answer:1. **Step 1: Analyze statement (1)** Given: [ A = left{ y mid y = x^2 + 1, , x in mathbf{N}_{+} right} ] [ B = left{ y mid y = x^2 - 2x + 2, , x in mathbf{N}_{+} right} ] - For ( y in A ): [ y = x^2 + 1 ] where ( x in mathbf{N}_{+} ). - For ( y in B ): [ y = x^2 - 2x + 2 ] which can be rewritten as: [ y = (x - 1)^2 + 1 ] where ( x - 1 geq 0 ) and hence ( x in mathbf{N}_{+} ). By transforming both expressions: [ B = left{ y mid y = (x-1)^2 + 1 ,text{for}, x in mathbf{N}_{+} right} ] Notice that, for all ( x in mathbf{N}_{+} ), ( (x-1) geq 0 ): [ (x-1) in mathbf{N}_{+} cup {0} = mathbf{N}_{0} ] Thus, every element of ( A ) can also be found in ( B ), but not vice versa for all ( x in mathbf{N}_{+} ). Therefore: [ A subseteq B ] But not ( A = B ). Thus, statement (1) is incorrect. 2. **Step 2: Analyze statement (2)** Given: [ A, , B neq varnothing, quad A cap B = varnothing ] Let's define: [ M = { ,text{subsets of}, A , } ] [ P = { ,text{subsets of}, B , } ] Since ( A cap B = varnothing ) and both ( A ) and ( B ) are non-empty, ( M ) and ( P ) are collections of subsets of completely disjoint sets. Thus, any subset of ( A ) will have no intersection with any subset of ( B ): [ M cap P = { varnothing } ] Therefore, statement (2) is correct. 3. **Step 3: Analyze statement (3)** Given: [ A = B ] Thus, for any set ( C ): [ A cap C = B cap C ] The equality ( A = B ) ensures that any common intersection with ( C ) will also be equal: [ (A cap C) = (B cap C) ] Therefore, statement (3) is correct. 4. **Step 4: Analyze statement (4)** Given: [ A cap C = B cap C ] We need to test whether this implies ( A = B ). Consider the case where: [ C = varnothing ] For any sets ( A ) and ( B ): [ A cap varnothing = varnothing = B cap varnothing ] Hence, if ( A = {1} ) and ( B = {2} ), the equality holds: [ (A cap varnothing) = (B cap varnothing) ] Yet, ( A neq B ). Thus, statement (4) is incorrect. # Conclusion The correct statements are (2) and (3). Thus, the correct number of true statements is [ boxed{2} ]

answer:We need to find all ordered triples of non-negative integers ((a, b, c)) such that (a^2 + 2b + c), (b^2 + 2c + a), and (c^2 + 2a + b) are all perfect squares. 1. **Assume (a leq b leq c):** - We start by analyzing the expression (c^2 + 2a + b). Since (a leq b leq c), we have (2a + b leq 3c). Therefore, [ c^2 leq c^2 + 2a + b leq c^2 + 3c. ] This implies that (c^2 + 2a + b) must be a perfect square between (c^2) and ((c+1)^2). Hence, [ c^2 leq c^2 + 2a + b < (c+1)^2. ] The only possibility is (c^2 + 2a + b = c^2), which implies (2a + b = 0). Since (a) and (b) are non-negative integers, the only solution is (a = b = 0). Thus, (c) can be any non-negative integer. 2. **Check the other expressions:** - For (a = b = 0), we have: [ a^2 + 2b + c = c, ] [ b^2 + 2c + a = 2c, ] [ c^2 + 2a + b = c^2. ] Since (c) is a non-negative integer, (c) must be a perfect square. Therefore, (c = 0) is the only solution. 3. **Consider the case where (2a + b = 2c + 1):** - If (c^2 + 2a + b = (c+1)^2), then: [ c^2 + 2a + b = c^2 + 2c + 1, ] [ 2a + b = 2c + 1. ] - Similarly, for (b^2 + 2c + a), we have: [ b^2 leq b^2 + 2c + a leq (b+1)^2, ] [ b^2 leq b^2 + 2c + a < (b+1)^2. ] The only possibility is (b^2 + 2c + a = b^2), which implies (2c + a = 0). Since (a) and (c) are non-negative integers, the only solution is (a = c = 0). Thus, (b) can be any non-negative integer. 4. **Combining the conditions:** - From the above analysis, we have: [ 2a + b = 2c + 1, ] [ 2c + a = 2b + 1. ] - Solving these equations, we get: [ b = 3a - 2, ] [ 2c = 5a - 3. ] - Substituting these into the first expression, we get: [ a^2 + frac{17}{2}a - frac{11}{2} = (a+1)^2, (a+2)^2, (a+3)^2, (a+4)^2. ] Solving these, we find the non-negative integer solutions for (a) are (a = 43, 1). These give the solution sets ((43, 127, 106)) and ((1, 1, 1)). The final answer is (boxed{(0, 0, 0), (1, 1, 1), (43, 127, 106)}).

answer:Let's assume the cost price of one article is C and the selling price of one article is S. According to the given information, the cost price of 22 articles is equal to the selling price of 16 articles. So we can write the equation: 22C = 16S Now, we want to find the profit or loss percentage, which is calculated based on the cost price. First, we need to find the relationship between the cost price and the selling price for one article. From the equation above, we can find the selling price in terms of the cost price: S = (22C / 16) Now, let's find the profit or loss per article. If the selling price is greater than the cost price, there is a profit. If the selling price is less than the cost price, there is a loss. Profit or Loss per article = S - C Substitute S with (22C / 16): Profit or Loss per article = (22C / 16) - C To simplify, we find a common denominator: Profit or Loss per article = (22C - 16C) / 16 Profit or Loss per article = 6C / 16 Profit or Loss per article = 3C / 8 Now, to find the percentage of profit or loss, we divide the profit or loss per article by the cost price and multiply by 100: Percentage of Profit or Loss = [(Profit or Loss per article) / C] * 100 Percentage of Profit or Loss = [(3C / 8) / C] * 100 C cancels out: Percentage of Profit or Loss = (3 / 8) * 100 Percentage of Profit or Loss = 0.375 * 100 Percentage of Profit or Loss = 37.5% Since the selling price is greater than the cost price (as seen from the original equation), the merchant makes a profit. Therefore, the merchant makes a profit of boxed{37.5%} .

answer:For the function f(x)=2sin (omega x+varphi) (omega > 0), given that fleft( frac {pi}{3}right)=0 and fleft( frac {pi}{2}right)=2, we have 2sin left(omega times frac {pi}{3}+varphiright)=0 and 2sin left(omega times frac {pi}{2}+varphiright)=2. From omega times frac {pi}{3}+varphi=kpi and omega times frac {pi}{2}+varphi=2kpi+ frac {pi}{2}, we get frac {pi}{6}omega=kpi+ frac {pi}{2}. Therefore, the minimum value of the real number omega is: 3. Hence, the answer is boxed{3}. By directly using fleft( frac {pi}{3}right)=0 and fleft( frac {pi}{2}right)=2, we set up equations and then solve for the value of omega, finding the minimum value. This problem examines the method of finding the analytical expression of trigonometric functions, the application of trigonometric function values, and the ability to analyze and solve problems.