由对称性解2-SAT问题

SAT Subject Test Practice - Results Summary Mathematics Level 21Your answer Omitted!What is the distance in space between the points with coordinates and ?(A)(B)(C)(D)(E)ExplanationDifficulty: EasyThe correct answer is D.The distance between the points with coordinates and is given by the distance formula: .Therefore, the distance between the points with coordinates and is:,which simplifies to .2Your answer Omitted!If , what value does approach as gets infinitely larger?(A)(B)(C)(D)(E)ExplanationDifficulty: EasyThe correct answer is E.One way to determine the value that approaches as gets infinitely larger is to rewrite the definition of the function to use only negative powers of and then reason about the behavior of negative powers of as gets infinitely larger. Since the question is only concerned with what happens to as gets infinitely larger, one can assume that is positive. For , theexpression is equivalent to the expression . As gets infinitely larger, the expression approaches the value , so as gets infinitely larger, the expression approaches the value . Thus, as gets infinitely larger, approaches .Alternatively, one can use a graphing calculator to estimate the height of the horizontal asymptote for the function . Graph the function on an interval with “large”, say, from to .By examining the graph, the all seem very close to . Graph the function again, from, say, to .The vary even less from . In fact, to the scale of the coordinate plane shown, the graph of the function is nearly indistinguishable from the asymptotic line . This suggests that as gets infinitely larger, approaches , that is, .Note: The algebraic method is preferable, as it provides a proof that guarantees that the value approaches is . Although the graphical method worked in this case, it does not provide a complete justification; for example, the graphical method does not ensure that the graph resembles a horizontal line for “very large”such as .3Your answer Omitted!If is a factor of , then(A)(B)(C)(D)(E)ExplanationDifficulty: EasyThe correct answer is A.By the Factor Theorem, is a factor of only when is a root ofthat is, , which simplifies to . Therefore, .Alternatively, one can perform the division of by and then find a value for so that the remainder of the division is .Since the remainder is , the value of must satisfy . Therefore, .4Your answer Omitted!Alison deposits into a new savings account that earns percent interest compounded annually. If Alison makes no additional deposits or withdrawals, how many years will it take for the amount in the account to double?(A)(B)(C)(D)(E)ExplanationDifficulty: MediumAfter year, the amount in the account is equal to . After years, the amount isequal to , and so on. After years, the amount is equal to . You needto find the value of for which . There are several ways to solve this equation. You can use logarithms to solve the equation as follows.Since , it will take more than years for the amount in the account to double. Thus, you need to round up to .Another way to find is to use your graphing calculator to graph and . From the answer choices, you know you need to set the viewing window with values from to about and values extending just beyond . The of the point of intersection is approximately . Thus you need to round up to .5Your answer Omitted!In the figure above, when is subtracted from , what is the length of the resultant vector?(A)(B)(C)(D)(E)ExplanationDifficulty: MediumThe resultant of can be determined by . The length of the resultant is:6Your answer Omitted!In the -plane, what is the area of a triangle whose vertices are , , and ?(A)(B)(C)(D)(E)ExplanationDifficulty: MediumIt is helpful to draw a sketch of the triangle:The length of the base of the triangle is and the height of the triangle is . Therefore, the area of the triangle is . The correct answer is B.7Your answer Omitted!A right circular cylinder has radius and height . If and are two points on its surface, what is the maximum possible straight-line distance between and ?(A)(B)(C)(D)(E)ExplanationDifficulty: MediumThe maximum possible distance occurs when and are on the circumference of opposite bases: You can use the Pythagorean Theorem:The correct answer is (B).8Your answer Omitted!Note: Figure not drawn to scale.In the figure above, and the measure of is . What is the value of ?