💡Okay, so I have this problem about Vasya buying marshmallows and meat pies over 15 school days. He can buy either 9 marshmallows, or 2 meat pies, or a combination of 4 marshmallows and 1 meat pie. Sometimes he doesn't buy anything. In total, over these 15 days, he bought 30 marshmallows and 9 meat pies. I need to find out how many days he didn't buy anything.First, I think I need to set up some variables to represent the different scenarios. Let me denote:- ( x ) = number of days he buys 9 marshmallows- ( y ) = number of days he buys 2 meat pies- ( z ) = number of days he buys 4 marshmallows and 1 meat pie- ( w ) = number of days he buys nothingSo, the total number of days is 15, which gives me the equation:( x + y + z + w = 15 )Next, I need to account for the total number of marshmallows and meat pies he bought. For marshmallows, he buys 9 on some days and 4 on others. So, the total marshmallows would be:( 9x + 4z = 30 )Similarly, for meat pies, he buys 2 on some days and 1 on others. So, the total meat pies would be:( 2y + z = 9 )Now, I have three equations:1. ( x + y + z + w = 15 )2. ( 9x + 4z = 30 )3. ( 2y + z = 9 )I need to solve these equations to find ( w ), the number of days he didn't buy anything.Let me start by solving the second equation for ( z ):( 9x + 4z = 30 )I can rearrange this to:( 4z = 30 - 9x )( z = frac{30 - 9x}{4} )Since ( z ) must be an integer (you can't have a fraction of a day), ( 30 - 9x ) must be divisible by 4. Let's see what values of ( x ) make this possible.Trying ( x = 0 ):( z = frac{30 - 0}{4} = 7.5 ) → Not an integer.Trying ( x = 1 ):( z = frac{30 - 9}{4} = frac{21}{4} = 5.25 ) → Not an integer.Trying ( x = 2 ):( z = frac{30 - 18}{4} = frac{12}{4} = 3 ) → Integer. Good.Trying ( x = 3 ):( z = frac{30 - 27}{4} = frac{3}{4} = 0.75 ) → Not an integer.So, the only valid value is ( x = 2 ) and ( z = 3 ).Now, let's use the third equation to find ( y ):( 2y + z = 9 )Substituting ( z = 3 ):( 2y + 3 = 9 )( 2y = 6 )( y = 3 )Now, we have ( x = 2 ), ( y = 3 ), and ( z = 3 ). Let's plug these into the first equation to find ( w ):( x + y + z + w = 15 )( 2 + 3 + 3 + w = 15 )( 8 + w = 15 )( w = 7 )So, Vasya didn't buy anything on 7 days.

💡Okay, so I've got this problem here: For how many real values of ( x ) does ( left|1 - frac{x}{2}iright| = 1 ) hold true? Hmm, let me try to figure this out step by step. First off, I remember that the absolute value (or modulus) of a complex number ( a + bi ) is calculated as ( sqrt{a^2 + b^2} ). So, in this case, the complex number is ( 1 - frac{x}{2}i ). That means the real part is 1 and the imaginary part is ( -frac{x}{2} ). So, applying the formula for the modulus, I should compute ( sqrt{1^2 + left(-frac{x}{2}right)^2} ). Let me write that down:[left|1 - frac{x}{2}iright| = sqrt{1^2 + left(-frac{x}{2}right)^2}]Simplifying inside the square root:[sqrt{1 + left(frac{x^2}{4}right)}]So, the equation given is:[sqrt{1 + frac{x^2}{4}} = 1]Now, to solve for ( x ), I can square both sides of the equation to eliminate the square root. Let's do that:[left(sqrt{1 + frac{x^2}{4}}right)^2 = 1^2]Which simplifies to:[1 + frac{x^2}{4} = 1]Hmm, okay, so subtracting 1 from both sides gives:[frac{x^2}{4} = 0]To solve for ( x ), I can multiply both sides by 4:[x^2 = 0]Taking the square root of both sides, I get:[x = 0]So, it looks like the only real value of ( x ) that satisfies the equation is 0. Wait, let me double-check to make sure I didn't make any mistakes. Starting from the beginning, the modulus of ( 1 - frac{x}{2}i ) is indeed ( sqrt{1 + frac{x^2}{4}} ). Setting that equal to 1, squaring both sides gives ( 1 + frac{x^2}{4} = 1 ), which simplifies correctly to ( x^2 = 0 ), so ( x = 0 ). Is there any other value of ( x ) that could satisfy this? Let me think. If ( x ) were any other real number, say positive or negative, then ( frac{x^2}{4} ) would be positive, making the modulus greater than 1. So, only when ( x = 0 ) does the modulus equal exactly 1. Therefore, I'm confident that the only real solution is ( x = 0 ), which means there's just one real value of ( x ) that satisfies the equation.**Final Answer**The number of real values of ( x ) is boxed{1}.

