💡First, I recognize that Jasmine's shoe size is 7, and the combined shoe size for both Jasmine and Alexa is 21.To find Alexa's shoe size, I subtract Jasmine's shoe size from the combined total:Alexa's shoe size = 21 - 7 = 14.Now, I have both shoe sizes: Alexa's is 14 and Jasmine's is 7.To determine the ratio of Alexa's shoe size to Jasmine's shoe size, I set up the ratio as 14:7.Next, I simplify this ratio by dividing both numbers by their greatest common divisor, which is 7:14 ÷ 7 = 27 ÷ 7 = 1Therefore, the simplified ratio is 2:1.

💡Alright, so I've got this problem here about a sports school with 55 people, and each person is either a tennis player or a chess player. The key point is that there are no four chess players who have the same number of friends among the tennis players. I need to figure out the maximum number of chess players possible in this school.Okay, let's break this down. First, let me define some variables to make it clearer. Let's say the number of tennis players is ( a ). That means the number of chess players is ( 55 - a ). So, our goal is to find the maximum value of ( 55 - a ), which would correspond to the minimum value of ( a ).Now, each chess player can have a certain number of friends who are tennis players. The number of friends a chess player can have ranges from 0 to ( a ), right? Because a chess player could potentially be friends with none of the tennis players or all of them. So, that gives us ( a + 1 ) possible different numbers of friends.The problem states that there are no four chess players with the same number of friends among the tennis players. This sounds like a pigeonhole principle problem. The pigeonhole principle basically says that if you have more pigeons than pigeonholes, at least one pigeonhole must contain more than one pigeon. In this case, the "pigeonholes" are the different numbers of friends (from 0 to ( a )), and the "pigeons" are the chess players.If we have too many chess players, some of them would have to share the same number of friends. Specifically, if we have more than ( 3(a + 1) ) chess players, then by the pigeonhole principle, at least four of them would have the same number of friends. But the problem says that this isn't allowed. So, the number of chess players must be less than or equal to ( 3(a + 1) ).Putting that into an inequality, we get:[55 - a leq 3(a + 1)]Let me solve this step by step. First, expand the right side:[55 - a leq 3a + 3]Now, let's get all the ( a ) terms on one side and the constants on the other:[55 - 3 leq 3a + a]Simplify both sides:[52 leq 4a]Divide both sides by 4:[13 leq a]So, ( a ) must be at least 13. That means the minimum number of tennis players is 13, which in turn means the maximum number of chess players is:[55 - 13 = 42]Hmm, does this make sense? Let me double-check. If there are 13 tennis players, then each chess player can have between 0 and 13 friends among the tennis players. That's 14 different possible numbers of friends. If we have 42 chess players, and we want to avoid having four chess players with the same number of friends, we can have at most three chess players for each number of friends.So, 14 different friend counts multiplied by 3 chess players each gives us ( 14 times 3 = 42 ) chess players. That fits perfectly. So, yes, 42 chess players is achievable without violating the condition.Is there a way to have more than 42 chess players? Let's see. If we tried to have 43 chess players, then we would have ( 55 - 43 = 12 ) tennis players. The number of possible friend counts would be 13 (from 0 to 12). Using the same logic, ( 3 times 13 = 39 ), which is less than 43. So, we would have more chess players than the maximum allowed without having four with the same number of friends. Therefore, 43 chess players wouldn't work.So, 42 seems to be the maximum number of chess players possible under the given conditions. I think that's the answer.**Final Answer**The maximum number of chess players is boxed{42}.