(A)(B)(C)(D)(E)ExplanationDifficulty: MediumThere are several ways to solve this problem. One way is to use the law of sines. Since ,the measure of is and the measure of is . Thus, and . (Make sure your calculator is in degree mode.)You can also use the law of cosines:Since is isosceles, you can draw the altitude to the triangle.9Your answer Omitted!The function is defined by for .What is the difference between the maximum and minimum values of ?(A)(B)(C)(D)(E)ExplanationDifficulty: MediumIt is necessary to use your graphing calculator for this question. First graph the function. It is helpful to resize the viewing window so the -values go fromto . On this interval the maximum value of is and the minimum value of is. The difference between these two values is , which rounds to .10Your answer Omitted!Suppose the graph of is translated units left and unit up. If the resulting graph represents , what is the value of ?(A)(B)(C)(D)(E)ExplanationDifficulty: MediumIt may be helpful to draw a graph of and .The equation for is . Therefore,. The correct answer is B.11Your answer Omitted!A sequence is recursively defined by , for . If and , what is the value of ?(A)(B)(C)(D)(E)ExplanationDifficulty: MediumThe values for and are given. is equal to . is equal to. is equal to . is equal to.If your graphing calculator has a sequence mode, you can define the sequence recursively and findthe value of . Let , since the first term is . Define . Let , since we have to define the first two terms and . Then examining a graph or table, you can find .12Your answer Omitted!The diameter and height of a right circular cylinder are equal. If the volume of the cylinder is , what is the height of the cylinder?(A)(B)(C)(D)(E)ExplanationDifficulty: MediumThe correct answer is A.To determine the height of the cylinder, first express the diameter of the cylinder in terms of theheight, and then express the height in terms of the volume of the cylinder.The volume of a right circular cylinder is given by , where is the radius of the circular base of the cylinder and is the height of the cylinder. Since the diameter and height are equal, . Thus . Substitute the expression for in the volume formula to eliminate :. Solving for gives . Since the volume of the cylinder is , theheight of the cylinder is .13Your answer Omitted!If ,then(A)(B)(C)(D)(E)ExplanationDifficulty: MediumThe correct answer is E.One way to determine the value of is to apply the sine of difference of two angles identity: . Since and , the identity gives . Therefore, .Another way to determine the value of is to apply the supplementary angle trigonometric identity for the sine: . Therefore, .14Your answer Omitted!A line has parametric equations and , where is the parameter. The slope of the line is(A)(B)(C)(D)(E)ExplanationDifficulty: MediumThe correct answer is B.One way to determine the slope of the line is to compute two points on the line and then use the slope formula. For example, letting gives the point on the line, and letting gives the point on the line. Therefore, the slope of the line is equal to .Alternatively, one can express in terms of . Since and , it follows that . Therefore, the slope of the line is .15Your answer Omitted!What is the range of the function defined by ?(A) All real numbers(B) All real numbers except(C) All real numbers except(D) All real numbers except(E) All real numbers between andExplanationDifficulty: MediumThe correct answer is D.The range of the function defined by is the set of such thatfor some .One way to determine the range of the function defined by is to solve the equation for and then determine which correspond to at least one . To solve for , first subtract from both sides to get and then take the reciprocal of both sides to get . The equation shows that for anyother than , there is an such that , and that there is no such for . Therefore, the range of the function defined by is all real numbers except .Alternatively, one can reason about the possible values of the term . The expression can take on any value except , so the expression can take on any value except . Therefore, the range of the function defined by is all real numbers except .16Your answer Omitted!The table above shows the number of digital cameras that were sold during a three-day sale. The prices of models , , and were , , and , respectively. Which of the following matrix representations gives the total income, in dollars, received from the sale of the cameras for each of the three days?