💡First, I need to determine how much money Roger had initially. Let's denote his initial amount of money as ( x ).Roger received 28 dollars for his birthday, so his total money became ( x + 28 ).He then spent 25 dollars on a new game, which means he had ( (x + 28) - 25 ) dollars left.According to the problem, after these transactions, Roger has 19 dollars. Therefore, I can set up the equation:[(x + 28) - 25 = 19]Simplifying the equation:[x + 3 = 19]Subtracting 3 from both sides:[x = 16]So, Roger initially had 16 dollars.

💡First, I need to simplify the fraction (frac{48}{72}). To do this, I'll start by finding the greatest common divisor (GCD) of the numerator and the denominator. I'll begin by listing the factors of 48 and 72. The factors of 48 are 1, 2, 3, 4, 6, 8, 12, 16, 24, and 48. The factors of 72 are 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, and 72. Looking at both lists, the common factors are 1, 2, 3, 4, 6, 8, 12, and 24. The greatest of these is 24, so the GCD of 48 and 72 is 24.Next, I'll divide both the numerator and the denominator of the fraction by the GCD. [frac{48}{72} = frac{48 div 24}{72 div 24} = frac{2}{3}]After simplifying, the fraction becomes (frac{2}{3}). Therefore, the simplified form of (frac{48}{72}) is (boxed{frac{2}{3}}).

💡Alright, so I have this problem where I need to find all natural numbers ( n geq 3 ) such that the sum of the number of positive divisors of ( n-1 ), ( n ), and ( n+1 ) is less than or equal to 8. That is, ( d(n-1) + d(n) + d(n+1) leq 8 ). First, I need to recall what the divisor function ( d(n) ) represents. It counts how many positive integers divide ( n ) without leaving a remainder. For example, ( d(6) = 4 ) because 6 has divisors 1, 2, 3, and 6.Since ( n geq 3 ), I'll start by checking small values of ( n ) to see if they satisfy the condition. Maybe there's a pattern or a specific set of numbers that work.Let me start with ( n = 3 ):- ( n-1 = 2 ), which has divisors 1 and 2, so ( d(2) = 2 ).- ( n = 3 ), which has divisors 1 and 3, so ( d(3) = 2 ).- ( n+1 = 4 ), which has divisors 1, 2, and 4, so ( d(4) = 3 ).Adding these up: ( 2 + 2 + 3 = 7 ), which is less than or equal to 8. So, ( n = 3 ) works.Next, ( n = 4 ):- ( n-1 = 3 ), ( d(3) = 2 ).- ( n = 4 ), ( d(4) = 3 ).- ( n+1 = 5 ), ( d(5) = 2 ).Sum: ( 2 + 3 + 2 = 7 ), which is also ≤8. So, ( n = 4 ) is good.Now, ( n = 5 ):- ( n-1 = 4 ), ( d(4) = 3 ).- ( n = 5 ), ( d(5) = 2 ).- ( n+1 = 6 ), ( d(6) = 4 ).Sum: ( 3 + 2 + 4 = 9 ), which is greater than 8. So, ( n = 5 ) doesn't work.Moving on to ( n = 6 ):- ( n-1 = 5 ), ( d(5) = 2 ).- ( n = 6 ), ( d(6) = 4 ).- ( n+1 = 7 ), ( d(7) = 2 ).Sum: ( 2 + 4 + 2 = 8 ), which is exactly 8. So, ( n = 6 ) is acceptable.Let me check ( n = 7 ):- ( n-1 = 6 ), ( d(6) = 4 ).- ( n = 7 ), ( d(7) = 2 ).- ( n+1 = 8 ), ( d(8) = 4 ).Sum: ( 4 + 2 + 4 = 10 ), which is too big. So, ( n = 7 ) is out.What about ( n = 8 ):- ( n-1 = 7 ), ( d(7) = 2 ).- ( n = 8 ), ( d(8) = 4 ).- ( n+1 = 9 ), ( d(9) = 3 ).Sum: ( 2 + 4 + 3 = 9 ), still too high.Hmm, seems like as ( n ) increases beyond 6, the sum tends to be larger than 8. Maybe I should look for a pattern or a reason why this happens.I notice that for even numbers ( n geq 6 ), ( n ) itself has at least 4 divisors (1, 2, ( n/2 ), and ( n )). Similarly, ( n-1 ) and ( n+1 ) are odd numbers. If ( n-1 ) or ( n+1 ) is a prime, it will have only 2 divisors, but if they are composite, they can have more.Wait, for ( n = 6 ), ( n+1 = 7 ) is prime, so it only adds 2. But for ( n = 8 ), ( n+1 = 9 ) is composite with 3 divisors. So, the sum increases.What about odd ( n )? Let me check ( n = 9 ):- ( n-1 = 8 ), ( d(8) = 4 ).- ( n = 9 ), ( d(9) = 3 ).- ( n+1 = 10 ), ( d(10) = 4 ).Sum: ( 4 + 3 + 4 = 11 ), which is way over.It seems that for ( n geq 7 ), whether odd or even, the sum ( d(n-1) + d(n) + d(n+1) ) tends to be greater than 8. Maybe I can generalize this.If ( n ) is even and greater than or equal to 6, ( d(n) geq 4 ). Then, ( n-1 ) and ( n+1 ) are odd. If either of them is composite, their divisor counts will be at least 3 or more, which would make the total sum exceed 8.For example, take ( n = 10 ):- ( n-1 = 9 ), ( d(9) = 3 ).- ( n = 10 ), ( d(10) = 4 ).- ( n+1 = 11 ), ( d(11) = 2 ).Sum: ( 3 + 4 + 2 = 9 ), which is still over.Wait, ( n = 10 ) gives a sum of 9, which is over. So, even if one of ( n-1 ) or ( n+1 ) is prime, the other being composite adds enough to make the total exceed 8.Similarly, for ( n = 12 ):- ( n-1 = 11 ), ( d(11) = 2 ).- ( n = 12 ), ( d(12) = 6 ).- ( n+1 = 13 ), ( d(13) = 2 ).Sum: ( 2 + 6 + 2 = 10 ), which is way over.So, it seems that once ( n ) is large enough, the divisor counts add up beyond 8. Therefore, the only possible solutions are the smaller numbers we checked earlier: 3, 4, and 6.Let me just verify ( n = 2 ) even though the problem states ( n geq 3 ):- ( n-1 = 1 ), ( d(1) = 1 ).- ( n = 2 ), ( d(2) = 2 ).- ( n+1 = 3 ), ( d(3) = 2 ).Sum: ( 1 + 2 + 2 = 5 ), which is ≤8, but since ( n geq 3 ), we don't include it.Thus, after checking all these cases, I conclude that the natural numbers ( n geq 3 ) satisfying the condition are 3, 4, and 6.**Final Answer**The natural numbers ( n geq 3 ) that satisfy the condition are boxed{3}, boxed{4}, and boxed{6}.**Final Answer**The natural numbers ( n geq 3 ) that satisfy the condition are boxed{3}, boxed{4}, and boxed{6}.

