answer:To evaluate which of the given expressions is correctly factored from left to right, we will analyze each option step by step: **Option A:** Given: (a+3)^{2}=a^{2}+6a+9 This represents the expansion of a squared binomial, which is in the form of (x+y)^2 = x^2 + 2xy + y^2. Applying this to (a+3)^2, we get: [ (a+3)^2 = a^2 + 2(a)(3) + 3^2 = a^2 + 6a + 9 ] This shows that option A is an expansion rather than a factorization. Therefore, option A is incorrect. **Option B:** Given: a^{2}-4a+4 The correct factorization of this expression is based on the formula (x-y)^2 = x^2 - 2xy + y^2. It should be: [ a^2 - 4a + 4 = (a-2)^2 ] Since the given factorization in option B is a(a-4) + 4, which does not match the correct factorization, option B is incorrect. **Option C:** Given: 5ax^{2}-5ay^{2} Factoring out the common factor 5a, we get: [ 5ax^2 - 5ay^2 = 5a(x^2 - y^2) ] Using the difference of squares formula, x^2 - y^2 = (x+y)(x-y), we further factorize it as: [ 5a(x^2 - y^2) = 5a(x+y)(x-y) ] This matches the given factorization in option C, indicating that option C is correctly factored. **Option D:** Given: a^{2}-2a-8 The correct factorization of this quadratic expression is based on finding two numbers that multiply to -8 and add to -2. These numbers are +2 and -4. Therefore, the correct factorization is: [ a^2 - 2a - 8 = (a+2)(a-4) ] Since the given factorization in option D is (a-2)(a+4), which does not match the correct factorization, option D is incorrect. Therefore, the correctly factored expression from the options given is: [ boxed{C} ]

answer:1. **Define the Partial Sums:** Define a sequence ( s_i ) which represents the partial sums of the sequence ( a_1, a_2, ldots, a_{100} ). [ s_1 = a_1, s_2 = a_1 + a_2, ldots, s_{100} = a_1 + a_2 + cdots + a_{100} ] As each ( a_i ) is a positive integer (either 1 or 2), the sequence ( s_i ) is strictly increasing. [ s_1 < s_2 < cdots < s_{100} ] 2. **Evaluate the Sum Bounds:** [ s_{100} = (a_1 + a_2 + cdots + a_{10}) + (a_{11} + a_{12} + cdots + a_{20}) + cdots + (a_{91} + a_{92} + cdots + a_{100}) ] According to the problem's condition, the sum of any 10 consecutive terms is at most 16: [ a_i + a_{i+1} + cdots + a_{i+9} leq 16 quad text{for} quad 1 leq i leq 91 ] Thus, each individual decagonal section is bounded as: [ s_{100} leq 10 times 16 = 160 ] 3. **Construct the Extended Sequence:** Construct the sequence that includes ( s_i ) and ( s_i + 39 ): [ underbrace{s_1, s_2, ldots, s_{100}, s_1 + 39, s_2 + 39, ldots, s_{100} + 39} ] This means there are 200 terms in total. The largest term in this augmented sequence is: [ s_{100} + 39 leq 160 + 39 = 199 ] 4. **Apply the Pigeonhole Principle:** The 200 terms are all distinct integers within the range from ( 1 ) to ( 199 ). By the pigeonhole principle, there must exist at least two terms in this range that are equal. Let ( s_h ) and ( s_k + 39 ) be such that: [ s_h = s_k + 39 quad text{for} quad 1 leq h, k leq 100 ] This implies: [ s_h - s_k = 39 ] 5. **Derive the Desired Sum:** Since: [ s_h = sum_{i=1}^{h} a_i quad text{and} quad s_k = sum_{i=1}^{k} a_i ] Therefore: [ sum_{i=1}^{h} a_i - sum_{i=1}^{k} a_i = 39 quad text{or} quad sum_{i=k+1}^{h} a_i = 39 ] # Conclusion: Thus, there exists a pair of indices ( h ) and ( k ), with ( k > h ), such that: [ a_h + a_{h+1} + cdots + a_k = 39 ] (blacksquare)

answer:Since (1-x)^4 = 1 - 4x + 6x^2 - 4x^3 + x^4, the coefficient of x^2 in the expansion of (x^2+x+1)(1-x)^4 is 1 - 4 + 6 = 3. Therefore, the answer is boxed{3}. **Analysis:** By expanding (1-x)^4 according to the binomial theorem, we can find the coefficient of x^2 in the expansion of (x^2+x+1)(1-x)^4.

answer:1. **Set up the Problem with Variables:** Let: - ( a ) be the number of campers who only go canoeing, - ( b ) be the number of campers who go canoeing and swimming but not fishing, - ( c ) be the number of campers who only go swimming, - ( d ) be the number of campers who go canoeing and fishing but not swimming, - ( e ) be the number of campers who go swimming and fishing but not canoeing, - ( f ) be the number of campers who only go fishing, - ( g ) be the number of campers who go canoeing, swimming, and fishing. We are given: - There are 15 campers who go canoeing: ( a + b + d = 15 ), - There are 22 campers who go swimming: ( b + c + e = 22 ), - There are 12 campers who go fishing: ( d + e + f = 12 ), - 9 campers do not take part in any of these activities. 2. **Express the Total Number of Campers:** The total number of campers is the sum of all the mentioned groups plus the 9 who don't participate: [ T = a + b + c + d + e + f + g + 9 ] Since no camper does all three activities, ( g = 0 ), so: [ T = a + b + c + d + e + f + 9 ] 3. **Combine the Given Equations to Find Total Constraints:** Add the equations for each activity: [ (a+b+d) + (b+c+e) + (d+e+f) = 15 + 22 + 12 ] Simplify: [ a + 2b + c + 2d + 2e + f = 49 ] To express this in terms of ( T ), which includes ( a + b + c + d + e + f ), we note: [ a + b + c + d + e + f = 49 - (b + d + e) ] 4. **Determine Constraints for Minimizing Number of Campers:** We express ( T ) as: [ T = (49 - (b + d + e)) + 9 = 58 - (b + d + e) ] To minimize ( T ), we need to maximize ( b + d + e ). 5. **Find the Maximum Value of ( b + d + e ) Under Given Constraints:** Use the original activity conditions: - ( b + d leq 15 ) - ( b + e leq 22 ) - ( d + e leq 12 ) Combine the constraints to solve for ( b + d + e ): [ 2b + 2d + 2e leq 49 implies b + d + e leq 24.5 ] Since ( b, d, e ) are integers, the actual maximum is: [ b + d + e leq 24 ] 6. **Verify Feasibility and Use Calculated Maximum Value:** Let: - ( b + d = 14 ) - ( b + e = 22 ) - ( d + e = 12 ) Solving these equations: - From ( b + d = 14 ) and ( d + e = 12 ), we get ( b + (12 - e) = 14 implies b - e = 2 ) - Combining with ( b + e = 22 ), solve ( (b - e) + (b + e) = 2 + 22 = 24 ) implies ( 2b = 24 ) thus ( b = 12 ), which in turn gives ( e = 10), ( d = 2 ). With ( b = 12, d = 2, e = 10 ), the minimum total number of campers is: [ T = 58 - 24 = 34 ] # Conclusion: [ boxed{34} ]