answer:**Analysis** This problem tests the understanding of the graph translation of trigonometric functions and the method of finding the monotonicity of composite functions related to the sine function. First, find omega from the function's period, then obtain the analytical expression of g(x) from the graph translation, and finally find the range of x in the increasing phase of the sine function to get the answer. **Step-by-step solution** 1. Given that the function f(x)=sin (omega x-frac{pi }{4})(omega > 0) has a minimum positive period of pi, therefore frac{2pi }{omega }=pi, which leads to omega =2. So, f(x)=sin (2x-frac{pi }{4}). 2. After shifting the graph to the left by frac{pi }{4} units, we obtain g(x)=sin [2(x+frac{pi }{4})-frac{pi }{4}]=sin (2x+frac{pi }{4}). 3. To find the monotonically increasing interval, we need to consider the following inequality: -frac{pi }{2}+2kpi leq 2x+frac{pi }{4}leq frac{pi }{2}+2kpi. 4. Solving the inequality for x, we get -frac{3pi }{8}+kpi leq xleq frac{pi }{8}+kpi ,kin mathbb{Z}. 5. Therefore, the monotonically increasing interval of the function y=g(x) is boxed{[-frac{3pi }{8}+kpi ,frac{pi }{8}+kpi ]},kin mathbb{Z}.

answer:First, let's calculate Scout's earnings from his base pay: On Saturday, he worked 4 hours at 10.00 per hour, so he earned: 4 hours * 10.00/hour = 40.00 On Sunday, he worked 5 hours at 10.00 per hour, so he earned: 5 hours * 10.00/hour = 50.00 Now, let's calculate his earnings from tips on Saturday: He delivered groceries to 5 people and earned 5.00 per person, so he earned: 5 people * 5.00/person = 25.00 Now, let's add his base pay and tips for Saturday: 40.00 (base pay) + 25.00 (tips) = 65.00 We know that he made 155 over the weekend, so let's subtract his earnings from Saturday to find out how much he made on Sunday: 155.00 (total weekend earnings) - 65.00 (Saturday earnings) = 90.00 (Sunday earnings) We already calculated his base pay for Sunday, which was 50.00, so the remaining 90.00 - 50.00 = 40.00 must have come from tips on Sunday. Since he earns 5.00 per tip, we can find out how many people he delivered to on Sunday by dividing the tip earnings by the amount per tip: 40.00 (tips) / 5.00 (per tip) = 8 people So, Scout delivered groceries to boxed{8} people on Sunday.

answer:**Analysis** This problem mainly examines the functional characteristics of sequences. **Solution** According to the problem, the sequence {a_n} has a period of 2, So a_3=a_1, that is, a_{2+1}=a_2+r=a_{1+1}+r=2a_1+r=a_1=m, solving this gives m+r=0. Therefore, the answer is boxed{m+r=0}.

answer:First, perform prime factorization for 48 and 180: 48 = 2^4 times 3^1 180 = 2^2 times 3^2 times 5^1 Next, determine the GCD: To find the GCD, take the lowest power of all primes appearing in both factorizations. The common primes are 2 and 3. For 2, the lowest power is 2^2, and for 3, it is 3^1. Therefore, the GCD is: [ text{GCD} = 2^2 times 3^1 = 4 times 3 = 12 ] Divisors of 12 (since 12 is the GCD of 48 and 180) are: 1, 2, 3, 4, 6, 12. There are boxed{6} positive integers that are divisors of both 48 and 180.