💡Okay, so I have this geometry problem here about triangle PQR. It's a right triangle with the right angle at R. The sides PR and QR are given as 9 and 12 units long, respectively. Points S and T are on sides PQ and QR, and angle RTS is 90 degrees. Also, the length of ST is given as 3 times the square root of 5. I need to find the length of RT.Alright, let me start by visualizing the triangle. Since it's a right triangle at R, PR and QR are the legs, and PQ is the hypotenuse. I remember that in a right triangle, the hypotenuse can be found using the Pythagorean theorem. So, let me calculate the length of PQ first.Using the Pythagorean theorem: PQ squared equals PR squared plus QR squared. Plugging in the numbers, that's 9 squared plus 12 squared. 9 squared is 81, and 12 squared is 144. Adding those together gives 225. Taking the square root of 225, I get 15. So, PQ is 15 units long.Now, points S and T are on PQ and QR, respectively. Angle RTS is 90 degrees, which means triangle RTS is also a right triangle. The length of ST is given as 3√5. I need to find RT.Hmm, maybe I can use similar triangles here. If triangle RTS is similar to triangle PQR, that could help me set up a proportion. Both triangles are right-angled, and if they share another angle, they would be similar by AA similarity.Looking at triangle PQR and triangle RTS, they both have a right angle. If angle at R is common, then they should be similar. Wait, but in triangle RTS, the right angle is at T, not at R. So, actually, the angle at R is not necessarily the same as in triangle PQR. Hmm, maybe I need to think differently.Alternatively, maybe I can assign coordinates to the triangle to make it easier. Let me place point R at the origin (0,0). Then, since PR is 9 and QR is 12, I can place point P at (0,9) and point Q at (12,0). That way, PQ will be the hypotenuse connecting (0,9) to (12,0).Point S is somewhere on PQ, and point T is somewhere on QR. Let me denote the coordinates of S as (x, y) and T as (a, 0), since T is on QR which lies along the x-axis from (0,0) to (12,0).Since S is on PQ, I can parametrize its coordinates. The line PQ goes from (0,9) to (12,0). The parametric equations for PQ can be written as x = 12t and y = 9 - 9t, where t ranges from 0 to 1. So, point S can be represented as (12t, 9 - 9t) for some t between 0 and 1.Point T is on QR, which is the x-axis from (0,0) to (12,0). So, T can be represented as (a, 0) where a is between 0 and 12.Now, angle RTS is 90 degrees, which means that vectors RT and ST are perpendicular. Let me express vectors RT and ST in terms of coordinates.Vector RT goes from R(0,0) to T(a,0), so it's (a, 0). Vector ST goes from S(12t, 9 - 9t) to T(a,0), so it's (a - 12t, -9 + 9t).Since RT and ST are perpendicular, their dot product should be zero. So, the dot product of RT and ST is:(a)(a - 12t) + (0)(-9 + 9t) = a(a - 12t) + 0 = a^2 - 12at = 0.So, a^2 - 12at = 0. Let me factor out a: a(a - 12t) = 0. This gives two possibilities: a = 0 or a = 12t.But a = 0 would mean point T is at the origin, which is point R. However, angle RTS would then be undefined because both R and T would coincide. So, a can't be 0. Therefore, a = 12t.So, point T is at (12t, 0).Now, I can express vector ST as (12t - 12t, 0 - (9 - 9t)) = (0, -9 + 9t). Wait, that can't be right because if a = 12t, then vector ST is (12t - 12t, 0 - (9 - 9t)) = (0, -9 + 9t). But that would mean vector ST is vertical, which would make angle RTS 90 degrees only if RT is horizontal. But RT is along the x-axis, so that makes sense.But wait, the length of ST is given as 3√5. Let me compute the length of ST using the coordinates.Point S is (12t, 9 - 9t) and point T is (12t, 0). So, the distance between S and T is the vertical distance, which is |9 - 9t - 0| = |9 - 9t|. Since t is between 0 and 1, 9 - 9t is positive, so ST = 9 - 9t.But ST is given as 3√5. So, 9 - 9t = 3√5.Let me solve for t:9 - 9t = 3√5Divide both sides by 3:3 - 3t = √5Then, 3t = 3 - √5So, t = (3 - √5)/3 = 1 - (√5)/3.Okay, so t is 1 - √5/3.Now, I need to find RT. RT is the distance from R(0,0) to T(12t, 0), which is just 12t, since it's along the x-axis.So, RT = 12t = 12*(1 - √5/3) = 12 - 4√5.Wait, but that can't be right because 12 - 4√5 is approximately 12 - 8.944 = 3.056, which seems too small given that ST is 3√5 ≈ 6.708.Wait, maybe I