answer：To determine the possible lengths of the third stick ( m ) that can form a triangle with the other two sticks, we use the Triangle Inequality theorem. The sum of the lengths of any two sides must be greater than the length of the third side. Therefore, we have:1. ( 7 + 12 > m ) which simplifies to ( m < 19 )2. ( 7 + m > 12 ) which rearranges to ( m > 5 )3. ( 12 + m > 7 ) which rearranges to ( m > -5 ) (this condition is already satisfied by ( m > 5 ))Thus, ( m ) must be an integer such that ( 6 leq m leq 18 ). The integers in this range are 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18.We calculate the sum of these integers:[ 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 = 153 ]Therefore, the sum of all possible whole number values of ( m ) is (boxed{153}).

answer：Okay, so I need to find the greatest common factor (GCF) of three numbers: 24, 72, and 60. Hmm, I remember that the GCF is the largest number that divides all the given numbers without leaving a remainder. But I'm a bit rusty on the exact method to find it for three numbers. Let me think.First, I think the best way to approach this is by using prime factorization. That means breaking down each number into its prime factors. Okay, let's start with 24. I know that 24 can be divided by 2, so 24 divided by 2 is 12. Then 12 divided by 2 is 6, and 6 divided by 2 is 3. So, 24 breaks down into 2 × 2 × 2 × 3, which is 2³ × 3¹.Next, let's do the same for 72. Dividing 72 by 2 gives me 36, then 36 divided by 2 is 18, and 18 divided by 2 is 9. Now, 9 isn't divisible by 2, so I'll switch to the next prime number, which is 3. Dividing 9 by 3 gives me 3, and then dividing by 3 again gives me 1. So, 72 breaks down into 2³ × 3².Finally, let's factorize 60. Dividing 60 by 2 gives me 30, then 30 divided by 2 is 15. 15 isn't divisible by 2, so I'll use 3 next. Dividing 15 by 3 gives me 5, and then dividing by 5 gives me 1. So, 60 breaks down into 2² × 3¹ × 5¹.Now that I have the prime factors for all three numbers:- 24 = 2³ × 3¹- 72 = 2³ × 3²- 60 = 2² × 3¹ × 5¹To find the GCF, I need to identify the common prime factors with the lowest exponents. So, looking at the prime number 2:- The exponents are 3, 3, and 2. The lowest exponent is 2.For the prime number 3:- The exponents are 1, 2, and 1. The lowest exponent is 1.The prime number 5 only appears in 60, so it's not common to all three numbers, so we can ignore it.Now, multiplying these together: 2² × 3¹ = 4 × 3 = 12.Wait, let me double-check to make sure I didn't make a mistake. If I divide each original number by 12:- 24 ÷ 12 = 2 (no remainder)- 72 ÷ 12 = 6 (no remainder)- 60 ÷ 12 = 5 (no remainder)Yes, that works. So, 12 is indeed the greatest common factor of 24, 72, and 60.

answer：Okay, so I have this problem here: If ( f(x) = sqrt{x - 5} ), what is the smallest real number ( x ) in the domain of ( f(f(x)) )? Hmm, let me try to figure this out step by step.First, I know that the function ( f(x) = sqrt{x - 5} ) is defined only when the expression inside the square root is non-negative. That means ( x - 5 geq 0 ), so ( x geq 5 ). So, the domain of ( f(x) ) is all real numbers ( x ) such that ( x ) is greater than or equal to 5.Now, the problem is asking about the domain of ( f(f(x)) ). That means I need to find all real numbers ( x ) for which ( f(f(x)) ) is defined. Let me write that out: ( f(f(x)) = f(sqrt{x - 5}) ). So, ( f(f(x)) ) is essentially applying the function ( f ) twice. First, I apply ( f ) to ( x ), which gives me ( sqrt{x - 5} ), and then I apply ( f ) again to that result. Let me break it down. The first application, ( f(x) = sqrt{x - 5} ), requires that ( x geq 5 ) as I already determined. Now, for the second application, ( f(sqrt{x - 5}) ), I need to make sure that the input to the second ( f ) is also within its domain. So, the input to the second ( f ) is ( sqrt{x - 5} ). For ( f ) to be defined here, ( sqrt{x - 5} ) must be greater than or equal to 5. That gives me the inequality:[ sqrt{x - 5} geq 5 ]To solve this inequality, I can square both sides to eliminate the square root. Squaring both sides gives:[ (sqrt{x - 5})^2 geq 5^2 ][ x - 5 geq 25 ]Now, solving for ( x ), I add 5 to both sides:[ x geq 25 + 5 ][ x geq 30 ]So, ( x ) must be greater than or equal to 30 for ( f(f(x)) ) to be defined. Therefore, the smallest real number ( x ) in the domain of ( f(f(x)) ) is 30.Wait, let me double-check that. If ( x = 30 ), then ( f(30) = sqrt{30 - 5} = sqrt{25} = 5 ). Then, ( f(f(30)) = f(5) = sqrt{5 - 5} = sqrt{0} = 0 ). So, ( f(f(30)) ) is defined and equals 0. That seems correct.What if ( x ) is less than 30, say 29? Then, ( f(29) = sqrt{29 - 5} = sqrt{24} ), which is approximately 4.899. Then, ( f(f(29)) = f(4.899) = sqrt{4.899 - 5} ). Wait, that would be the square root of a negative number, which is not a real number. So, ( f(f(29)) ) is undefined. That confirms that ( x ) must be at least 30.Therefore, my conclusion is that the smallest real number ( x ) in the domain of ( f(f(x)) ) is 30.

