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高中数学竞赛 历届IMO竞赛试题(1-46届完整中文版).doc

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1、According to Office Party of mass line education practice activities led group of unified arrangements, units main led to for cadres workers Shang a times party lecture, due to himself level limited, only put this stage concentrated learning of experience today and everyone common with learning exch

2、ange, purpose is let we further deep understanding mass line of connotation, insisted party of mass line, do masses work, following main from four a aspects told: a, and full awareness party of mass line education practice activities of major meaning Partys 18 major distinct proposed to focus on the

3、 partys advanced nature and purity, in the party, with “pragmatic and clean for the people“ as the main content of the partys mass line and educational practice. This is comrade XI Jinping as party General Secretary and adhere to strictly administering the party, strengthen the partys major decision

4、 is in line with the public expectations and strengthening the construction of learning-oriented service innovation of Marxist partys major deployment, is the important measure to advance the great cause of socialism with Chinese characteristics. First, carry out the partys mass line in education pr

5、actice, was carrying 18 of the party spirit, summon a powerful force for realizing the great rejuvenation of the Chinese nation the Chinese dream of urgent needs. Partys 18 to promote strategic deployment to building socialism with Chinese characteristics, presents two 100-year goal, namely, by 2020

6、 Chinas gross domestic product and per capita income of urban and rural residents in 2010 on the basis of double complete the building of a well-off society; the middle of this century into a prosperous, democratic, civilized and harmonious Socialist modern country, and on this basis, the Chinese dr

7、eam of realizing the great rejuvenation of the Chinese nation. 18 Congress of the party, Comrade XI Jinping repeatedly expounded the Chinese dream. The basic connotation of the Chinese dream, is the national prosperity, national revitalization, the happiness of the people, China important to follow

8、is to keep China Road, carry forward the spirit and cohesion strength of China, dream dreams of the people of China, but determined confidence, enhance self-awareness, achieve self-reliance, to build a stronger China, civilization, harmony, beauty, China, China, China. Strategic conception of China

9、was 18 the spirit of the enrichment and development of the party, conscientiously study the Chinese dream, leading people to hard to realize the Chinese dream, mass education activities in depth the fundamental starting point and ending point. Strengthen the education of the partys mass line, helps

10、to train the collective consciousness of the unity of the party and the people, playing the Chinese dream ideological bases. Comrade Mao Zedong stressed that parties should have “common language“ socialist countries must have “unity of will“. History hasproved that a people and a nation, if it does

11、not have its own spiritual pillar, there is no unity of spiritual support, it means that no soul, will lose cohesion and vitality. Figuratively speaking, a sack of potatoes, quantity cannot be too a team, form a joint force; only like granite, so closely United to form invincible, no difficult means

12、 of strength. National rejuvenation of Chinese dream, not only embodies the common interests and pursuits, and covers a variety of groups and classes, has a wide applicability and inclusive, with a strong integrating force and lead the force. She embodies the aspirations of several generations of Ch

13、inese, reflecting today . Into sex and purity, and the inevitable requirement of constantly improving the partys governing capacity. Strengthening the construction of the advanced nature and purity, the core problem is always maintaining the partys flesh-and-blood ties with the masses, so that our p

14、arty will always be the most wholehearted support of the masses. Must take the fundamental interests of the overwhelming majority of the people as the party all the starting point and ending point, the party has always been a common fate with the masses, this is the magic weapon for our party to alw

15、ays maintain the advanced nature and purity. Strengthening the construction of partys advanced nature and purity and enhancing the partys ruling ability will eventually be implemented to realize, safeguard and develop the fundamental interests of the overwhelming majority of the people. All the work

16、 merits this is a measure of the partys fundamental standards, is a measure of the partys advanced nature and purity standards. Believe who, depending on who, who, standing on the position of the overwhelming majority of the people are always, always realize, safeguard and develop the fundamental in

17、terests of the overwhelming majority of the people, it is a touchstone of the Marxist party of judgement, is also sign of Marxist party is different from other political parties. Marxist political party has a clear political position and historical mission: for the interests of the vast majority of