made a mistake in calculating the coordinates.Let me double-check. Point S is on PQ, which I parametrized as (12t, 9 - 9t). Point T is on QR, which I set as (a, 0). Then, I found that a = 12t.So, point T is (12t, 0). Then, vector RT is (12t, 0), and vector ST is (12t - 12t, 0 - (9 - 9t)) = (0, -9 + 9t). So, the length of ST is | -9 + 9t | = 9 - 9t, since t < 1.Given that ST = 3√5, so 9 - 9t = 3√5, which simplifies to t = 1 - (√5)/3.Therefore, RT is 12t = 12*(1 - √5/3) = 12 - 4√5.Wait, but 12 - 4√5 is approximately 12 - 8.944 = 3.056, which is less than the length of ST, which is about 6.708. That doesn't make sense because in a right triangle, the legs can't be longer than the hypotenuse. Wait, but in this case, ST is one leg and RT is the other leg. So, actually, it's possible for one leg to be longer than the other.But let me check if my parametrization is correct. Maybe I should have used a different parameter.Alternatively, perhaps I should use similar triangles. Since both triangles PQR and RTS are right-angled, and they share angle at R, they should be similar.Wait, but earlier I thought the right angle in RTS is at T, not at R. So, actually, angle at R is not the right angle in triangle RTS. So, maybe they are not similar by AA similarity.Wait, let me think again. Triangle PQR has right angle at R, and triangle RTS has right angle at T. So, they don't share the same right angle. Therefore, they might not be similar.Hmm, so maybe my initial approach with coordinates is better.Wait, but according to my calculation, RT is 12 - 4√5, which is approximately 3.056. But let's compute ST: 3√5 is approximately 6.708. So, in triangle RTS, the legs are RT ≈ 3.056 and ST ≈ 6.708, so the hypotenuse RT would be sqrt(3.056² + 6.708²) ≈ sqrt(9.34 + 45) ≈ sqrt(54.34) ≈ 7.37, which is less than PQ, which is 15. That seems plausible.But let me check if my calculation of RT is correct. RT is 12t, and t = 1 - √5/3. So, 12*(1 - √5/3) = 12 - 4√5. That seems correct.Wait, but 12 - 4√5 is approximately 12 - 8.944 = 3.056, which is positive, so that's fine.Alternatively, maybe I can use the Pythagorean theorem in triangle RTS. Since it's a right triangle, RT² + ST² = RS².Wait, no, because the right angle is at T, so RT and ST are the legs, and RS is the hypotenuse. So, RT² + ST² = RS².But I don't know RS yet. Alternatively, maybe I can express RS in terms of t.Point S is (12t, 9 - 9t), so RS is the distance from R(0,0) to S(12t, 9 - 9t). So, RS² = (12t)² + (9 - 9t)² = 144t² + 81 - 162t + 81t² = 225t² - 162t + 81.But from triangle RTS, RT² + ST² = RS². RT is 12t, ST is 3√5, so:(12t)² + (3√5)² = RS²144t² + 45 = 225t² - 162t + 81Let me bring all terms to one side:225t² - 162t + 81 - 144t² - 45 = 0Simplify:(225t² - 144t²) + (-162t) + (81 - 45) = 081t² - 162t + 36 = 0Divide all terms by 9:9t² - 18t + 4 = 0Now, solve for t using quadratic formula:t = [18 ± sqrt(324 - 144)] / 18sqrt(324 - 144) = sqrt(180) = 6√5So, t = [18 ± 6√5]/18 = [3 ± √5]/3So, t = 1 + (√5)/3 or t = 1 - (√5)/3But t must be between 0 and 1 because it's a parameter along PQ. Let's compute both:1 + (√5)/3 ≈ 1 + 2.236/3 ≈ 1 + 0.745 ≈ 1.745, which is greater than 1, so invalid.1 - (√5)/3 ≈ 1 - 0.745 ≈ 0.255, which is between 0 and 1, so valid.So, t = 1 - √5/3, which matches my earlier result.Therefore, RT = 12t = 12*(1 - √5/3) = 12 - 4√5.Wait, but earlier I thought this was approximately 3.056, but let me compute 12 - 4√5:√5 ≈ 2.236, so 4√5 ≈ 8.94412 - 8.944 ≈ 3.056So, RT ≈ 3.056 units.But the problem states that ST = 3√5 ≈ 6.708 units, which is longer than RT. That's fine because in a right triangle, one leg can be longer than the other.But let me check if this makes sense in the context of the triangle. Since S is closer to R than to Q, because t is about 0.255, so S is closer to P. Therefore, T is at 12t ≈ 3.056 on QR, which is closer to R. So, ST is the vertical distance from S to T, which is 9 - 9t ≈ 9 - 2.3 ≈ 6.7, which matches 3√5.So, everything seems consistent.Therefore, the length of RT is 12 - 4√5.Wait, but let me rationalize or simplify this expression. 