💡Alright, so I have this problem with two lines given in parametric form, and I need to find where they intersect. Hmm, okay. Let me try to understand what's going on here.First, let me write down the equations again to make sure I have them right. The first line is parameterized as:[begin{pmatrix} x y end{pmatrix} = begin{pmatrix} 1 4 end{pmatrix} + s begin{pmatrix} -2 6 end{pmatrix}]So, breaking that down, the x-component is (1 - 2s) and the y-component is (4 + 6s). Got it. That means as the parameter (s) changes, the point ((x, y)) moves along the line.The second line is parameterized as:[begin{pmatrix} x y end{pmatrix} = begin{pmatrix} 3 10 end{pmatrix} + v begin{pmatrix} -1 3 end{pmatrix}]Similarly, the x-component here is (3 - v) and the y-component is (10 + 3v). So, as the parameter (v) changes, this point moves along the second line.Now, I need to find if there's a point where these two lines cross each other. That means I need to find values of (s) and (v) such that both the x and y components are equal for both lines. In other words, I need to solve the system of equations:1. (1 - 2s = 3 - v)2. (4 + 6s = 10 + 3v)Okay, let's tackle the first equation. So, (1 - 2s = 3 - v). I can rearrange this to solve for one of the variables. Let me solve for (v) because it seems straightforward.Subtracting 1 from both sides gives:(-2s = 2 - v)Then, adding (v) to both sides and adding (2s) to both sides:(v = 2 + 2s)Wait, is that right? Let me check:Starting from (1 - 2s = 3 - v), subtract 1 from both sides:(-2s = 2 - v)Then, add (v) to both sides:(-2s + v = 2)Then, add (2s) to both sides:(v = 2 + 2s)Yes, that seems correct. So, (v = 2 + 2s). Okay, now I can substitute this expression for (v) into the second equation to solve for (s).The second equation is (4 + 6s = 10 + 3v). Substituting (v = 2 + 2s) into this:(4 + 6s = 10 + 3(2 + 2s))Let me compute the right side:(10 + 3*2 + 3*2s = 10 + 6 + 6s = 16 + 6s)So now, the equation becomes:(4 + 6s = 16 + 6s)Hmm, that's interesting. If I subtract (6s) from both sides:(4 = 16)Wait, that can't be right. 4 doesn't equal 16. That doesn't make sense. Did I make a mistake somewhere?Let me go back and check my steps.First, the first equation:(1 - 2s = 3 - v)Solving for (v):(v = 3 - (1 - 2s) = 3 - 1 + 2s = 2 + 2s)Yes, that's correct.Then, substituting into the second equation:(4 + 6s = 10 + 3v)Which becomes:(4 + 6s = 10 + 3(2 + 2s))Calculating the right side:(10 + 6 + 6s = 16 + 6s)So, (4 + 6s = 16 + 6s)Subtracting (6s) from both sides:(4 = 16)Hmm, that's definitely a problem. It suggests that there's no solution because 4 is not equal to 16. So, does that mean the lines don't intersect?Wait, but maybe I made a mistake in setting up the equations. Let me double-check the parametric equations.First line:(x = 1 - 2s)(y = 4 + 6s)Second line:(x = 3 - v)(y = 10 + 3v)So, setting x's equal: (1 - 2s = 3 - v)Setting y's equal: (4 + 6s = 10 + 3v)That seems correct.So, solving the first equation for (v), I get (v = 2 + 2s), which I substitute into the second equation.But when I do that, I end up with (4 = 16), which is impossible. That suggests that there is no solution, meaning the lines do not intersect.But wait, could it be that the lines are parallel? Let me check the direction vectors.The direction vector of the first line is (begin{pmatrix} -2 6 end{pmatrix}), and the direction vector of the second line is (begin{pmatrix} -1 3 end{pmatrix}).Is one a scalar multiple of the other? Let's see:If I multiply the second direction vector by 2, I get (begin{pmatrix} -2 6 end{pmatrix}), which is exactly the direction vector of the first line. So, yes, the direction vectors are scalar multiples, meaning the lines are parallel.If the lines are parallel, they can either be coincident (lying on top of each other) or they are distinct and never intersect.To check if they are coincident, let's see if a point from one line lies on the other line.Take a point from the first line when (s = 0): ((1, 4)).Does this point satisfy the equation of the second line? Let's see.For the second line, (x = 3 - v) and (y = 10 + 3v).If (x = 1), then (1 = 3 - v) implies (v = 2).Then, plugging (v = 2) into the y-component: (10 + 3*2 = 16), but the y-coordinate is 4, not 16. So, the point (1, 4) is not on the second line.Therefore, the lines are parallel and distinct, meaning they never intersect.So, the conclusion is that there is no intersection point.Wait, but just to make sure, let me try another point. Let's take (s = 1) on the first line: (x = 1 - 2*1 = -1), (y = 4 + 6*1 = 10). So, the point (-1, 10).Now, check if this lies on the second line.For the second line, (x = 3 - v). If (x = -1), then ( -1 = 3 - v ) implies (v = 4).Then, plugging (v = 4) into the y-component: (10 + 3*4 = 22). But the y-coordinate is 10, not 22. So, again, the point is not on the second line.Therefore, confirming that the lines are parallel and distinct, so they do not intersect.I think that's solid. So, the answer is that there is no intersection.

