Development history
Hope Cup Mathematics Invitational Tournament has been held for more than ten times since 1990. 10 For many years, the organizers have insisted that the contest is open to most schools and students. Every link from proposition, award to organization revolves around one purpose: to stimulate middle school students' interest in learning, cultivate their self-confidence and constantly improve their ability and quality. This activity only involves Grade One, Grade Two, Grade One and Grade Two, not Grade Three and Grade Three. It is not repeated with the Olympic Games, linked with the senior high school entrance examination and college entrance examination, and does not increase the burden on teachers and students, so it is welcomed by teachers and students. The competition was affirmed by the former State Education Commission and included in the list of national competitions approved by the former State Education Commission. At the same time, more and more mathematicians and mathematicians give enthusiastic care and support to the invitational tournament. By the tenth session, there were more than 500 participating cities and the total number of students reached 5.98 million. "Hope Cup" national mathematics invitational tournament has become one of the largest and most influential extracurricular activities among middle school students.
Edit the competition system of this paragraph
The competition is divided into two parts. The preliminary examination (conducted in March every year) is organized by all localities (provinces, cities, counties, [districts] and schools) and conducted at the same time in all participating schools in the country. Each test point is graded and graded according to the grading standard issued by the proposition Committee, and the players who take part in the second test are selected according to the proportion of one seventh. The second test (conducted in May every year) is organized by the local branch of the Editorial Committee of Mathematics and Physics, or by the teaching and research offices or education colleges, teaching institutes and teachers' further education schools in various cities. After the examination, the examination papers will be sealed in each test center, sent to the organizing committee after registration, and marked by the proposition committee, from which the first, second and third prizes will be awarded according to the results, and gold, silver and bronze prizes and award certificates will be awarded respectively. The organizing committee will award the "Hope Cup" prize for organizing the mathematical invitational tournament to the regions or schools that have done a good job in organizing it.
Edit the impact of this event.
The Japanese Mathematical Olympic Committee is very concerned about this competition. Mr. Wakayama Rongji, Director of the Committee Affairs Bureau, made a special trip to China to discuss the exhibition with the organizing committee of the invitational tournament. Since 1996, some Japanese middle school students have been organized to participate in the competition for three consecutive years, which is the first time for chinese social organizations to hold similar competitions abroad. In recent years, relevant organizations in the United States and Germany have also contacted and cooperated with the Organizing Committee. A middle school student who has signed up for the Hope Cup Mathematics Invitational Tournament said, "After signing up, I have already started reviewing for the exam. Through preparing for the exam, I found that my problem-solving ability has improved and my interest in mathematics has increased. " The organizer said that the test questions of Hope Cup are very regular and the knowledge points of the test are impartial, which is very beneficial for students who are not necessarily gifted in mathematics but study hard. [ 1]
It is illegal to edit this paragraph.
20 1 1,110/in October, the Municipal Education Commission issued the Notice on Prohibiting Students from Participating in Subject Competitions in Compulsory Education, pointing out that some non-governmental organizations and groups have recently organized city-wide subject competitions for schools and students in the compulsory education stage in the name of holding the "Hope Cup" math competition. At the same time, parents are misled by various means that excellent students will be selected from the winners of such competitions and recommended to higher-level schools, which leads many parents to blindly register for the exam. The head of the Organizing Committee of Hope Cup Beijing Division said that the students have been refunded according to the requirements of the Municipal Education Commission. 2011165438+1On October 26th, Zhou Guozhen, head of the National Organizing Committee of Hope Cup, said that the organizing right of Beijing Organizing Committee 20 12 had been revoked, and the illegal behaviors of the organizing committee had been investigated. Zhou Guozhen said that the National Organizing Committee will conduct a detailed investigation on the Beijing Organizing Committee from two aspects. First, whether the Beijing Organizing Committee charges high fees; Secondly, have you used the Hope Cup platform for other profitable activities? "If the investigation results show that it is indeed illegal, the organizing committee of the competition area will deal with it, and the worst result is to disqualify it from hosting the event in Beijing." Zhou Guozhen said that the 20 12 competition in Beijing will be held in March and April as scheduled, just like other national competitions. The National Organizing Committee has entrusted Qixing and other training institutions to undertake the registration work, and the National Organizing Committee will directly supervise the competition of the 20 12 Beijing Division.