(A)(B)(C)(D)(E)ExplanationDifficulty: MediumThe correct answer is C.A correct matrix representation must have exactly three entries, each of which represents the total income, in dollars, for one of the three days. The total income for Day is given by the arithmetic expression , which is the single entry of the matrix product; in the same way, the total income for Day is given by, the single entry of ; and the total income for Day is given by , the single entry of. Therefore, the matrix representationgives the total income, in dollars, received from the sale of the cameras for each of the three days. Although it is not necessary to compute the matrix product in order to answer the question correctly, equals .17Your answer Omitted!The right circular cone above is sliced horizontally forming two pieces, each of which has the sameheight. What is the ratio of the volume of the smaller piece to the volume of the larger piece?(A)(B)(C)(D)(E)ExplanationDifficulty: HardIt is helpful to label the figure.The top piece is a cone whose height is one-half the height of the original cone . Using the properties of similar right triangles, you should realize the radii of these two cones must be in the same ratio. So if the top cone has radius , the original cone has radius .The volume of the top piece is equal to . The volume of the bottom piece is equal to the volume of the original cone minus the volume of the top piece.The ratio of the volume of the smaller piece to the volume of the larger piece is .18Your answer Omitted!In the figure above, is a regular pentagon with side of length . What is the -coordinate of ?(A)(B)(C)(D)(E)ExplanationDifficulty: HardThe sum of the measures of the interior angles of a regular pentagon is equal to . Each interior angle has a measure of . Using supplementary angles, has a measure of . You can use right triangle trigonometry to find the -coordinate of point .Since , is about . Since the length of each side of the pentagon is , the -coordinate of point is . Putting the information together tells us that the -coordinate of point is . The correct answer is (B).19Your answer Omitted!For a class test, the mean score was , the median score was , and the standard deviation of the scores was . The teacher decided to add points to each score due to a grading error. Which of the following statements must be true for the new scores?I. The new mean score is .II. The new median score is .III. The new standard deviation of the scores is .(A) None(B) only(C) only(D) and only(E) , , andExplanationDifficulty: HardFor this type of question you need to evaluate each statement separately. Statement is true. If you add to each number in a data set, the mean will also increase by . Statement is also true. The relative position of each score will remain the same. Thus, the new median score will be equal to more than the old median score. Statement is false. Since each new score is more than the old score, the spread of the scores and the position of the scores relative to the mean remain the same. Thus, the standard deviation of the new scores is the same as the standard deviation of the old scores.20Your answer Omitted!A game has two spinners. For the first spinner, the probability of landing on blue is . Independently, for the second spinner, the probability of landing on blue is What is the probability that the first spinner lands on blue and the second spinner does not land on blue?(A)(B)(C)(D)(E)ExplanationDifficulty: HardSince the two events are independent, the probability that the first spinner lands on blue and the second spinner does not land on blue is the product of the two probabilities. The first probability is given. Since the probability that the second spinner lands on blue is the probability that thesecond spinner does not land on blue is Therefore, . The correct answer is (E).21Your answer Omitted!In January the world’s population was billion. Assuming a growth rate of percent per year, the world’s population, in billions, for years after can be modeled by theequation . According to the model, the population growth from January to January was(A)(B)(C)(D)(E)ExplanationDifficulty: HardThe correct answer is C.According to the model, the world’s population in January was and in January was . Therefore, according to the model, the population growth from January to January , in billions, was , or equivalently,.22Your answer Omitted!What is the measure of one of the larger angles of a parallelogram in the that has vertices with coordinates , , and ?