💡Okay, so I have this math problem here: (2 + 3 cdot 4 - 5 cdot 6 + 7). Hmm, I need to figure out what this equals. Alright, let's break it down step by step.First, I remember that in math problems like this, I should follow the order of operations, which is often remembered by the acronym PEMDAS—Parentheses, Exponents, Multiplication and Division, and then Addition and Subtraction. Since there are no parentheses or exponents in this problem, I should start with the multiplication parts.Looking at the expression, I see two multiplication operations: (3 cdot 4) and (5 cdot 6). Let me calculate those first.So, (3 cdot 4) is 12, and (5 cdot 6) is 30. Now, I can substitute these back into the original expression to simplify it. That gives me: (2 + 12 - 30 + 7).Now, I need to handle the addition and subtraction from left to right. Let's start with (2 + 12). That equals 14. Next, I subtract 30 from 14, which gives me (-16). Finally, I add 7 to (-16), resulting in (-9).Wait, did I do that right? Let me double-check. Starting over: (2 + 3 cdot 4 - 5 cdot 6 + 7). Multiplications first: (3 cdot 4 = 12) and (5 cdot 6 = 30). So, substituting back: (2 + 12 - 30 + 7). Then, (2 + 12 = 14), (14 - 30 = -16), and (-16 + 7 = -9). Yeah, that seems consistent.Is there another way to look at this? Maybe by grouping the positive and negative numbers separately. Let's see: (2 + 12 + 7) is (21), and (-30) is the only negative number. So, (21 - 30 = -9). That matches my previous result. Okay, I'm more confident now that (-9) is the correct answer.I guess it's important to remember to do the multiplications before the additions and subtractions to avoid mistakes. If I had just gone from left to right without considering the order of operations, I might have added 2 and 3 first, which would have given me a different and incorrect result. So, following PEMDAS really helps in getting the right answer here.Another thing to note is that subtraction is the same as adding a negative number. So, (2 + 12 - 30 + 7) can also be thought of as (2 + 12 + (-30) + 7). Adding these up: (2 + 12 = 14), (14 + (-30) = -16), and (-16 + 7 = -9). This alternative way of thinking reinforces that the answer is indeed (-9).I wonder if there's a way to visualize this problem to make it clearer. Maybe by drawing a number line. Starting at 2, then moving up by 12 (which gets me to 14), then moving back by 30 (which takes me to (-16)), and finally moving up by 7 (ending at (-9)). That visual representation also leads me to the same conclusion.It's also interesting to see how the positive and negative numbers balance out. The positive numbers add up to (2 + 12 + 7 = 21), and the negative number is (-30). So, the total is (21 - 30 = -9). This method of grouping positives and negatives can be helpful in more complex expressions too.I think I've covered all the ways to approach this problem. I've double-checked my calculations, used different methods like the number line and grouping, and each time I arrive at the same answer. It's reassuring to see consistency across different methods. So, I'm pretty sure that the value of the expression (2 + 3 cdot 4 - 5 cdot 6 + 7) is (-9).

