answer:Try to express the difference in purchases of two ladies "in small birds".## SolutionThe first lady paid for her purchase as for 13 small birds, and the second as for 11 small birds. The difference in purchases is 2 small birds, and the difference in price is 20 rubles. Therefore, a small bird costs 10 rubles, and a large one costs 20 rubles.## AnswerA small bird costs 10 rubles, and a large one costs 20 rubles.

answer:First, find the derivative of the function y = frac{2 - cos x}{sin x}:y' = frac{sin x cdot sin x - (2 - cos x) cos x}{sin^2 x}Next, evaluate the derivative at the point x = frac{pi}{2}:y'(frac{pi}{2}) = 1So, the slope of the tangent line to the curve at the point (frac{pi}{2}, 2) is 1. Since the tangent line is perpendicular to the line x + ay + 1 = 0, the product of their slopes should be equal to -1:-1 = - frac{1}{a} cdot 1Solving for a:a = 1Therefore, the value of a is boxed{1}.To solve this problem, we first found the derivative of the given function, which represents the slope of the tangent line at any point on the curve. We then used this slope and the property of perpendicular lines to find the value of a.

answer:Let us rewrite the given equation as follows:a b+sqrt{a^{2}+b} sqrt{b^{2}+a}=-sqrt{a b+1} .Squaring this gives usbegin{aligned}a^{2} b^{2}+2 a b sqrt{a^{2}+b} sqrt{b^{2}+a}+left(a^{2}+bright)left(b^{2}+aright) & =a b+1 left(a^{2} b^{2}+a^{3}right)+2 a b sqrt{a^{2}+b} sqrt{b^{2}+a}+left(a^{2} b^{2}+b^{3}right) & =1 left(a sqrt{b^{2}+a}+b sqrt{a^{2}+b}right)^{2} & =1 a sqrt{b^{2}+a}+b sqrt{a^{2}+b} & = pm 1 .end{aligned}Next, we show that ( a sqrt{b^{2}+a}+b sqrt{a^{2}+b} > 0 ). Note thata b=-sqrt{a b+1}-sqrt{a^{2}+b} cdot sqrt{b^{2}+a} < 0 implies a b < 0 implies a < 0 text{ and } b > 0 text{ or } a > 0 text{ and } b < 0.Without loss of generality, assume ( a < 0 ) and ( b > 0 ). Then rewritea sqrt{b^{2}+a}+b sqrt{a^{2}+b}=aleft(sqrt{b^{2}+a}+bright)-bleft(a-sqrt{a^{2}+b}right)and, since ( sqrt{b^{2}+a}+b ) and ( a-sqrt{a^{2}+b} ) are both positive, the expression above is positive. Therefore,a sqrt{b^{2}+a}+b sqrt{a^{2}+b}=1,and the proof is finished.

answer:Answer: 526.Solution. Let r and ell denote the number of knights and liars, respectively. Note that a knight will say to another knight and a liar will say to another liar: "You are a knight!", while a knight will say to a liar and a liar will say to a knight: "You are a liar!" Therefore, the number of liar-knight pairs is frac{230}{2}=115=r ell. Since r ell=115=5 cdot 23 and r, ell geqslant 2, either r=5 and ell=23, or r=23 and ell=5. In either case, the number of knight-knight and liar-liar pairs is frac{5 cdot 4}{2}+frac{23 cdot 22}{2}=263. Therefore, the phrase "You are a knight!" was said 526 times.

answer:Since Chris spent 68 in total and 25 on the helmet, then he spent 68- 25= 43 on the two hockey sticks.Since the two sticks each cost the same amount, then this cost was 43 div 2= 21.50.ANSWER: (C)

answer:［Solution］ Let the real-coefficient cubic equation bex^{3}+a x^{2}+b x+c=0with three real roots x_{0}-y, x_{0}, x_{0}+y forming an arithmetic sequence. By the relationship between roots and coefficients, we haveleft(x_{0}-yright)+x_{0}+left(x_{0}+yright)=-a,so x_{0}=-frac{1}{3} a.Substituting x = -frac{1}{3} a into equation (1), we getleft(-frac{1}{3} aright)^{3}+aleft(-frac{1}{3} aright)^{2}+bleft(-frac{1}{3} aright)+c=0which simplifies to 2 a^{3}-9 a b+27 c=0.Conversely, if a, b, c satisfy equation (2), then equation (1) has a root x_{0}=-frac{1}{3} a. Let the other two roots of the equation be x_{1}, x_{2}. By the relationship between roots and coefficients, we havex_{0}+x_{1}+x_{2}=-a, quad x_{0} x_{1}+x_{1} x_{2}+x_{2} x_{0}=b,and since x_{0}=-frac{1}{3} a, we getx_{1}+x_{2}=-frac{2}{3} a, quad x_{1} x_{2}=b-frac{2}{9} a^{2}Thus, x_{1}, x_{0}, x_{2} form an arithmetic sequence, and the numbers x_{1} and x_{2} satisfy the quadratic equationx^{2}+frac{2}{3} a x+left(b-frac{2}{9} a^{2}right)=0Clearly, x_{1} and x_{2} are real numbers if and only if Delta=left(frac{2}{3} aright)^{2}-4left(b-frac{2}{9} a^{2}right) geqslant 0, which is equivalent toa^{2}-3 b geqslant 0Therefore, the cubic equation (1) has three real roots forming an arithmetic sequence if and only if both conditions (2) and (3) are satisfied simultaneously.