(A)(B)(C)(D)(E)ExplanationDifficulty: HardThe correct answer is C.First, note that the angle of the parallelogram with vertex is one of the two larger angles of the parallelogram: Looking at the graph of the parallelogram in the makes this apparent. Alternatively, the sides of the angle of the parallelogram with vertex are a horizontal line segment with endpoints and and a line segment of positive slope with endpoints and that intersects the horizontal line segment at its left endpoint , so the angle must measure more than Since the sum of the measures of the four angles of aparallelogram equals , the angle with vertex must be one of the larger angles.One way to determine the measure of the angle of the parallelogram with vertex is to apply the Law of Cosines to the triangle with vertices , , and . The length of the two sides of the angle with vertex are and; the length of the side opposite the angle is . Let represent the angle with vertex and apply the Law of Cosines: , so. Therefore, the measure of one of the larger angles of the parallelogram is .Another way to determine the measure of the angle of the parallelogram with vertex is to consider the triangle , , and . The measure of the angle of this triangle with vertex is less than the measure of the angle of the parallelogram with vertex . The angle of the triangle has opposite side of length and adjacent side of length , so the measure of this angle is . Therefore, the measure of the angle of the parallelogram withvertex is .Yet another way to determine the measure of the angle of the parallelogram with vertex is to use trigonometric relationships to find the measure of one of the smaller angles, and then use the fact that each pair of a larger and smaller angle is a pair of supplementary angles. Consider the angle of the parallelogram with vertex ; this angle coincides with the angle at vertex of the right triangle with vertices at , , and , with opposite side of lengthand adjacent side of length , so the measure of this angle is . This angle, together with the angle of the parallelogram with vertex , form a pair of interior angles on the same side of a transversal that intersects parallel lines, so the sum of the measures of the pair of angles equals . Therefore, the measure of the angle of the parallelogram with vertex is.23Your answer Omitted!For some real number , the first three terms of an arithmetic sequence are, and . What is the numerical value of the fourth term?(A)(B)(C)(D)(E)ExplanationDifficulty: HardThe correct answer is E.To determine the numerical value of the fourth term, first determine the value of and then apply the common difference.Since , and are the first three terms of an arithmetic sequence, it must be true that, that is, Solving for gives . Substituting for in the expressions of the three first terms of the sequence, one sees that they are , , and , respectively. The common difference between consecutive terms for this arithmetic sequence is , and therefore, the fourth term is .24Your answer Omitted!In a group of people, percent have brown eyes. Two people are to be selected at random from the group. What is the probability that neither person selected will have brown eyes?(A)(B)(C)(D)(E)ExplanationDifficulty: HardThe correct answer is A.One way to determine the probability that neither person selected will have brown eyes is to count both the number of ways to choose two people at random from the people who do not have brown eyes and the number of ways to choose two people at random from all people, and then compute the ratio of those two numbers.Since percent of the people have brown eyes, there are people with brown eyes, and people who do not have brown eyes. The number of ways of choosing two people, neither of whom has brown eyes, is : there are ways to choose a first person and ways to choose a second person, but there are ways in which that same pair of people could be chosen. Similarly, the number of ways of choosing two people at random from the people is . Therefore, the probability that neither of the two people selected has brown eyes is.Another way to determine the probability that neither person selected will have brown eyes is to multiply the probability of choosing one of the people who does not have brown eyes at random from the people times the probability of choosing one of the people who does not have brown eyes at random from the remaining people after one of the people who does not have brown eyes has been chosen.Since percent of the people have brown eyes, the probability of choosing one of the people who does not have brown eyes at random from the people is . If one of the people who does not have brown eyes has been chosen, there remain people who do not have brown eyes out of a total of people; the probability of choosing one of the people who does not have brown eyes at random from the people is . Therefore, if two people are to be selected from the group at random, the probability that neither person selected will have brown eyes is .25Your answer Omitted!If , what is ?