💡Okay, so I have this problem where there's a right circular cylinder inscribed in a hemisphere. The cylinder has a radius of 3, and the hemisphere has a radius of 7. The bases of the cylinder are parallel to the base of the hemisphere. I need to find the height of the cylinder. Hmm, let me try to visualize this.First, I know that a hemisphere is like half of a sphere. So, if the hemisphere has a radius of 7, that means the flat base of the hemisphere is a circle with radius 7, and the curved part is half of a sphere with radius 7. Now, inside this hemisphere, there's a cylinder. The cylinder is right and circular, which means its bases are circles, and it's standing straight up, not leaning to any side.The cylinder has a radius of 3, so each of its circular bases has a radius of 3. Since the cylinder is inscribed in the hemisphere, it must fit perfectly inside without extending beyond the hemisphere. Also, the bases of the cylinder are parallel to the base of the hemisphere, which means the cylinder is sitting straight up, aligned with the center of the hemisphere.I think drawing a diagram might help. Let me imagine the hemisphere with its flat base on the ground. The center of the hemisphere, let's call it point O, is at the center of the flat base. Now, the cylinder is inside this hemisphere, touching it at the top and around the sides. The top of the cylinder will be somewhere inside the hemisphere, not reaching the top of the hemisphere because the hemisphere's radius is larger than the cylinder's radius.Since the cylinder is inscribed, the top circular face of the cylinder must touch the inner surface of the hemisphere. So, if I consider a cross-sectional view of the hemisphere and the cylinder, it should look like a semicircle (the cross-section of the hemisphere) with a rectangle inscribed inside it (the cross-section of the cylinder). The rectangle will have a height equal to the height of the cylinder and a width equal to twice the radius of the cylinder, which is 6.Let me denote the height of the cylinder as h. The center of the hemisphere is at point O, and the center of the cylinder's base is also at O because the cylinder is inscribed and symmetric. The top of the cylinder is then h units above the base.Now, considering the cross-sectional view, the semicircle has a radius of 7, and the rectangle has a height h and a width of 6. The top corner of the rectangle touches the semicircle. So, if I consider the coordinates of that top corner, it should satisfy the equation of the semicircle.Let me set up a coordinate system where the center O is at (0,0). The semicircle is the upper half of the circle with radius 7, so its equation is x² + y² = 7², which simplifies to x² + y² = 49. The top corner of the rectangle is at (3, h) because the width of the rectangle is 6, so half of that is 3, and the height is h.Since this point (3, h) lies on the semicircle, it must satisfy the equation x² + y² = 49. Plugging in the coordinates, we get:3² + h² = 49Calculating 3 squared is 9, so:9 + h² = 49Subtracting 9 from both sides:h² = 40Taking the square root of both sides:h = √40Simplifying √40, which can be written as √(4*10) = 2√10.So, the height of the cylinder is 2√10 units.Wait, let me double-check my steps to make sure I didn't make a mistake. I set up the coordinate system with the center at (0,0). The top corner of the cylinder is at (3, h), which lies on the hemisphere's surface. Plugging into the equation x² + y² = 49 gives 9 + h² = 49, so h² = 40, and h = √40. That seems correct.Alternatively, I can think about this using the Pythagorean theorem. The radius of the hemisphere is 7, and the radius of the cylinder is 3. If I imagine a right triangle where one leg is the radius of the cylinder (3), another leg is the height of the cylinder (h), and the hypotenuse is the radius of the hemisphere (7), then by Pythagoras:3² + h² = 7²Which again gives 9 + h² = 49, leading to h² = 40 and h = √40.Yes, that confirms my earlier result. So, the height of the cylinder is √40, which simplifies to 2√10.I think that's the correct answer. It makes sense because the cylinder is inscribed, so it's as tall as it can be without exceeding the hemisphere's boundaries. The height is determined by the Pythagorean relationship between the hemisphere's radius, the cylinder's radius, and the cylinder's height.Just to recap:1. Hemisphere radius (R) = 72. Cylinder radius (r) = 33. Height of cylinder (h) = ?Using Pythagoras:r² + h² = R²3² + h² = 7²9 + h² = 49h² = 40h = √40 = 2√10Yep, that all checks out. I don't see any mistakes in my reasoning, so I'm confident that the height of the cylinder is 2√10.**Final Answer**The height of the cylinder is boxed{2sqrt{10}}.