12 - 4√5 can be factored as 4*(3 - √5). But I don't think that's necessary unless the problem expects a specific form.Alternatively, maybe I made a mistake in assuming the coordinates. Let me try another approach using similar triangles.Since triangle RTS is right-angled at T, and triangle PQR is right-angled at R, maybe they are similar in some way.Wait, but the right angles are at different vertices, so they might not be similar. Alternatively, maybe triangle RTS is similar to another triangle in the figure.Alternatively, perhaps I can use the concept of projections or similar triangles along the sides.Wait, another idea: Since S is on PQ and T is on QR, and angle RTS is 90 degrees, maybe we can use the property that in such cases, the triangles formed are similar.Wait, let me consider the coordinates again. I think my earlier approach was correct, and the answer is RT = 12 - 4√5.But let me compute 12 - 4√5 numerically to see if it makes sense. 4√5 ≈ 8.944, so 12 - 8.944 ≈ 3.056. As I calculated before, ST is 3√5 ≈ 6.708, so in triangle RTS, the legs are approximately 3.056 and 6.708, and the hypotenuse RS would be sqrt(3.056² + 6.708²) ≈ sqrt(9.34 + 45) ≈ sqrt(54.34) ≈ 7.37, which is less than PQ = 15, so that seems plausible.Alternatively, maybe I can express RT in terms of the sides of the main triangle. Since RT is along QR, which is 12 units, and T is somewhere between R and Q, RT must be less than 12. My answer of approximately 3.056 is less than 12, so that's fine.Wait, but let me check if there's another way to approach this problem without coordinates. Maybe using similar triangles or proportions.Since angle RTS is 90 degrees, and S is on PQ, T is on QR, perhaps we can set up proportions based on the areas or something else.Alternatively, maybe using the concept of similar triangles where the ratio of sides is preserved.Wait, another idea: Let me denote RT as x. Then, since T is on QR, which is 12 units, QT would be 12 - x.Now, since S is on PQ, let me denote PS as y and SQ as 15 - y, since PQ is 15 units.Now, in triangle RTS, which is right-angled at T, we have RT = x, ST = 3√5, and RS as the hypotenuse.So, by Pythagoras, RS² = RT² + ST² = x² + (3√5)² = x² + 45.But RS is also the distance from R to S, which is a point on PQ. So, RS can be expressed in terms of y.Wait, but RS is the length from R to S, which is a point on PQ. Since PQ is the hypotenuse of triangle PQR, which is 15 units, and S divides PQ into segments PS = y and SQ = 15 - y.But RS is not along PQ; it's a straight line from R to S. So, RS can be found using coordinates or vectors, but maybe I can express it in terms of y.Alternatively, maybe I can use the concept that in triangle PQR, the length of RS can be found using the formula for the length of a median or something similar, but I'm not sure.Wait, perhaps using coordinate geometry again. Let me assign coordinates as before: R(0,0), P(0,9), Q(12,0). Then, point S is on PQ, which can be parametrized as (12t, 9 - 9t), as before. Then, RS is the distance from R(0,0) to S(12t, 9 - 9t), which is sqrt((12t)^2 + (9 - 9t)^2) = sqrt(144t² + 81 - 162t + 81t²) = sqrt(225t² - 162t + 81).From triangle RTS, RS² = RT² + ST² = (12t)^2 + (3√5)^2 = 144t² + 45.So, setting them equal:225t² - 162t + 81 = 144t² + 45Simplify:225t² - 162t + 81 - 144t² - 45 = 081t² - 162t + 36 = 0Divide by 9:9t² - 18t + 4 = 0Which is the same quadratic equation as before. Solving this gives t = [18 ± sqrt(324 - 144)] / 18 = [18 ± sqrt(180)] / 18 = [18 ± 6√5]/18 = [3 ± √5]/3.So, t = 1 - √5/3, as before.Therefore, RT = 12t = 12*(1 - √5/3) = 12 - 4√5.So, after verifying through two different approaches, I'm confident that RT is 12 - 4√5 units long.But wait, let me check if 12 - 4√5 is positive. Since √5 ≈ 2.236, 4√5 ≈ 8.944, so 12 - 8.944 ≈ 3.056, which is positive, so that's fine.Alternatively, maybe I can rationalize or present it differently, but I think 12 - 4√5 is the simplest form.Wait, another thought: Maybe I can express RT in terms of the sides of the main triangle. Since RT is along QR, which is 12 units, and T divides QR into RT and TQ, which are x and 12 - x respectively.But I don't see an immediate way to relate x to the given ST length without going through the coordinate method or quadratic equation.So, I think my answer is correct: RT = 12 - 4√5.But let me compute 12 - 4√5 numerically to see if it makes sense in the context of the problem. 4√5 ≈ 8.944, so 12 - 8.944 ≈ 3.056. So, RT ≈ 3.056 units, which seems reasonable given that ST is 3√5 ≈ 6.708 units.Therefore, I think I've arrived at the correct answer.