(A)(B)(C)(D)(E)ExplanationDifficulty: HardThe correct answer is E.One way to determine the value of is to solve the equation for . Since , start with the equation , and cube both sides to get. Isolate to get , and apply the cube root to both sides of the equation to get .Another way to determine the value of is to find a formula for and then evaluate at Let and solve for : cubing both sides gives , so , and. Therefore, , and .26Your answer Omitted!Which of the following equations best models the data in the table above?(A)(B)(C)(D)(E)ExplanationDifficulty: HardThe correct answer is D.One way to determine which of the equations best models the data in the table is to use a calculator that has a statistics mode to compute an exponential regression for the data.The specific steps to be followed depend on the model of calculator, but can be summarized as follows: Enter the statistics mode, edit the list of ordered pairs to include only the four points givenin the table and perform an exponential regression. The coefficients are, approximately, for the constant and for the base, which indicates that the exponential equation is the result of performing the exponential regression. If the calculator reports a correlation, it should be a number that is very close , to which indicates that the data very closely matches the exponential equation. Therefore, of the given models, best fits the data.Alternatively, without using a calculator that has a statistics mode, one can reason about the data given in the table.The data indicates that as increases, increases; thus, options A and B cannot be candidates for such a relationship. Evaluating options C, D and E at shows that option D is the one that gives a value of that is closest to In the same way, evaluating options C, D and E at each of the other given data points shows that option D is a better model for that one data point than either option C or option E. Therefore, is the best of the given models for the data.27Your answer Omitted!The linear regression model above is based on an analysis of nutritional data from 14 varieties of cereal bars to relate the percent of calories from fat to the percent of calories from carbohydrates . Based on this model, which of the following statements must be true?I. There is a positive correlation between and .II. When percent of calories are from fat, the predicted percent of calories from carbohydrates is approximately .III. The slope indicates that as increases by , decreases by .(A) II only(B) I and II only(C) I and III only(D) II and III only(E) I, II, and IIIExplanationDifficulty: HardThe correct answer is D.Statement I is false: Since , high values of are associated with low values of which indicates that there is a negative correlation between and .Statement II is true: When percent of calories are from fat, and the predicted percent of calories from carbohydrates is .Statement III is true: Since the slope of the regression line is , as increases by , increases by ; that is, decreases by .28Your answer Omitted!The number of hours of daylight, , in Hartsville can be modeled by , where is the number of days after March . The day with the greatest number of hours of daylight has how many more daylight hours than May ? (March and May have days each. April and June have days each.)(A) hr(B) hr(C) hr(D) hr(E) hrExplanationDifficulty: HardThe correct answer is A.To determine how many more daylight hours the day with the greatest number of hours of daylight has than May , find the maximum number of daylight hours possible for any day and then subtract from that the number of daylight hours for May .To find the greatest number of daylight hours possible for any day, notice that the expressionis maximized when , which corresponds to , so. However, for this problem, must be a whole number, as it represents a count of days after March . From the shape of the graph of the sine function, either or corresponds to the day with the greatest number of hours of daylight, and since, the expression is maximized when days after March . (It is not required to find the day on which the greatest number of hours of daylight occurs, but it is days after March ,that is, June .)Since May is days after March , the number of hours of daylight for May is .Therefore, the day with the greatest number of hours of daylight hasmore daylight hours than May .。