answer：Okay, so I have this function ( f(x) = sqrt{x - 1} + sqrt{x + 24 - 10sqrt{x - 1}} ), and I need to find the largest interval where it's real and constant. Hmm, let's break this down step by step.First, I know that for square roots to be real numbers, the expressions inside them must be non-negative. So, let's look at each square root separately.Starting with ( sqrt{x - 1} ), the expression inside is ( x - 1 ). For this to be non-negative, ( x - 1 geq 0 ), which means ( x geq 1 ). So, the domain of this part is ( [1, infty) ).Next, the second square root is ( sqrt{x + 24 - 10sqrt{x - 1}} ). The expression inside here is ( x + 24 - 10sqrt{x - 1} ). For this to be non-negative, we need:( x + 24 - 10sqrt{x - 1} geq 0 )Hmm, this looks a bit more complicated. Maybe I can simplify this expression. Let me set ( y = sqrt{x - 1} ). Then, ( y^2 = x - 1 ), so ( x = y^2 + 1 ). Let's substitute this back into the second square root:( x + 24 - 10sqrt{x - 1} = (y^2 + 1) + 24 - 10y = y^2 + 25 - 10y )So, the second square root becomes ( sqrt{y^2 + 25 - 10y} ). Let me see if I can simplify this further. Maybe the expression inside is a perfect square?Looking at ( y^2 + 25 - 10y ), I notice that it resembles the expansion of ( (y - 5)^2 ), which is ( y^2 - 10y + 25 ). Hey, that's exactly the same as ( y^2 + 25 - 10y )! So, ( y^2 + 25 - 10y = (y - 5)^2 ).Therefore, the second square root simplifies to ( sqrt{(y - 5)^2} ), which is the absolute value of ( y - 5 ), so ( |y - 5| ).Now, let's rewrite the original function ( f(x) ) in terms of ( y ):( f(x) = y + |y - 5| )Okay, so ( f(x) ) is now expressed as ( y + |y - 5| ). Let's analyze this expression. The absolute value function ( |y - 5| ) can be split into two cases:1. When ( y geq 5 ), ( |y - 5| = y - 5 ). So, ( f(x) = y + (y - 5) = 2y - 5 ).2. When ( y < 5 ), ( |y - 5| = 5 - y ). So, ( f(x) = y + (5 - y) = 5 ).Interesting! So, depending on the value of ( y ), ( f(x) ) can either be a linear function ( 2y - 5 ) or a constant function ( 5 ).But we want ( f(x) ) to be constant. So, when is ( f(x) ) constant? From the above, it's constant when ( y < 5 ), which means ( sqrt{x - 1} < 5 ). Let's solve for ( x ):( sqrt{x - 1} < 5 )Squaring both sides (since both sides are non-negative):( x - 1 < 25 )So,( x < 26 )But we also have the domain restriction from the first square root, which is ( x geq 1 ). Therefore, combining these, the interval where ( f(x) ) is constant is ( [1, 26) ).Wait, but the question asks for the largest interval where ( f(x) ) is real and constant. So, we need to check if at ( x = 26 ), ( f(x) ) is still real and constant.At ( x = 26 ):( sqrt{26 - 1} = sqrt{25} = 5 )So, ( y = 5 ). Then, the second square root becomes ( |5 - 5| = 0 ), so ( f(26) = 5 + 0 = 5 ).So, at ( x = 26 ), ( f(x) ) is still 5. Therefore, the interval should include 26.Wait a second, earlier I thought the function is constant when ( y < 5 ), which is ( x < 26 ). But at ( x = 26 ), ( y = 5 ), and ( f(x) = 5 ) as well. So, does that mean the function is constant over ( [1, 26] )?Let me double-check:For ( x ) in ( [1, 26) ), ( y = sqrt{x - 1} < 5 ), so ( f(x) = 5 ).At ( x = 26 ), ( y = 5 ), so ( f(x) = 5 + |5 - 5| = 5 ).Therefore, ( f(x) = 5 ) for all ( x ) in ( [1, 26] ).But wait, what happens beyond ( x = 26 )? Let's test ( x = 27 ):( y = sqrt{27 - 1} = sqrt{26} approx 5.1 )So, ( f(27) = sqrt{26} + | sqrt{26} - 5 | = sqrt{26} + (sqrt{26} - 5) = 2sqrt{26} - 5 approx 2*5.1 - 5 = 10.2 - 5 = 5.2 )So, ( f(27) approx 5.2 ), which is greater than 5. Hence, beyond ( x = 26 ), ( f(x) ) is no longer constant.Therefore, the largest interval where ( f(x) ) is real and constant is ( [1, 26] ).I think that's it. So, the function is constant on ( [1, 26] ) and beyond that, it increases. So, the interval is from 1 to 26, including both endpoints.**Final Answer**The largest interval over which ( f(x) ) is real and constant is boxed{[1, 26]}.