💡Okay, so I have this problem with five statements, and I need to figure out which ones are correct. Let me go through each one step by step.**Statement 1:** The negation of the proposition "∀x ∈ ℝ, 2x ≤ 3" is "∃x ∈ ℝ such that 2^x > 3".Hmm, okay. So the original statement is saying that for all real numbers x, 2x is less than or equal to 3. The negation of a universal statement is an existential statement. So the negation should be that there exists some real number x where 2x is greater than 3. Wait, but the statement says 2^x > 3 instead of 2x > 3. That seems different. Is that a typo or intentional?Let me think. If the original statement was "∀x ∈ ℝ, 2x ≤ 3", then the negation should be "∃x ∈ ℝ, 2x > 3". But here, it's written as 2^x > 3. That changes the meaning because 2^x is an exponential function, not a linear function. So, is this a mistake? Or is there a reason for the exponent?Well, if it's 2x, that's linear, but 2^x is exponential. The negation should maintain the same structure, right? So if the original was about 2x, the negation should also be about 2x, not 2^x. So I think this might be incorrect. Maybe Statement 1 is false.But wait, maybe I'm overcomplicating it. Let me check the logic. The negation of "for all x, P(x)" is "there exists x such that not P(x)". So if P(x) is "2x ≤ 3", then not P(x) is "2x > 3". So the negation should be "∃x ∈ ℝ, 2x > 3". But the statement says "2^x > 3". So unless there's a typo, Statement 1 is incorrect because it changes the expression from linear to exponential.**Statement 2:** The function f(x) = (m² - m - 1)x^m is a power function, and it is increasing on x ∈ (0, +∞), then m = 2.Alright, so f(x) is a power function, which generally is of the form f(x) = kx^n. For it to be a power function, the coefficient (m² - m - 1) should be a constant, which it is. Now, it's increasing on (0, +∞). For a power function to be increasing, its derivative should be positive in that interval.Let's compute the derivative. f'(x) = (m² - m - 1) * m * x^(m - 1). For f'(x) to be positive on (0, +∞), we need two things: the coefficient (m² - m - 1) * m should be positive, and x^(m - 1) should be positive. Since x is positive, x^(m - 1) is positive as long as m - 1 is defined, which it is for real exponents. So the sign depends on (m² - m - 1) * m.We need (m² - m - 1) * m > 0. Let's solve for m.First, factor m² - m - 1. The roots are [1 ± sqrt(1 + 4)] / 2 = [1 ± sqrt(5)] / 2. So m² - m - 1 = (m - (1 + sqrt(5))/2)(m - (1 - sqrt(5))/2).So the critical points are m = (1 + sqrt(5))/2 ≈ 1.618 and m = (1 - sqrt(5))/2 ≈ -0.618.Now, let's analyze the sign of (m² - m - 1) * m.We can make a sign chart:- For m < -0.618: Let's pick m = -1. Then (m² - m - 1) = 1 + 1 - 1 = 1, which is positive. m is negative. So positive * negative = negative.- For -0.618 < m < 0: Let's pick m = -0.5. (m² - m - 1) = 0.25 + 0.5 - 1 = -0.25, negative. m is negative. Negative * negative = positive.- For 0 < m < 1.618: Let's pick m = 1. (m² - m - 1) = 1 - 1 -1 = -1, negative. m is positive. Negative * positive = negative.- For m > 1.618: Let's pick m = 2. (m² - m - 1) = 4 - 2 -1 = 1, positive. m is positive. Positive * positive = positive.So the product is positive in intervals (-0.618, 0) and (1.618, +∞). But since m is an exponent in a power function, it can be any real number, but in the context of being increasing on (0, +∞), we probably consider positive m because negative exponents would make the function decreasing.Wait, for m positive, the function f(x) = kx^m is increasing if k > 0 and m > 0. But in our case, k = (m² - m - 1). So we need k > 0 and m > 0.From the sign chart, k = (m² - m - 1) is positive when m < (1 - sqrt(5))/2 ≈ -0.618 or m > (1 + sqrt(5))/2 ≈ 1.618.But since m must be positive, we consider m > 1.618.But the statement says m = 2. Let's check m = 2.k = (4 - 2 -1) = 1, which is positive. So f(x) = x², which is indeed increasing on (0, +∞). So m = 2 is a solution.But are there other solutions? For example, m = 3. Then k = 9 - 3 -1 = 5, positive. So f(x) = 5x³, which is increasing. So m can be any number greater than approximately 1.618, not just 2. So the statement says "then m = 2", which is not necessarily the only solution. So Statement 2 is false.Wait, but maybe the problem implies that m is an integer? The problem doesn't specify, so I think m can be any real number. So since m can be greater than 1.618, not just 2, Statement 2 is incorrect.**Statement 3:** The proposition "If the function f(x) has an extremum at x = x₀, then f'(x₀) = 0" has a true contrapositive.Okay, the contrapositive of "If P, then Q" is "If not Q, then not P". So the contrapositive here would be "If f'(x₀) ≠ 0, then the function f(x) does not have an extremum at x = x₀".Is this true? Well, in calculus, we know that if a function has an extremum at a point, then the derivative at that point is zero (assuming the function is differentiable there). So the contrapositive would be: if the derivative is not zero, then there is no extremum. That seems true because if the derivative isn't zero, the function isn't stationary there, so it can't have a maximum or minimum.Wait, but isn't there a case where the derivative doesn't exist? For example, |x| at x=0 has an extremum but the derivative doesn't exist. But the contrapositive is about f'(x₀) ≠ 0, not about the derivative not existing. So if the derivative exists and is not zero, then there's no extremum. That's correct.So Statement 3 is true.