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2024 SAT考试历年真题数学专题全解2024年SAT考试数学部分依然是考生们最为担心和重视的科目之一。

为了帮助广大考生更好地应对考试，本文将为大家提供全面的2024 SAT考试历年真题数学专题全解。

通过对历年真题的详细解析，希望能够帮助考生们更好地掌握数学知识和解题技巧。

一、整数与小数整数与小数是SAT数学中一个重要的基础知识点。

在解题过程中，考生需要灵活运用整数与小数之间的转换以及四则运算等概念。

在解题过程中，考生应注意以下几点：1.了解整数与小数之间的转换关系。

2.掌握四则运算的基本规则。

3.注意小数位数计算和精确度问题。

二、代数与方程代数与方程是SAT数学中的核心内容之一。

考生需要熟练掌握代数运算的基本规则，灵活运用代数方程知识解题。

在解题过程中，考生应注意以下几点：1.理解代数方程的含义和定义。

2.熟悉代数运算的基本规则。

3.运用代数方程的性质和解题技巧。

三、几何与三角学几何与三角学是SAT数学中的另一个重要内容。

考生需要掌握几何图形的性质和运算规则，灵活运用三角学知识解题。

在解题过程中，考生应注意以下几点：1.掌握几何图形的基本性质和定义。

2.熟练运用三角学的相关概念和运算规则。

3.注意几何图形的变换和投影等问题。

四、数据与统计数据与统计是SAT数学中的重要内容之一。

考生需要了解数据分析和统计学的基本概念，掌握数据处理和统计方法。

在解题过程中，考生应注意以下几点：1.熟悉数据分析和统计学的基本概念。

2.掌握数据处理和统计方法。

3.灵活运用数据与统计知识解题。

五、概率与排列组合概率与排列组合是SAT数学中的难点之一。

考生需要掌握概率和排列组合的基本概念，灵活运用相关知识解题。

在解题过程中，考生应注意以下几点：1.理解概率和排列组合的基本概念。

2.熟悉概率和排列组合的运算规则。

3.注意概率和排列组合在实际问题中的应用。

通过对以上五个数学专题的全面解析与讲解，相信考生们已经对2024 SAT考试数学部分有了更深入的理解与掌握。

如何利用对称性解决问题对称性是自然界中普遍存在的一种现象，它在数学、物理、化学等领域都有广泛的应用。

利用对称性解决问题可以简化计算过程，提高效率，并且有助于发现问题的本质。

本文将介绍如何利用对称性解决问题的方法和技巧。

对称性的定义与分类对称性是指在某种变换下保持不变的性质。

根据变换的类型，对称性可以分为以下几类：空间对称性空间对称性是指在空间变换下保持不变的性质。

常见的空间对称性包括平移对称、旋转对称和镜像对称等。

例如，在几何学中，正方形具有旋转对称和镜像对称，这意味着无论如何旋转或翻转正方形，它都保持不变。

时间对称性时间对称性是指在时间变换下保持不变的性质。

在物理学中，时间对称性是一个重要的概念。

根据物理定律，自然界中的大部分过程在时间上都是可逆的，即过程的演化无论是向前还是向后都是一样的。

这种时间对称性的存在使得我们可以利用对称性简化物理问题的求解过程。

对称群对称群是指保持某种结构不变的所有变换构成的集合。

对称群在数学中有着重要的地位，它可以描述各种几何结构和物理系统的对称性。

例如，正方形的对称群包括四个旋转变换和四个镜像变换，它们组成了一个八元素的群。

利用对称性解决问题的方法利用对称性解决问题可以通过以下几个步骤进行：1. 确定问题中存在的对称性首先，我们需要仔细观察问题，找出其中可能存在的对称性。

这可能涉及到几何结构、物理规律或数学模型等方面。

通过分析问题的特点，我们可以确定问题中存在的对称性类型，并进一步研究其性质和特征。

2. 利用对称性简化问题一旦确定了问题中的对称性，我们可以利用这些对称性来简化问题。

例如，在求解几何问题时，如果问题具有旋转对称性，我们可以选择一个合适的旋转角度，将问题转化为更简单的形式。

类似地，在物理计算中，如果问题具有时间对称性，我们可以利用这一性质简化计算过程。

3. 发现问题的本质通过利用对称性解决问题，我们可以更深入地理解问题的本质。

对称性的存在意味着系统具有某种稳定性和不变性，这对于研究问题的规律和特征非常重要。

2-SAT（⼼累时学习的算法）今年noi考了⼀道2-SAT裸题，害怕今年省选会出到，只能填坑SAT是适定性(Satisfiability)问题的简称。

⼀般形式为k-适定性问题，简称 k-SAT。

当k>2时，k-SAT是NP完全的。

因此⼀般讨论的是k=2的情况，即2-SAT问题。

2-SAT，简单的说就是给出n个集合，每个集合有两个元素，已知若⼲个2-SAT问题现有⼀个由N个布尔值组成的序列A，给出⼀些限制关系，⽐如A[x] AND A[y]=0、A[x] OR A[y] OR A[z]=1等，要确定A[0..N-1]的值，使得其满⾜所有限制关系。