18、people, committed to the fundamental interests of the overwhelming majority of the people. From the day of the establishment of the Chinese Communist Party, always adhere to basic principles of historical materialism, for the interests of the most people as their goal, always serving as its fundamen

19、tal purpose and values. Fully trust the masses, closely rely on the masses maintain flesh-and-blood ties with the masses is always the source of our party full of vigor and vitality, is always the most determined in the development of our party and our country. Conscientiously accept supervision by

20、the masses, pay attention to listen to the masses assessment, which itself was the partys advanced nature and purity into practical action to realize the fundamental interests of the overwhelming majority of the people. At present, the party committees and leading cadres and the Government generally

21、 value the close relations with the masses, do a lot of productive work, but there are still some第 1 届 IMO1. 求证(21n+4)/(14n+3) 对每个自然数 n 都是最简分数。 2. 设(x+(2x-1)+(x-(2x-1)=A,试在以下 3 种情况下分别求出 x 的实数解:(a) A=2;(b)A=1;(c)A=2。 3. a、b、c 都是实数,已知 cos x 的二次方程 a cos2x + b cos x + c = 0,试用 a,b,c 作出一个关于 cos 2x 的二次方程,

22、使它的根与原来的方程一样。当a=4,b=2,c=-1 时比较 cos x 和 cos 2x 的方程式。4. 试作一直角三角形使其斜边为已知的 c,斜边上的中线是两直角边的几何平均值。 5. 在线段 AB 上任意选取一点 M,在 AB 的同一侧分别以 AM、MB 为底作正方形AMCD、MBEF,这两个正方形的外接圆的圆心分别是 P、Q,设这两个外接圆又交于M、N, (a.) 求证 AF、BC 相交于 N 点;(b.) 求证 不论点 M 如何选取 直线 MN 都通过一定点 S;(c.) 当 M 在 A 与 B 之间变动时,求线断 PQ 的中点的轨迹。6. 两个平面 P、Q 交于一线 p,A 为 p

23、 上给定一点,C 为 Q 上给定一点,并且这两点都不在直线 p 上。试作一等腰梯形 ABCD(AB 平行于 CD) ,使得它有一个内切圆,并且顶点 B、D 分别落在平面 P 和 Q 上。 第 2 届 IMO1. 找出所有具有下列性质的三位数 N:N 能被 11 整除且 N/11 等于 N 的各位数字的平方和。 According to Office Party of mass line education practice activities led group of unified arrangements, units main led to for cadres workers Sha

24、ng a times party lecture, due to himself level limited, only put this stage concentrated learning of experience today and everyone common with learning exchange, purpose is let we further deep understanding mass line of connotation, insisted party of mass line, do masses work, following main from fo

25、ur a aspects told: a, and full awareness party of mass line education practice activities of major meaning Partys 18 major distinct proposed to focus on the partys advanced nature and purity, in the party, with “pragmatic and clean for the people“ as the main content of the partys mass line and educ

26、ational practice. This is comrade XI Jinping as party General Secretary and adhere to strictly administering the party, strengthen the partys major decision is in line with the public expectations and strengthening the construction of learning-oriented service innovation of Marxist partys major depl

27、oyment, is the important measure to advance the great cause of socialism with Chinese characteristics. First, carry out the partys mass line in education practice, was carrying 18 of the party spirit, summon a powerful force for realizing the great rejuvenation of the Chinese nation the Chinese drea

28、m of urgent needs. Partys 18 to promote strategic deployment to building socialism with Chinese characteristics, presents two 100-year goal, namely, by 2020 Chinas gross domestic product and per capita income of urban and rural residents in 2010 on the basis of double complete the building of a well

29、-off society; the middle of this century into a prosperous, democratic, civilized and harmonious Socialist modern country, and on this basis, the Chinese dream of realizing the great rejuvenation of the Chinese nation. 18 Congress of the party, Comrade XI Jinping repeatedly expounded the Chinese dre