**Statement 4:** The function y = tan(2x + π/6) is increasing in the interval (-π/3, π/12).Alright, tan function has a period of π, and it's increasing on each interval between its vertical asymptotes. The standard tan(x) is increasing on (-π/2, π/2), but here we have tan(2x + π/6). Let's find its period and where it's increasing.The period of tan(Bx + C) is π / |B|. Here, B = 2, so the period is π / 2.To find where it's increasing, we need to look at the intervals between its vertical asymptotes. The function tan(θ) has vertical asymptotes at θ = π/2 + kπ, where k is integer.So for tan(2x + π/6), set 2x + π/6 = π/2 + kπ.Solving for x: 2x = π/2 - π/6 + kπ = (3π/6 - π/6) + kπ = (2π/6) + kπ = π/3 + kπ.So x = π/6 + kπ/2.So the vertical asymptotes are at x = π/6 + kπ/2.Therefore, the function is increasing on intervals between these asymptotes. Let's find the asymptotes around the given interval (-π/3, π/12).Let's compute for k = 0: x = π/6 ≈ 0.523.For k = -1: x = π/6 - π/2 = -π/3 ≈ -1.047.So the asymptotes near our interval are at x = -π/3 and x = π/6.Therefore, the function tan(2x + π/6) is increasing on (-π/3, π/6).The given interval is (-π/3, π/12). Since π/12 ≈ 0.2618, which is less than π/6 ≈ 0.523, the interval (-π/3, π/12) is entirely within (-π/3, π/6). Therefore, the function is increasing on (-π/3, π/12).So Statement 4 is true.**Statement 5:** "log₂(x) > log₃(x)" is a necessary and sufficient condition for "2^x > 3^x" to hold.Hmm, let's analyze both inequalities.First, "2^x > 3^x". Let's solve this inequality.Divide both sides by 3^x: (2/3)^x > 1.Since 2/3 < 1, (2/3)^x is decreasing. So (2/3)^x > 1 when x < 0.So "2^x > 3^x" holds when x < 0.Now, "log₂(x) > log₃(x)". Let's analyze this.First, the domain of log₂(x) and log₃(x) is x > 0.So we're comparing log₂(x) and log₃(x) for x > 0.Let me rewrite both in terms of natural logs:log₂(x) = ln(x)/ln(2)log₃(x) = ln(x)/ln(3)So log₂(x) > log₃(x) implies ln(x)/ln(2) > ln(x)/ln(3).Assuming ln(x) ≠ 0, which is when x ≠ 1.Case 1: ln(x) > 0, which is x > 1.Then, since ln(2) < ln(3), 1/ln(2) > 1/ln(3). So ln(x)/ln(2) > ln(x)/ln(3) is true because 1/ln(2) > 1/ln(3). So for x > 1, log₂(x) > log₃(x).Case 2: ln(x) < 0, which is x < 1.Then, ln(x) is negative. So ln(x)/ln(2) > ln(x)/ln(3) would be equivalent to multiplying both sides by ln(2)ln(3), which are positive, so inequality remains the same.But since ln(x) is negative, dividing by ln(2) and ln(3) (both positive) reverses the inequality.Wait, let's think differently.log₂(x) > log₃(x)Multiply both sides by ln(2)ln(3):ln(x) * ln(3) > ln(x) * ln(2)Which simplifies to ln(x)(ln(3) - ln(2)) > 0.Since ln(3) - ln(2) > 0, this reduces to ln(x) > 0, which is x > 1.So "log₂(x) > log₃(x)" holds when x > 1.But "2^x > 3^x" holds when x < 0.So the two conditions are not related in terms of necessary and sufficient. One holds for x < 0, the other for x > 1. They don't overlap.Therefore, "log₂(x) > log₃(x)" is neither necessary nor sufficient for "2^x > 3^x". So Statement 5 is false.**Summary:**1. False (negation changes 2x to 2^x)2. False (m can be any number > 1.618, not just 2)3. True (contrapositive is true)4. True (function is increasing in the given interval)5. False (conditions don't overlap)So the correct statements are 3 and 4.Wait, but earlier I thought Statement 1 was false because of the 2^x vs 2x. Let me double-check that.Original statement: "∀x ∈ ℝ, 2x ≤ 3"Negation should be "∃x ∈ ℝ, 2x > 3"But the given negation is "∃x ∈ ℝ, 2^x > 3"So unless there's a typo, it's incorrect. So Statement 1 is false.So the correct statements are 3 and 4.But wait, in my initial analysis, I thought Statement 2 was false because m can be any number greater than approximately 1.618, not just 2. But maybe the problem assumes m is an integer? Let me check.The function is f(x) = (m² - m - 1)x^m. If m is an integer, then m² - m - 1 must be 1 for it to be a power function? Wait, no, a power function is just any function of the form kx^n, where k is a constant. So (m² - m - 1) can be any constant, not necessarily 1. So the function is a power function regardless of m.But the problem says "the function f(x) = (m² - m - 1)x^m is a power function, and it is increasing on x ∈ (0, +∞), then m = 2".So it's given that it's a power function, which it is for any m. Then, it's increasing on (0, +∞). So we need to find m such that f(x) is increasing there.As I analyzed earlier, f'(x) = (m² - m - 1)*m*x^(m-1). For f'(x) > 0 on (0, +∞), we need (m² - m - 1)*m > 0.From the sign chart, this is positive when m ∈ (-0.618, 0) or m > 1.618.But since m is an exponent, and for the function to be increasing, we probably consider m > 0. So m > 1.618.But the statement says m = 2. Is m = 2 the only solution? No, because m could be 3, 4, etc. So unless m is restricted to integers, m = 2 is just one solution. So the statement is not necessarily true because m could be other values greater than 1.618.Therefore, Statement 2 is false.So the correct statements are 3 and 4.Wait, but in my initial analysis, I thought Statement 3 was