这个称为SAT问题，特别的，若每种限制关系中最多只对两个元素进⾏限制，则称为2-SAT问题。

由于在2-SAT问题中，最多只对两个元素进⾏限制，所以可能的限制关系共有11种：A[x]NOT A[x]A[x] AND A[y]A[x] AND NOT A[y]A[x] OR A[y]A[x] OR NOT A[y]NOT (A[x] AND A[y])NOT (A[x] OR A[y])A[x] XOR A[y]NOT (A[x] XOR A[y])A[x] XOR NOT A[y]注意：这⾥的OR是指两个条件⾄少有⼀个是正确的⽐如x1和x2⼀共有三种组合满⾜“x1为真或x2为假”：x1=1，x2=1x1=1，x2=0x1=0，x2=02-SAT的解决⽅法有很多，由于博主⽐较蒟，所以就选择⼀种简单易懂的介绍⼀下：算法构造⼀个有向图G，每个变量xi拆成两个点2i和2i+1分别表⽰xi为假，xi为真那么对于“xi为真或xj为假”这样的条件我们就需要连接两条边2*i —>2*j(表⽰如果i为假，那么j必须是假)2*j+1—>2*i+1(表⽰如果j为真，那么i必须是真)这就有点像推导的过程实际上每⼀个限制条件都会对应两条“对称”的边接下来逐考虑每个没有赋值的变量，设为x我们先假定x为假（为什么⼀定是假，约定俗成了）之后沿着从ta出发的有向边进⾏标记如果在标记过程中，发现有⼀个点的两种状态都被标记过了那么我们之前的假设就被推翻了需要改成x为真，重新标记如果发现⽆论这个点赋值成真还是假，都会引起⽭盾可以证明这个2-SAT⽆解可能我的叙述有点容易让读者yy但是⼀定要注意：这个算法没有回溯过程下⾯给出代码：struct TwoSAT{int n;vector<int> G[N*2];bool mark[N*2];int S[N*2],c;int dfs(int x){if (mark[x^1]) return 0;if (mark[x]) return 1; //和假设的值⼀样mark[x]=1;S[c++]=x;for (int i=0;i<G[x].size;i++)if (!dfs(G[x][i])) return 0;return 1;}//x=xval or y=yvalvoid add_clause(int x,int xv,int y,int yv){x=x*2+xv;y=y*2+yv;G[x^1].push_back(y);G[y^1].push_back(x);}void init(int n){this->n=n;for (int i=0;i<2*n;i++) G[i].clear();memset(mark,0,sizeof(mark));}bool solve(){for (int i=0;i<2*n;i+=2) //枚举每⼀个点if (!mark[i]&&!mark[i+1]) //没有标记{c=0;if (!dfs(i)){while (c>0) mark[S[--c]]=0; //清空标记 if (!dfs(i+1)) return 0;}}return 1;}};。

MAX2SAT近似算法一、引言MAX2SAT问题是一种典型的NP困难问题，它是2SAT问题的扩展。

在2SAT问题中，给定一个包含n个变量和m个约束的集合，每个约束都是一个包含2个变量的布尔表达式，目标是确定这些变量的值，使得尽可能多的约束得到满足。

由于2SAT问题是NP困难的，MAX2SAT问题同样也是NP困难的。

在实际应用中，我们经常需要处理大规模的MAX2SAT问题，而这些问题的精确解通常是不可行的或成本高昂的。

因此，研究和设计近似算法来近似地解决MAX2SAT问题是具有重要的理论和应用价值。

二、近似算法的基本概念近似算法是一种可以近似解决NP困难问题的算法。

这类算法通常不能保证找到最优解，但可以在多项式时间内找到一个近似最优解，其性能接近于最优解。

近似算法的性能通常使用一个性能比率来度量，它表示算法找到的解与最优解的比值。

如果一个近似算法的性能比率接近于1，则认为该算法的性能较好。

三、常见的Max2SAT近似算法1.贪心算法：贪心算法是一种常见的近似算法，它在每一步选择中都采取当前状态下最好或最优（即最有利）的选择，从而希望导致结果是最好或最优的。

