answer:【Solution】Solution: According to the problem, we know:The numbers retained in the first round are 2,5,9, cdots Therefore, the maximum number retained in the first round is:begin{array}{l}2+3+4+cdots+n =frac{2+n}{2} times(n-1)end{array}When n=63, the sum of the sequence is 2015. This indicates that 2015 is a retained number.At this point, not all numbers have been crossed out and further crossing is required. However, since it is a circle, the order of crossing continues as 2016, 2,5,9 cdots, this being the 63rd operation, 2015 is the last one to be crossed out.Therefore, the answer is: 2015.

answer:## SolutionSince cos left(frac{1}{x}right)_{text {- is bounded, and }}begin{aligned}& lim _{x rightarrow 0} e^{x^{2}}-cos x=e^{0^{2}}-cos 0=e^{0}-1=1-1=0 & left(e^{x^{2}}-cos xright) cos left(frac{1}{x}right) rightarrow 0, text { then } & text {, as } x rightarrow 0end{aligned}begin{aligned}& lim _{x rightarrow 0} ln left(left(e^{x^{2}}-cos xright) cos left(frac{1}{x}right)+operatorname{tg}left(x+frac{pi}{3}right)right)=ln left(0+operatorname{tg}left(0+frac{pi}{3}right)right)= & =ln left(operatorname{tg}left(frac{pi}{3}right)right)=ln sqrt{3}end{aligned}

answer:Since the base edge length of the square pyramid is 2 and the lateral edge length is sqrt{3}, and its front view and side view are congruent isosceles triangles; thus, the legs in the front and side views are the slant heights of the square pyramid. The length of the slant height is: sqrt{2}. Therefore, the perimeter of the front view is: 2 + 2sqrt{2}. Hence, the answer is 2 + 2sqrt{2}.The fact that the front and side views of the solid are congruent isosceles triangles indicates that the legs are the slant heights of the square pyramid. By finding the slant height, we can determine the perimeter of the front view. This question tests the understanding of simple solid's orthographic projections, and a common mistake is not recognizing that the legs in the congruent isosceles triangles of the front and side views are the slant heights of the square pyramid.Therefore, the perimeter of the front view is boxed{2 + 2sqrt{2}}.

answer:Given that each term in the geometric sequence {a_n} is positive and the geometric mean of a_5 and a_15 is 2 sqrt {2},begin{cases} (a_1q^{4})(a_1q^{14})=(2 sqrt {2})^{2} q > 0 end{cases}Hence, a_{10}=a_1q^{9}=2 sqrt {2},begin{align*} log _{2}a_{4}+log _{2}a_{16} &=log _{2}(a_{4}a_{16}) &=log _{2}(a_{10}^{2}) &=log _{2}8 &=3 end{align*}Therefore, the answer is boxed{B}.By using the formula for the general term of a geometric sequence and the geometric mean, we find a_{10}. Then, using the properties of logarithms, we can find the value of log _{2}a_{4}+log _{2}a_{16}. This problem tests the understanding of logarithmic values and requires careful reading and the reasonable application of geometric sequence properties.

answer:Solution: Given vector overrightarrow{a}, we have overrightarrow{a}+2overrightarrow{a}=3overrightarrow{a}. Therefore, the correct choice is boxed{B}. This can be directly calculated using vector addition. This question tests the addition of vectors and their geometric meaning, and it is a basic question.

answer:To solve this problem, let's break down the solution into detailed steps:1. Assumption: We assume that the magnitudes of the vectors overrightarrow{a}, overrightarrow{b}, and their sum overrightarrow{a}+overrightarrow{b} are all equal to 1. That is, we have |overrightarrow{a}|=|overrightarrow{b}|=|overrightarrow{a}+overrightarrow{b}|=1.2. Expanding the Square of the Sum: We know that the square of the magnitude of a vector sum can be expanded as follows: [ (overrightarrow{a}+overrightarrow{b})^2 = overrightarrow{a}^2 + 2overrightarrow{a}cdotoverrightarrow{b} + overrightarrow{b}^2 ] Given our assumption, this equation simplifies to 1 = 1 + 2overrightarrow{a}cdotoverrightarrow{b} + 1.3. Solving for the Dot Product: From the equation above, we can solve for the dot product overrightarrow{a}cdotoverrightarrow{b}: [ 2overrightarrow{a}cdotoverrightarrow{b} = 1 - 2 implies overrightarrow{a}cdotoverrightarrow{b} = -frac{1}{2} ]4. Finding the Cosine of the Angle: The cosine of the angle between two vectors is given by the formula: [ cosangle(overrightarrow{a},overrightarrow{b}) = frac{overrightarrow{a}cdotoverrightarrow{b}}{|overrightarrow{a}||overrightarrow{b}|} ] Substituting the values we have, we get: [ cosangle(overrightarrow{a},overrightarrow{b}) = frac{-frac{1}{2}}{1cdot1} = -frac{1}{2} ]5. Determining the Angle: Given that the cosine of the angle is -frac{1}{2} and considering the range of vector angles is [0,pi], the only angle that satisfies this condition is frac{2pi}{3}.Therefore, the angle between vectors overrightarrow{a} and overrightarrow{b} is boxed{frac{2pi}{3}}.