💡To determine how many digits ( D ) make the four-digit number ( 10D4 ) a multiple of both 3 and 4, we need to check the divisibility rules for these numbers.**Divisibility by 3:**A number is divisible by 3 if the sum of its digits is a multiple of 3. For ( 10D4 ), the sum of the digits is:[ 1 + 0 + D + 4 = 5 + D ]We need ( 5 + D ) to be a multiple of 3. The possible values of ( D ) that satisfy this are:[ D = 1, 4, 7 ]**Divisibility by 4:**A number is divisible by 4 if the number formed by its last two digits is a multiple of 4. For ( 10D4 ), the last two digits are ( D4 ). We need ( D4 ) to be a multiple of 4. Checking the possible values:- ( 04 ) is a multiple of 4.- ( 14 ) is not a multiple of 4.- ( 24 ) is a multiple of 4.- ( 34 ) is not a multiple of 4.- ( 44 ) is a multiple of 4.- ( 54 ) is not a multiple of 4.- ( 64 ) is a multiple of 4.- ( 74 ) is not a multiple of 4.- ( 84 ) is a multiple of 4.- ( 94 ) is not a multiple of 4.Thus, the possible values of ( D ) are:[ D = 0, 2, 4, 6, 8 ]**Finding Common Values:**From the two sets of possible values:- Divisibility by 3: ( D = 1, 4, 7 )- Divisibility by 4: ( D = 0, 2, 4, 6, 8 )The common value is:[ D = 4 ]Therefore, there is only one digit ( D ) that satisfies both conditions.

💡Alright, so I've got this problem about a cubic function, f(x) = x³ + a x² + b x + c. There are four statements given, and I need to figure out how many of them are correct. Let's take them one by one.1. **There exists an x₀ in real numbers such that f(x₀) = 0.**Hmm, okay. Since f(x) is a cubic polynomial, I remember that cubic functions always have at least one real root. That's because as x approaches positive infinity, f(x) goes to positive infinity, and as x approaches negative infinity, f(x) goes to negative infinity. So, by the Intermediate Value Theorem, there must be some real number x₀ where f(x₀) = 0. That makes sense. So, statement 1 is correct.2. **The necessary and sufficient condition for the function y = f(x) to have two extreme points is a² > 3b.**Alright, let's think about this. To find extreme points, I need to take the derivative of f(x). The derivative f'(x) = 3x² + 2a x + b. For there to be two extreme points, the derivative must have two real roots. That happens when the discriminant of the quadratic is positive.The discriminant D of f'(x) is (2a)² - 4 * 3 * b = 4a² - 12b. For two real roots, D must be greater than 0. So, 4a² - 12b > 0 ⇒ a² > 3b. That's exactly what statement 2 says. So, statement 2 is correct.3. **If x₀ is a local maximum point of y = f(x), then f(x) is monotonically decreasing in the interval (x₀, +∞).**Okay, so if x₀ is a local maximum, that means the function changes from increasing to decreasing at x₀. But does that mean it's decreasing all the way to infinity? Let's think about the behavior of a cubic function.A cubic function has an inflection point where the concavity changes. After the local maximum, the function might start decreasing, but since it's a cubic, it will eventually turn around and start increasing again as x approaches infinity. So, it's not necessarily monotonically decreasing on the entire interval (x₀, +∞). It might decrease for a while, reach a minimum, and then start increasing. Therefore, statement 3 is incorrect.4. **The center of symmetry of the function y = f(x) is (-a/3, f(-a/3)).**Hmm, centers of symmetry for cubic functions. I recall that cubic functions are symmetric about their inflection points. To find the inflection point, we need to find where the second derivative changes sign.The second derivative f''(x) = 6x + 2a. Setting f''(x) = 0 gives x = -a/3. So, the inflection point is at x = -a/3, and the y-coordinate is f(-a/3). Since the function is symmetric about this point, the center of symmetry is indeed (-a/3, f(-a/3)). So, statement 4 is correct.Putting it all together:- Statement 1: Correct- Statement 2: Correct- Statement 3: Incorrect- Statement 4: CorrectSo, out of the four statements, three are correct. Therefore, the number of correct statements is 3.**Final Answer**boxed{C}

💡To determine ( N ) in the equation ( 995 + 997 + 999 + 1001 + 1003 = 5005 - N ), we can follow these steps:1. **Express each term as a transformation of 1000:** [ 995 = 1000 - 5, quad 997 = 1000 - 3, quad 999 = 1000 - 1, quad 1001 = 1000 + 1, quad 1003 = 1000 + 3 ]2. **Rewrite the sum using these transformations:** [ (1000 - 5) + (1000 - 3) + (1000 - 1) + (1000 + 1) + (1000 + 3) ]3. **Simplify the expression by factoring out 1000 and combining like terms:** [ 5 times 1000 - (5 + 3 + 1) + (1 + 3) ] Notice that the negative and positive terms cancel each other out: [ 5 times 1000 - 5 + 5 = 5 times 1000 = 5000 ]4. **Set the simplified sum equal to the given equation:** [ 5000 = 5005 - N ]5. **Solve for ( N ):** [ 5005 - 5000 = N implies N = 5 ]Thus, the value of ( N ) is (boxed{B}).