在MAX2SAT问题中，贪心算法通常从约束集合中选择一个或多个满足度最高的约束进行满足，直到无法再满足更多的约束为止。

这种算法的时间复杂度为O(n^3)，其中n是变量的数量。

2.遗传算法：遗传算法是一种基于生物进化原理的优化算法，它通过模拟自然选择和遗传机制来寻找最优解。

在MAX2SAT问题中，遗传算法通常将约束表示为个体，并通过选择、交叉和变异等操作来产生新的个体，逐步接近最优解。

遗传算法的性能取决于许多参数的选择和设置，但通常来说，它在处理大规模MAX2SAT问题时表现较好。

3.模拟退火算法：模拟退火算法是一种基于物理退火过程的优化算法，它通过随机地接受或拒绝状态转移来寻找最优解。

在MAX2SAT问题中，模拟退火算法通常从一个随机解开始，通过不断接受或拒绝满足更多约束的状态转移来逐步接近最优解。

2.21 某办公楼工程地质勘探中取原状土做试验。

用天平称50cm3湿土质量为，烘干后质量为，土粒比重为。

计算此土样的天然密度、干密度、饱和密度、天然含水率、孔隙比、孔隙率以及饱和度。

【解】m，m s，m w，V = 50.0 cm3，d s。

V1.0) = 28.1 cm3s取g = 10 m/s2，那么V w = 20.1 cm3V= 50.0 - 28.1 = 21.9 cm3vV= 50.0 – 28.1 – 20.1 = 1.8 cm3a于是，= m / V3= m s / V3d= (m s +w V v)/ V = (75.05 + 1.0 3s a tw = m/ m s = 20.1 / 75.05 = 0.268 = 26.8%we = V/ V svn = V/ V = 21.9 / 50 = 0.438 = 43.8%vS= V w / V vr2.22 一厂房地基表层为杂填土，厚，第二层为粘性土，厚5m，地下水位深。

在粘性土中部取土样做试验，测得天然密度3，土粒比重为。

计算此土样的天然含水率w、干密度d、孔隙比e和孔隙率n。

【解】依题意知，S r，s a t= 3。

由，得n = e /(1 + eg/cm3。

3，含水率w= 19.3%，2.23 某宾馆地基土的试验中，已测得土样的干密度d土粒比重为。

计算土的孔隙比e、孔隙率n和饱和度S r。

又测得该土样的液限与塑限含水率分别为w L = 28.3%，w p = 16.7%。

计算塑性指数I p和液性指数I L，并描述土的物理状态，为该土定名。

【解】〔1〕 =(1 + w) = 1.54 3dn = e /(1 + e〔2〕I p = w L - w pIL= (w L - w) / I p = (28.3 – 19.3)/11.6 = 0.7760.75 < I L < 1，那么该土样的物理状态为软塑。

由于10 < I p < 17，那么该土应定名为粉质粘土。

SAT考试2024数学历年题目精讲在本篇文章中，我们将重点讲解SAT考试2024年数学部分的历年题目。

我们将按照题目类型进行分类，并为每个题型提供详细的解答和解题技巧，帮助考生更好地应对这些题目。

一、单选题1. 题目描述：某汽车展厅共展出了150辆汽车，其中的三分之一是SUV车型，四分之一是轿车车型，其余的是其他车型。

问展厅中轿车车型的数量是多少？解答与技巧：首先，计算出SUV车型的数量：150 * (1/3) = 50辆。

然后，计算出其他车型的数量：150 - 50 - 150 * (1/4) = 50辆。

所以，轿车车型的数量是50辆。

2. 题目描述：某商场举办了一次打折活动，原价100元的商品现在只需80元购买。

如果小明购买了3件该商品，他需要支付多少钱？解答与技巧：首先，计算出每件商品的折扣金额：100 - 80 = 20元。

然后，计算出小明需要支付的金额：3 * 20 = 60元。

所以，小明需要支付60元。

二、多选题1. 题目描述：以下哪些数是正整数？（A）-1（B）0（C）1（D）2解答与技巧：在SAT考试中，如果题目要求选择多个选项，我们需要仔细审题。

在这个题目中，需要选择正整数，所以选项B和A都不是正整数。

所以正确答案是（C）和（D）。

2. 题目描述：以下哪些图形具有对称性？（A）正方形（B）长方形（C）圆形（D）三角形解答与技巧：我们需要判断每个选项是否具有对称性。

在这个题目中，正方形和圆形都具有对称轴，所以正确答案是（A）和（C）。

三、填空题1. 题目描述：若a + a^-1 = 5，求a^2 + a^-2的值。

解答与技巧：首先，我们可以对等式两边进行平方操作，得到a^2+ 2 + a^(-2) = 25。

然后，我们需要解方程，将等式左边与右边的常数项进行抵消，得到a^2 + a^(-2) = 23。

2. 题目描述：某比赛共有10个选手参加，其中3个选手退出比赛，剩余的选手中将决出第一名、第二名和第三名。