true, but let me double-check.Statement 3: "If the function f(x) has an extremum at x = x₀, then f'(x₀) = 0" has a true contrapositive.The contrapositive is "If f'(x₀) ≠ 0, then f(x) does not have an extremum at x = x₀". This is true because if the derivative isn't zero, the function isn't stationary there, so it can't have a local maximum or minimum. So yes, the contrapositive is true.Therefore, Statement 3 is true.So overall, Statements 3 and 4 are correct.But wait, in the initial analysis, I thought Statement 1 was false, Statement 2 false, 3 true, 4 true, 5 false. So correct statements are 3 and 4.But the user initially thought the answer was 1,2,4, but after analysis, it's 3,4.Wait, maybe I made a mistake in Statement 1.Let me re-examine Statement 1.Original statement: "∀x ∈ ℝ, 2x ≤ 3"Negation: "∃x ∈ ℝ, 2^x > 3"Is this correct?Wait, the original statement is about 2x, which is linear. The negation changes it to 2^x, which is exponential. So unless it's a typo, the negation is incorrect. So Statement 1 is false.Therefore, the correct statements are 3 and 4.But in the initial analysis, I thought Statement 2 was false because m can be any number greater than 1.618, not just 2. So unless m is restricted, Statement 2 is false.So the correct statements are 3 and 4.Wait, but in the initial problem, the user wrote:"The correct statement numbers are ______."So I need to list the correct ones. Based on my analysis, 3 and 4 are correct.But let me check Statement 2 again.Statement 2: The function f(x) = (m² - m - 1)x^m is a power function, and it is increasing on x ∈ (0, +∞), then m = 2.So f(x) is a power function, which it is for any m. It is increasing on (0, +∞). So we need to find m such that f'(x) > 0 on (0, +∞).f'(x) = (m² - m - 1)*m*x^(m-1)We need (m² - m - 1)*m > 0.From earlier analysis, m > (1 + sqrt(5))/2 ≈ 1.618.So m can be 2, 3, etc. So m = 2 is a solution, but not the only one. Therefore, the statement says "then m = 2", which is not necessarily true because m could be other values greater than 1.618. So Statement 2 is false.Therefore, the correct statements are 3 and 4.But wait, in the initial analysis, I thought Statement 1 was false, Statement 2 false, 3 true, 4 true, 5 false. So correct statements are 3 and 4.But the user initially thought the answer was 1,2,4, but after analysis, it's 3,4.Wait, but in the initial problem, the user wrote:"The correct statement numbers are ______."So I need to list the correct ones. Based on my analysis, 3 and 4 are correct.But let me check Statement 5 again.Statement 5: "log₂(x) > log₃(x)" is a necessary and sufficient condition for "2^x > 3^x" to hold.We saw that "2^x > 3^x" holds when x < 0, while "log₂(x) > log₃(x)" holds when x > 1. So they don't overlap. Therefore, neither is necessary nor sufficient for the other. So Statement 5 is false.So final answer: Statements 3 and 4 are correct.

💡Okay, so I have this problem about triangle ABC with vertices A(-2,4), B(4,-6), and C(5,1). There are two parts to the problem. Let me try to tackle them one by one.Starting with part (1): Find the equation of the median on side AB.Hmm, okay. A median of a triangle is a line segment joining a vertex to the midpoint of the opposite side. So, in this case, the median on side AB would be the line segment from point C to the midpoint of AB.First, I need to find the midpoint of side AB. The coordinates of A are (-2,4) and B are (4,-6). To find the midpoint, I can use the midpoint formula, which is the average of the x-coordinates and the average of the y-coordinates.So, the midpoint M of AB would be:M_x = (-2 + 4)/2 = 2/2 = 1M_y = (4 + (-6))/2 = (-2)/2 = -1So, the midpoint M is (1, -1).Now, the median is the line from point C(5,1) to this midpoint M(1,-1). To find the equation of this line, I need to find the slope first.The slope formula is (y2 - y1)/(x2 - x1). Let's take point C as (x1, y1) = (5,1) and midpoint M as (x2, y2) = (1,-1).So, slope k = (-1 - 1)/(1 - 5) = (-2)/(-4) = 1/2Okay, so the slope is 1/2. Now, using the point-slope form of the equation of a line, which is y - y1 = k(x - x1). Let's use point C(5,1):y - 1 = (1/2)(x - 5)Let me simplify this:y - 1 = (1/2)x - 5/2Multiply both sides by 2 to eliminate the fraction:2(y - 1) = x - 52y - 2 = x - 5Bring all terms to one side:x - 2y - 3 = 0So, the equation of the median on side AB is x - 2y - 3 = 0.Wait, let me double-check my calculations. The midpoint seems correct: (1, -1). The slope calculation: (-1 -1)/(1 -5) = (-2)/(-4) = 1/2. That seems right. Then using point C(5,1):y - 1 = (1/2)(x - 5)Yes, that's correct. Then multiplying by 2:2y - 2 = x - 5Then rearranged to x - 2y - 3 = 0. Yeah, that looks correct.Okay, moving on to part (2): Find the equation of the line passing through point A with intercepts on the x-axis and y-axis being equal.Hmm, so the line passes through A(-2,4) and has equal x-intercept and y-intercept. Let's denote the x-intercept as 'a' and the y-intercept as 'a' as well.The general equation of a line with x-intercept 'a' and y-intercept 'a' is x/a + y/a = 1, which simplifies to x + y = a.Alternatively, it can be written as y = -x + a.Since the line passes through point A(-2,4), we can substitute x = -2 and y = 4 into the equation to find 'a'.So, substituting:4 = -(-2) + a4 = 2 + aTherefore, a = 4 - 2 = 2So, the equation of the line is x + y = 2, or in standard form, x + y - 2 = 0.Wait, let me think again. Is there another possibility? Because sometimes, lines can pass through the origin as well, but in that case, the intercepts would both be zero, which is a special case. But the problem says intercepts on the x-axis and y-axis being equal, so they can be any equal value, not necessarily non-zero.But in this case, when we found a = 2, the intercepts are both 2. So, the line is x + y = 2.Alternatively, if we consider the line passing through the origin, it would have intercepts at (0,0), but that's a different case. Since the problem doesn't specify whether the intercepts are non-zero, but in general, when they say intercepts, they usually mean non-zero unless specified otherwise.But let me check if the line passes through the origin as well. If the line passes through the origin, then its equation would be y = mx. Since it passes through A(-2,4), we can find the slope m.So, substituting x = -2 and y = 4:4 = m*(-2)So, m = 4/(-2) = -2Therefore, the equation would be y = -2x, or 2x + y = 0.But wait, does this line have equal intercepts? Let's see. The x-intercept is when y=0: 2x + 0 = 0 => x=0. Similarly, the y-intercept is when x=0: 0 + y = 0 => y=0. So, both intercepts are zero, which are equal.But in this case, the intercepts are both zero, which is a special case. However, the problem says "intercepts on the x-axis and y-axis being equal." So, technically, both zero is equal, but usually, when problems mention intercepts, they mean non-zero intercepts unless specified otherwise.Therefore, the line x + y = 2 is the more appropriate answer because it has non-zero equal intercepts at (2,0) and (0,2).Wait, but let me verify if the line y = -2x passes through A(-2,4):Substituting x = -2, y = -2*(-2) = 4. Yes, it does pass through A. So, both lines pass through A, but one has intercepts at (2,0) and (0,2), and the other has intercepts at (0,0). So, depending on the interpretation, both could be correct.But since the problem says "intercepts on the x-axis and y-axis being equal," and doesn't specify non-zero, both could be considered. However, usually, when intercepts are mentioned, they refer to non-zero intercepts. So, I think the answer is x + y = 2.But just to be thorough, let me check both possibilities.First, the line with intercepts at (2,0) and (0,2):Equation: x + y = 2Second, the line passing through the origin:Equation: y = -2xBoth pass through A(-2,4). So, both are valid. But the problem says "intercepts on the x-axis and y-axis being equal." If we consider the intercepts being equal in value, then both are equal to 2 or both are equal to 0. So, both are correct.But in the context of the problem, since it's a triangle with vertices not at the origin, I think the intended answer is the line with intercepts at (2,0) and (0,2), which is x + y = 2.Alternatively, if the problem expects a single answer, it's safer to provide both possibilities, but I think x + y = 2 is the primary answer.Wait, let me think again. The problem says "intercepts on the x-axis and y-axis being equal." So, if the intercepts are equal, they can be any equal value, positive or negative, but in this case, since the line passes through A(-2,4), which is in the second quadrant, the intercepts would have to be positive if they are non-zero.Wait, actually, if the line passes through (-2,4), which is in the second quadrant, and has equal intercepts, the intercepts could be positive or negative. Let me check.If the intercepts are both positive, the line would intersect the positive x-axis and positive y-axis. If the intercepts are both negative, the line would intersect the negative x-axis and negative y-axis.But in our case, when we found a = 2, the intercepts are (2,0) and (0,2), which are positive. If we consider a negative intercept, say a = -2, then the equation would be x + y = -2, but does this line pass through A(-2,4)?Let's check:x + y = -2Substituting x = -2, y = 4:-2 + 4 = 2 ≠ -2So, no, it doesn't pass through A. Therefore, the only possible non-zero equal intercepts are positive, which is a = 2.Therefore, the equation is x + y = 2.Alternatively, if we consider the line passing through the origin, which has intercepts at (0,0), which are equal, but as I thought earlier, that might be a special case.But since the problem doesn't specify non-zero intercepts, both could be considered, but in the context of the problem, I think the intended answer is x + y = 2.So, to summarize:Part (1): The median on side AB has equation x - 2y - 3 = 0.Part (2): The line passing through A with equal x and y intercepts has equation x + y - 2 = 0.I think that's it. Let me just double-check my calculations to make sure I didn't make any mistakes.For part (1):Midpoint of AB: (-2 + 4)/2 = 1, (4 + (-6))/2 = -1. Correct.Slope from C(5,1) to M(1,-1): (-1 -1)/(1 -5) = (-2)/(-4) = 1/2. Correct.Equation using point C: y -1 = (1/2)(x -5). Multiply by 2: 2y -2 = x -5. Rearranged: x -2y -3 = 0. Correct.For part (2):Line with equal intercepts: x/a + y/a = 1 => x + y = a.Passing through A(-2,4): -2 + 4 = a => a = 2.Equation: x + y = 2 => x + y -2 = 0. Correct.Yes, everything checks out.