Find the minimum of formula_27.Below is the 29th problem in Mathematics subject for all applicants of CSAT held in 1996 that is considered to be the hardest one in history. It had correct response rates of 0.08%.If two equations formula_35 and formula_36 have 7 and 9 solutions respectively and a setformula_37 is an infinite set, formula_38, the number of elements in formula_39's subsetformula_40, varies according to the values of formula_41 and formula_42. Find the maximum of formula_38. Below is the 29th problem in Mathematics subject type B, previous name of type Ga, of CSAT held in 2013.formula_44 and formula_45 are points on the sphere formula_46. formula_47 and formula_48 are the foots of two perpendiculars from formula_44 and formula_45 to the plane formula_51 respectively. formula_52 and formula_53 are the foots of two prependiculars from formula_44 and formula_45 to the plane formula_56 respectively. Find the maximum of formula_57. Below is a notorious problem in English subject of CSAT held in 2010 which had correct response rates of 9.77%. The article is extracted from J. L. Stocks's 'The Limits of Purpose'. So far as you are wholly concentrated on bringing about a certain result, clearly the quicker and easier it is brought about the better. Your resolve to secure a sufficiency of food for yourself and your family will induce you to spend weary days in tilling the ground and tending livestock; but if Nature provided food and meat in abundance ready for the table, you would thank Nature for sparing you much labor and consider yourself so much the better off. An executed purpose, in short, is a transaction in which the time and energy spent on the execution are balanced against the resulting assets, and the ideal case is one in which__________________. Purpose, then, justifies the efforts it exacts only conditionally, by their fruits.  ① demand exceeds supply, resulting in greater returns  ② life becomes fruitful with our endless pursuit of dreams  ③ the time and energy are limitless and assets are abundant  ④ Nature does not reward those who do not exert efforts  ⑤ the former approximates to zero and the latter to infinity  The Preliminary College Scholastic Ability Test (PCSAT) is a preliminary examination for the CSAT that is hosted nationally.