answer：Alright, so I have this problem here that asks me to prove that the square root of (1 plus x) is less than 1 plus half of x, given that x is greater than 0. And I need to figure out which of the options A, B, C, or D is the correct cause for this inequality. The options are about x squared being greater than 2, 4, 0, or 1.Okay, let me start by understanding what the problem is asking. It's about inequalities involving square roots and linear expressions. I remember that sometimes when dealing with square roots, squaring both sides can be helpful because it gets rid of the square root, making the inequality easier to handle.So, if I have sqrt(1 + x) < 1 + x/2, and x > 0, maybe I can square both sides to simplify this. Let me try that.Squaring both sides gives me (sqrt(1 + x))^2 < (1 + x/2)^2. Simplifying both sides, that becomes 1 + x < 1 + x + (x^2)/4.Hmm, okay, so on the left side, I have 1 + x, and on the right side, I have 1 + x plus x squared over 4. If I subtract 1 + x from both sides, I get 0 < x^2 / 4. Multiplying both sides by 4, that simplifies to 0 < x^2.So, x squared is greater than 0. That makes sense because x is greater than 0, so squaring it would definitely make it positive. So, x^2 > 0 is always true in this case.Looking back at the options, option C is x^2 > 0, which seems to be the condition we arrived at. The other options are x^2 > 2, x^2 > 4, and x^2 > 1. But in our case, we just needed x^2 to be greater than 0, which is a weaker condition than the others. So, I think option C is the correct cause.Wait, let me double-check. If I choose x = 1, then sqrt(1 + 1) = sqrt(2) ≈ 1.414, and 1 + 1/2 = 1.5. So, 1.414 < 1.5, which is true. And x^2 = 1, which is not greater than 2, 4, or 1, but it is greater than 0. So, that still holds.If I pick x = 0.5, sqrt(1 + 0.5) = sqrt(1.5) ≈ 1.225, and 1 + 0.5/2 = 1.25. So, 1.225 < 1.25, which is also true. Here, x^2 = 0.25, which is still greater than 0, but not greater than 1, 2, or 4.So, it seems like the only necessary condition is that x^2 > 0, which is always true for x > 0. The other options are too restrictive because they require x to be larger than certain values, but the inequality holds even for small positive x.Therefore, I'm confident that option C is the correct answer.