The minimum value of. The function on is (c)
A.0 B. 1 C.2 D.3
[Solution] Therefore, when
If and only if the above equations are equal and this equation has a solution, the minimum value in is 2.
2. Let, if, then the range of real numbers is (d)
A.B. C. D。
[Solution] Because there are two real roots.
, ,
So it's equivalent to and, that is
Besides,
Get a solution.
3. When Party A and Party B play table tennis, it is agreed that the winner will get 1 point and the loser will get 0 point. When one of them has 2 points more than the other or has played 6 rounds, the game will stop. Assuming that the probability of Party A winning each game is 0, the probability of Party B winning each game is 0, and the outcome of each game is independent, then the expected number of games when the game stops is (b).
A.B. C. D。
[Answer 1] According to the meaning of the question, all possible values of are 2, 4 and 6.
Let every two games be a round, then the probability that the game will stop at the end of the round is
.
If the game will continue at the end of this round, both parties must score one point in this round. At this point, the result of this round has no effect on whether the next round of competition will stop.
,
,
,
Therefore.
[Solution 2] According to the meaning of the question, all possible values of are 2, 4 and 6.
Order means that A wins in the first game and B wins in the second game.
Through independence and mutual incompatibility
,
,
,
Therefore.
4. If the sum of the surface areas (unit: cm) of a cube whose three sides are integers is 564 cm2, the sum of the volumes of these three cubes is (a).
A.764 cubic centimeters or 586 cubic centimeters.
C.586 cubic centimeters or 564 cubic centimeters deep 586 cubic centimeters
[Solution] If the side lengths of these three cubes are respectively 0, then there is, and it may be set, so, therefore, we can only take 9, 8, 7 and 6.
If, then, it is easy to know and get a solution.
If, then, but, therefore, it is still 5. If, then no solution, if, then no solution. At this time, there is no solution.
If there is only one solution.
If, then, at this time, therefore, but, therefore, there is no solution at this time.
To sum up, * * * has two solutions or.
The volume is cm3 or cm3.
5. The number of rational number solutions of the equation is (B)
A. 1 B. 2 C. 3 D. 4
[Solution] If, then the solution is or.
If there is, you will get it.
Jed. ②
Replace ③ with ②.
From ① to ③, simplify.
It is easy to know that there is no rational number root, so it is contradictory to get from ① and ②, so the system of equations * * * has two rational number solutions or.
6. If the opposite side of the inner angle is a geometric series, the range of values is
(3)
A.B.
C.D.
[Solution] Set the common ratio, and then, and
.
Therefore, only a range of values is needed.
Because the largest side of a geometric series can only be or, one and only one inequality group is needed to form three sides of a triangle.
that is
solve
Therefore, the range of values sought is.
2. Fill in the blanks (the full score of this question is 54 points, and each small question is 9 points)
7. Let, where is a real number,,, If, then 5.
[Solution] Judging from the meaning of the question,
,
Therefore, from …
8. Then, set the minimum value to.
[Solution]
,
(1), the minimum value;
(2) When the minimum value is 1;
(3), the minimum value.
Or, the minimum value of cannot be,
So, solve (give up).
9. If 24 volunteer places are allocated to 3 schools, each school will have at least one place, and there are 222 different allocation methods.
[Solution 1] Use the gap between four sticks to represent three schools, and use the number of delegates. take for example
It means that the first, second and third schools have 4 18 and 2 places respectively.
If each ""and each ""are regarded as a position, because the left and right ends must be | ",different distribution methods are equivalent to an" occupation method "in which a position (excluding both ends) is occupied by two|".
