The reflection condition leads to a circle as the locus of projections. The center is shifted along the x-axis from the origin, and the radius is proportional to a and k²/(1 + k²). Since k = tan(α), this can also be written in terms of the angle α.So, the final answer should be the equation of this circle. Let me write it again:(x - (a k²)/(1 + k²))² + y² = (a k²/(1 + k²))².Alternatively, expanding it:x² + y² - 2 a k² x / (1 + k²) = 0.Yeah, that's a cleaner way to write it. So, the locus is the circle defined by x² + y² - 2 a k² x / (1 + k²) = 0.I think that's it. Let me just double-check my steps. I set up the coordinate system, found the normal vector, applied the reflection formula, set the z-component to zero, substituted the cone equation, and ended up with a circle equation. Seems solid to me.

💡First, I need to determine the total price of the bicycle based on the information provided. Lily paid 20% of the cost upfront, which amounted to 200.To find the total price, I'll set up an equation where 20% of the total price equals 200. This can be written as:0.20 × P = 200Next, I'll solve for P by dividing both sides of the equation by 0.20:P = 200 / 0.20Calculating this gives:P = 1000Therefore, the total price of the bicycle is 1000.