answer：Okay, so I've got this problem here about two shooters, A and B, who are shooting at the same target at the same time. Shooter A has a hit rate of 0.9, which means they hit the target 90% of the time. Shooter B has a hit rate of 0.8, so they hit the target 80% of the time. The problem says that their shootings are independent, and I need to find the probability that the target will be hit when both A and B aim at and shoot the target simultaneously.Alright, let's break this down. First, I need to understand what it means for two events to be independent. In probability, two events are independent if the occurrence of one doesn't affect the probability of the other. So, in this case, whether shooter A hits the target doesn't affect the probability of shooter B hitting the target, and vice versa.Now, the question is asking for the probability that the target is hit when both shooters shoot at the same time. That means I need to find the probability that at least one of them hits the target. Hmm, okay. So, it's not about both hitting the target necessarily, but just that the target is hit by at least one of them.I remember that sometimes it's easier to calculate the probability of the complementary event and then subtract it from 1 to get the desired probability. The complementary event in this case would be that neither shooter hits the target. So, if I can find the probability that both shooters miss the target, I can subtract that from 1 to find the probability that at least one hits the target.Let's denote the probability that shooter A hits the target as P(A) = 0.9, and the probability that shooter B hits the target as P(B) = 0.8. Then, the probability that shooter A misses the target would be 1 - P(A) = 1 - 0.9 = 0.1, and similarly, the probability that shooter B misses the target would be 1 - P(B) = 1 - 0.8 = 0.2.Since the shootings are independent, the probability that both shooters miss the target simultaneously is the product of their individual probabilities of missing. So, P(both miss) = P(A misses) * P(B misses) = 0.1 * 0.2 = 0.02.Therefore, the probability that at least one shooter hits the target is 1 - P(both miss) = 1 - 0.02 = 0.98.Wait, let me double-check that. If both shooters have high hit rates, it makes sense that the probability of the target being hit is quite high. 0.98 seems reasonable because even if one of them misses, the other is likely to hit. Only in the case where both miss, which is a rare event, does the target not get hit.Just to make sure I didn't make a mistake, let me think about another approach. Instead of calculating the complementary probability, I could calculate the probability that at least one hits directly. That would involve considering three scenarios: A hits and B misses, A misses and B hits, or both hit. Then, I would add those probabilities together.So, P(A hits and B misses) = P(A) * P(B misses) = 0.9 * 0.2 = 0.18.P(A misses and B hits) = P(A misses) * P(B) = 0.1 * 0.8 = 0.08.P(both hit) = P(A) * P(B) = 0.9 * 0.8 = 0.72.Adding these up: 0.18 + 0.08 + 0.72 = 0.98.Okay, that's the same result as before. So, whether I calculate it directly or use the complementary probability, I end up with 0.98. That gives me more confidence that the answer is correct.Another way to think about it is to visualize a probability tree. The first branch represents shooter A's outcome: hit or miss. The second branch represents shooter B's outcome: hit or miss. Each branch has its respective probability. Then, I can multiply the probabilities along the branches to find the probabilities of each outcome.So, starting with shooter A:- If A hits (0.9), then B can either hit (0.8) or miss (0.2). So, the probabilities here are 0.9 * 0.8 = 0.72 (both hit) and 0.9 * 0.2 = 0.18 (A hits, B misses).- If A misses (0.1), then B can either hit (0.8) or miss (0.2). So, the probabilities here are 0.1 * 0.8 = 0.08 (A misses, B hits) and 0.1 * 0.2 = 0.02 (both miss).Adding up all the probabilities: 0.72 + 0.18 + 0.08 + 0.02 = 1, which checks out because all possible outcomes are accounted for.Now, focusing on the outcomes where the target is hit: both hit (0.72), A hits and B misses (0.18), and A misses and B hits (0.08). Adding these together: 0.72 + 0.18 + 0.08 = 0.98.So, again, I get the same result. This reinforces that the probability of the target being hit is 0.98.I think I've approached this problem from a couple of different angles now: calculating the complementary probability, calculating the direct probability by considering all favorable outcomes, and even visualizing it with a probability tree. All methods lead me to the same conclusion, which is that the probability of the target being hit is 0.98.Just to be thorough, let me consider if there's any other way this could be approached. Maybe using probability formulas or rules. I recall that for two independent events, the probability of their intersection is the product of their probabilities. So, P(A and B) = P(A) * P(B). But in this case, I'm not just looking for the intersection; I'm looking for the union of A hitting or B hitting.The formula for the probability of the union of two independent events is P(A ∪ B) = P(A) + P(B) - P(A ∩ B). Plugging in the values, that would be 0.9 + 0.8 - (0.9 * 0.8) = 0.9 + 0.8 - 0.72 = 1.7 - 0.72 = 0.98.Yes, that's another way to get the same result. So, using the inclusion-exclusion principle also gives me 0.98.I think I've covered all the bases here. I've used multiple methods to calculate the probability, and they all consistently give me 0.98. I feel confident that this is the correct answer.One last thing to consider: does this make intuitive sense? Shooter A has a 90% chance to hit, and Shooter B has an 80% chance. If they both shoot at the same target, it's highly likely that at least one of them will hit. The only way the target isn't hit is if both miss, which is a 10% chance for A and a 20% chance for B. Multiplying those gives a 2% chance that both miss, which is pretty low. So, the probability that the target is hit is 98%, which seems very high but makes sense given their high hit rates.Yeah, I think that's solid. I don't see any flaws in my reasoning, and all the methods I've used confirm the same result. So, I'm pretty sure the probability is 0.98.