💡Okay, so I have this problem here asking me which of the given radicals is the simplest quadratic radical. The options are A: √(1/2), B: √12, C: √4.5, and D: √5. Hmm, I need to figure out which one is the simplest. Let me think about what a quadratic radical is. I believe it refers to a square root that can't be simplified any further, meaning there are no perfect square factors inside the radical, and it's not a fraction or decimal inside the square root.Starting with option A: √(1/2). That looks like a fraction inside the square root. I remember that you can simplify square roots of fractions by taking the square root of the numerator and the denominator separately. So √(1/2) would be √1 over √2, which is 1/√2. But usually, we don't like having radicals in the denominator, so we rationalize it by multiplying numerator and denominator by √2, resulting in √2/2. Okay, so √(1/2) simplifies to √2/2. That means it's not in its simplest form because we can write it without a radical in the denominator. So A is not the simplest quadratic radical.Moving on to option B: √12. I know that 12 can be factored into 4 and 3, and 4 is a perfect square. So √12 is √(4*3), which is √4 times √3, which simplifies to 2√3. So √12 simplifies to 2√3. That means B is also not in its simplest form because it can be simplified further.Option C: √4.5. Hmm, 4.5 is a decimal. I think it might be easier to convert that into a fraction. 4.5 is the same as 9/2. So √4.5 is √(9/2). Similar to option A, I can separate the square root of the numerator and the denominator. So that would be √9 over √2, which is 3/√2. Again, we don't like radicals in the denominator, so rationalizing it by multiplying numerator and denominator by √2 gives us (3√2)/2. So √4.5 simplifies to (3√2)/2. Therefore, C is also not in its simplest form.Finally, option D: √5. Let me see if this can be simplified. 5 is a prime number, so it doesn't have any perfect square factors other than 1. That means √5 cannot be simplified any further. There are no perfect squares inside the radical, and it's not a fraction or decimal. So √5 is already in its simplest form.So, going back through all the options:- A simplifies to √2/2.- B simplifies to 2√3.- C simplifies to (3√2)/2.- D is already simplified as √5.Therefore, the simplest quadratic radical among the options is D: √5.

💡Okay, so I have this problem about polynomials with complex coefficients. The polynomial is given as ( f(z) = C_0 z^n + C_1 z^{n-1} + dots + C_{n-1} z + C_n ). I need to prove that there exists a complex number ( z_0 ) with ( |z_0| leq 1 ) such that ( |f(z_0)| geq |C_0| + |C_n| ).Hmm, let me think about how to approach this. I know that for polynomials, evaluating them on the unit circle can sometimes give useful information. Maybe I can use some properties of complex numbers or roots of unity here.First, let me recall that the roots of unity are complex numbers that are evenly spaced around the unit circle. They have the property that their sum is zero, which might be useful. Maybe I can use some kind of averaging argument over these roots.Let me define ( omega = e^{2pi i / n} ), which is a primitive nth root of unity. Then, the roots of unity are ( omega^k ) for ( k = 0, 1, 2, dots, n-1 ). If I take ( eta ) as another complex number on the unit circle, so ( |eta| = 1 ), then ( omega^k eta ) is also on the unit circle.What if I evaluate ( f ) at each of these points ( omega^k eta ) and then sum them up? Let me try that.So, ( f(omega^k eta) = C_0 (omega^k eta)^n + C_1 (omega^k eta)^{n-1} + dots + C_{n-1} (omega^k eta) + C_n ).If I sum this over ( k = 1 ) to ( n ), I get:[sum_{k=1}^{n} f(omega^k