answer: Solution:# Part 1: Finding angle AGiven that in triangle ABC, the sides opposite angles A, B, and C are a, b, and c respectively, and they satisfy the equation:[b^{2}ccos C + c^{2}bcos B = ab^{2} + ac^{2} - a^{3}]We can rewrite this equation using the law of cosines, which states that in any triangle, the square of one side is equal to the sum of the squares of the other two sides minus twice the product of those sides and the cosine of the included angle. Therefore, we have:[b^{2}ccos C + c^{2}bcos B = a(b^{2} + c^{2} - a^{2})]This simplifies to:[a cdot 2bccos A = b^{2}ccos C + c^{2}bcos B]Dividing both sides by bc, we get:[2acos A = bcos C + ccos B]Using the sine rule and trigonometric identities, we can rewrite bcos C + ccos B as sin Bcos C + sin Ccos B, which equals 2sin Acos A due to the sum-to-product identities. This leads to:[sin A = 2sin Acos A]Solving for cos A, we find:[cos A = frac{1}{2}]Given that A is an angle in a triangle and thus must be between 0 and pi radians (0 and 180^circ), we conclude that:[A = frac{pi}{3}]Therefore, the angle A is boxed{frac{pi}{3}}.# Part 2: Finding the minimum value of a when b + c = 2Given b + c = 2, we apply the cosine rule to find a^2:[a^2 = b^2 + c^2 - 2bccos A]Since A = frac{pi}{3}, cos A = frac{1}{2}, and thus:[a^2 = b^2 + c^2 - bc]Substituting b + c = 2, we get:[a^2 = (b + c)^2 - 3bc]Applying the AM-GM inequality, where the arithmetic mean is greater than or equal to the geometric mean, we have:[bc leq left(frac{b + c}{2}right)^2]Therefore:[a^2 geq (b + c)^2 - 3left(frac{b + c}{2}right)^2 = 1]Equality holds when b = c = 1, which implies the minimum value of a is sqrt{1} = 1.Hence, the minimum value of a is boxed{1}.

answer:33. How many people got lost in the mountains? Let's call one portion the amount of supplies one person consumes in a day. Initially, 9 people had 45 portions (a 5-day food supply). On the second day, they had only 36 portions left. On the same day, they met a second group, and the 36 remaining portions were enough for everyone for 3 days. Therefore, there should have been 12 people in total.This means that the second group had 3 people.

answer:Since P and Q are points on the curve y = 1 - x^2 on opposite sides of the y-axis, let's consider P with coordinates (a, 1 - a^2) where a > 0, and Q with coordinates (b, 1 - b^2) where b 0 > b_0, we find: a_0 = frac{sqrt{3}}{3} , quad b_0 = -frac{sqrt{3}}{3} Substituting these values back into our expression for f(a, b) gives us the minimum area: f_{text{min}}(a_0, b_0) = frac{8 sqrt{3}}{9} Therefore, the minimum area of the triangle is boxed{frac{8 sqrt{3}}{9}}.

answer:Solution: 326% = 3.26, 3 frac {1}{5} = 3.2, frac {15}{4} = 3.75Hence, 3.2 < 3.25 < 3.26 < 3.75Or, 3 frac {1}{5} < 3.25 < 326% < frac {15}{4}Therefore, the smallest number is 3 frac {1}{5}.So, the answer is: boxed{text{C}}.To find the smallest number, convert all the options into decimals and then compare them. This way, you can avoid the hassle of converting fractions to a common denominator. When comparing decimals, fractions, or percentages, it's common to convert them into decimals (rounded to a certain number of decimal places) and then compare them using the standard method for comparing decimals.

answer:2. 176.Substituting 1-x for x in equation (1) yieldsbegin{array}{l}2 fleft(x^{2}-3 x+2right)+fleft(x^{2}+xright) =9 x^{2}-15 x .end{array}Combining equations (1) and (2) givesbegin{array}{l}fleft(x^{2}+xright)=3 x^{2}+3 x-4 =3left(x^{2}+xright)-4 .end{array}Therefore, f(60)=3 times 60-4=176.

answer:Given that the lengths of the three face diagonals of a rectangular parallelepiped are 2, sqrt{3}, and sqrt{5}, we aim to find the surface area of the circumscribed sphere of the rectangular parallelepiped.Let's denote the length, width, and height of the rectangular parallelepiped as x, y, and z, respectively. The radius of the circumscribed sphere is represented by R.The relationships between the face diagonals and the dimensions of the parallelepiped are given by the equations:[begin{align*}x^2 + y^2 &= 4,y^2 + z^2 &= 3,z^2 + x^2 &= 5.end{align*}]Adding these equations together, we get:[2(x^2 + y^2 + z^2) = 4 + 3 + 5 = 12.]Solving for x^2 + y^2 + z^2, we find:[x^2 + y^2 + z^2 = frac{12}{2} = 6.]The radius of the circumscribed sphere, R, is half the square root of the sum of the squares of the dimensions of the parallelepiped:[R = frac{sqrt{x^2 + y^2 + z^2}}{2} = frac{sqrt{6}}{2}.]The surface area of a sphere is given by the formula 4pi R^2. Substituting the value of R, we get:[begin{align*}text{Surface area} &= 4pi left(frac{sqrt{6}}{2}right)^2 &= 4pi times frac{6}{4} &= 6pi.end{align*}]Therefore, the surface area of the circumscribed sphere of the rectangular parallelepiped is boxed{6pi}.