The division of "at least one quota per school" is equivalent to selecting two gaps from 23 gaps among 24 ""and inserting "|", so there are two kinds.
In the method of "at least one quota per school", the allocation method of "at least two schools have the same quota" is 3 1.
To sum up, there are 253-3 1 = 222 eligible distribution modes.
[Solution 2] If the number of places allocated to three schools is 0, then the score of at least one place in each school is an indefinite equation.
.
The number of positive integer solutions of, that is, the number of non-negative integer solutions of the equation is equal to 2 1 element recombination from three different elements:
.
In the method of "at least one quota per school", the allocation method of "at least two schools have the same quota" is 3 1.
To sum up, there are 253-3 1 = 222 eligible distribution modes.
10. Let the sum of the first few terms of the series satisfy:, then the general term =.
[Solution],
Namely 2
= ,
This leads to 2.
Order, (),
Yes, so, so.
1 1. Let the function be defined in, if and for any, satisfy.
, then =
[Solution 1] We know from the conditions of the topic.
,
Therefore, there is, therefore.
.
[Solution 2] So, the order
,
,
That is to say,
Therefore,
It must be a periodic function with a period of 2,
So ...
12. A ball with a radius of 1 can move freely in all directions in a regular tetrahedron container with an inner wall length of, so the inner wall area of the container that the ball can never touch is.
[Solution] If the answer is 12, figure 1, consider the situation that the ball is squeezed into the corner. Let the radius of the ball be//plane and tangent to this point, then the center of the ball is the center of the regular tetrahedron and the vertical foot is the center.
because
,
Therefore, therefore.
Remember that the tangent point of the ball and the surface is connected at this time, then
.
Considering that the ball is tangent to a surface of a regular tetrahedron, it is easy to know that the trajectory of the tangent point of the ball closest to the edge on the surface is still a regular triangle, which is recorded as, for example, the answer 12 Figure 2. Record regular tetrahedron.
The side length of is too long.
Because, yes, the side length of a small triangle.
The area of the part of the ball that does not touch the surface is (for example, the shaded part in Figure 2 in the answer 12).
.
Again, so
.
By symmetry and regular tetrahedron * * *, the area of the inner wall of the container that the ball can't touch is * * *.
Third, solve the problem (the full score of this question is 60 points, and each small question is 20 points)
13. It is known that there are only three intersections between the image of the function and the straight line, and the maximum abscissa of the intersection is, so it is verified that:
.
The three intersections of the ID image and the straight line are shown in the answer 13, and they are tangent inside, and the tangent point is.
... five points
Because, so, that's ... 10.
therefore
... 15 point
... 20 points
14. Solving inequalities
.
[Solution 1] Because there is increasing function in the world, the original inequality is equivalent to
.
That is ... five points.
Grouping decomposition
, … 10 integral
So,
... 15 point
So, this is either.
Therefore, the solution set of the original inequality is ... 20 points.
[Solution 2] Because there is increasing function in the world, the original inequality is equivalent to
... five points
that is
,
, … 10 integral
Order, inequality is
,
Obviously, it is an increasing function in the world, so the above inequality is equivalent to
, … 15 point
The solution (discard),
Therefore, the solution set of the original inequality is ... 20 points.
15. As shown in figure 15, it is the moving point on a parabola, the point is on the axis, and the circle is inscribed, so as to find the minimum area.
[Solution] Set, you might as well set.
Linear equation:
Simplify it.
The distance from the center of the circle is 1,
, ... 5 points
Therefore,
It is easy to know that the above formula is simplified,
Similarly, there are ... 10 points.
So, then,
.
Because it is a point on a parabola, if there is, then
, ... 15 o'clock
therefore
.
When, the above formula takes the equal sign, at this time.
So the minimum value is 8.
"Qian jiao, you go. Com "and look for it. I remember there were many questions about the Hope Cup.
You can find it in the "competition questions of various subjects", which is free of charge.
Come on, it's good to participate in the Hope Cup.