eta) = C_0 sum_{k=1}^{n} (omega^k eta)^n + C_1 sum_{k=1}^{n} (omega^k eta)^{n-1} + dots + C_{n-1} sum_{k=1}^{n} (omega^k eta) + n C_n]Now, let's analyze each term in this sum. The term with ( C_0 ) is ( C_0 sum_{k=1}^{n} (omega^k eta)^n ). Since ( omega^n = 1 ), this simplifies to ( C_0 eta^n sum_{k=1}^{n} 1 = n C_0 eta^n ).For the other terms, like ( C_j sum_{k=1}^{n} (omega^k eta)^{n-j} ) where ( j = 1, 2, dots, n-1 ), we have ( sum_{k=1}^{n} omega^{k(n-j)} ). But ( omega^{n-j} ) is another root of unity, and the sum of all roots of unity is zero. So each of these sums is zero.Therefore, the entire sum simplifies to:[sum_{k=1}^{n} f(omega^k eta) = n C_0 eta^n + n C_n]Dividing both sides by ( n ), we get:[frac{1}{n} sum_{k=1}^{n} f(omega^k eta) = C_0 eta^n + C_n]Taking the magnitude on both sides:[frac{1}{n} sum_{k=1}^{n} |f(omega^k eta)| geq |C_0 eta^n + C_n|]Because the magnitude of the average is less than or equal to the average of the magnitudes.Now, I need to find ( eta ) such that ( |C_0 eta^n + C_n| ) is maximized. Since ( |eta| = 1 ), ( eta^n ) is also on the unit circle. Let me write ( C_0 ) and ( C_n ) in polar form.Let ( C_0 = r_0 e^{itheta_0} ) and ( C_n = r_n e^{itheta_n} ). Then,[C_0 eta^n + C_n = r_0 e^{itheta_0} eta^n + r_n e^{itheta_n}]I want to choose ( eta ) such that ( eta^n ) aligns the two terms in the same direction. That is, I want ( theta_0 + nphi = theta_n ) where ( eta = e^{iphi} ). This way, both terms are in the same direction, and their magnitudes add up.So, set ( phi = frac{theta_n - theta_0}{n} ). Then, ( eta = e^{iphi} ) and ( eta^n = e^{i(theta_n - theta_0)} ). Substituting back,[C_0 eta^n + C_n = r_0 e^{itheta_0} e^{i(theta_n - theta_0)} + r_n e^{itheta_n} = r_0 e^{itheta_n} + r_n e^{itheta_n} = (r_0 + r_n) e^{itheta_n}]Thus, the magnitude is ( |C_0 eta^n + C_n| = r_0 + r_n = |C_0| + |C_n| ).So, substituting back into the inequality,[frac{1}{n} sum_{k=1}^{n} |f(omega^k eta)| geq |C_0| + |C_n|]This implies that the average of ( |f(omega^k eta)| ) is at least ( |C_0| + |C_n| ). Therefore, at least one of the terms ( |f(omega^k eta)| ) must be at least ( |C_0| + |C_n| ).Hence, there exists some ( z_0 = omega^k eta ) with ( |z_0| = 1 ) such that ( |f(z_0)| geq |C_0| + |C_n| ).Wait, but the problem states ( |z_0| leq 1 ). However, in my construction, ( |z_0| = 1 ). So, does this mean the result is also true for ( |z_0| leq 1 )?Well, if I can achieve ( |f(z_0)| geq |C_0| + |C_n| ) on the boundary ( |z_0| = 1 ), then certainly it's true for ( |z_0| leq 1 ) because the maximum modulus principle tells us that the maximum of ( |f(z)| ) on the closed unit disk occurs on the boundary.Therefore, the conclusion holds.I think this makes sense. I used the roots of unity to average out the polynomial evaluations, then by choosing a specific ( eta ) to align the leading and constant terms, I was able to get the desired inequality. The key was that the sum of the polynomial evaluations simplifies nicely due to the properties of roots of unity.Yeah, I think that works. So, to summarize:1. Evaluated the polynomial at points ( omega^k eta ) for ( k = 1, dots, n ).2. Summed these evaluations, which simplified due to the roots of unity summing to zero except for the leading and constant terms.3. Chose ( eta ) such that the leading and constant terms add constructively, maximizing their combined magnitude.4. Concluded that at least one evaluation must meet or exceed this maximum, hence proving the existence of ( z_0 ).I think that covers it. I don't see any gaps in the logic, so I feel confident about this proof.**Final Answer**boxed{|f(z_0)| geq |C_0| + |C_n|}