30、am. The basic connotation of the Chinese dream, is the national prosperity, national revitalization, the happiness of the people, China important to follow is to keep China Road, carry forward the spirit and cohesion strength of China, dream dreams of the people of China, but determined confidence,

31、enhance self-awareness, achieve self-reliance, to build a stronger China, civilization, harmony, beauty, China, China, China. Strategic conception of China was 18 the spirit of the enrichment and development of the party, conscientiously study the Chinese dream, leading people to hard to realize the

32、 Chinese dream, mass education activities in depth the fundamental starting point and ending point. Strengthen the education of the partys mass line, helps to train the collective consciousness of the unity of the party and the people, playing the Chinese dream ideological bases. Comrade Mao Zedong

33、stressed that parties should have “common language“ socialist countries must have “unity of will“. History hasproved that a people and a nation, if it does not have its own spiritual pillar, there is no unity of spiritual support, it means that no soul, will lose cohesion and vitality. Figuratively

34、speaking, a sack of potatoes, quantity cannot be too a team, form a joint force; only like granite, so closely United to form invincible, no difficult means of strength. National rejuvenation of Chinese dream, not only embodies the common interests and pursuits, and covers a variety of groups and cl

35、asses, has a wide applicability and inclusive, with a strong integrating force and lead the force. She embodies the aspirations of several generations of Chinese, reflecting today . Into sex and purity, and the inevitable requirement of constantly improving the partys governing capacity. Strengtheni

36、ng the construction of the advanced nature and purity, the core problem is always maintaining the partys flesh-and-blood ties with the masses, so that our party will always be the most wholehearted support of the masses. Must take the fundamental interests of the overwhelming majority of the people

37、as the party all the starting point and ending point, the party has always been a common fate with the masses, this is the magic weapon for our party to always maintain the advanced nature and purity. Strengthening the construction of partys advanced nature and purity and enhancing the partys ruling

38、 ability will eventually be implemented to realize, safeguard and develop the fundamental interests of the overwhelming majority of the people. All the work merits this is a measure of the partys fundamental standards, is a measure of the partys advanced nature and purity standards. Believe who, dep

39、ending on who, who, standing on the position of the overwhelming majority of the people are always, always realize, safeguard and develop the fundamental interests of the overwhelming majority of the people, it is a touchstone of the Marxist party of judgement, is also sign of Marxist party is diffe

40、rent from other political parties. Marxist political party has a clear political position and historical mission: for the interests of the vast majority of people, committed to the fundamental interests of the overwhelming majority of the people. From the day of the establishment of the Chinese Comm

41、unist Party, always adhere to basic principles of historical materialism, for the interests of the most people as their goal, always serving as its fundamental purpose and values. Fully trust the masses, closely rely on the masses maintain flesh-and-blood ties with the masses is always the source of

42、 our party full of vigor and vitality, is always the most determined in the development of our party and our country. Conscientiously accept supervision by the masses, pay attention to listen to the masses assessment, which itself was the partys advanced nature and purity into practical action to re

43、alize the fundamental interests of the overwhelming majority of the people. At present, the party committees and leading cadres and the Government generally value the close relations with the masses, do a lot of productive work, but there are still some2. 寻找使下式成立的实数 x: 4x2/(1 - (1 + 2x) 2 = 43 A. 并求

44、出等号何时成立。 3. 解方程 cos nx - sinnx = 1, 其中 n 是一个自然数。 4. P 是三角形 ABC 内部一点,PA 交 BC 于 D,PB 交 AC 于 E,PC 交 AB 于 F,求证AP/PD, BP/PE, CP/PF 中至少有一个不大于 2,也至少有一个不小于 2。 5. 作三角形 ABC 使得 AC=b, AB=c,锐角 AMB = ,其中 M 是线断 BC 的中点。求证这个三角形存在的充要条件是 b tan(/2) 1/2.3. 正方体 ABCDABCD(ABCD、ABCD分别是上下底) 。一点 x 沿着正方形 ABCD的边界以方向 ABCDA 作匀速运动

45、;一点 Y 以同样的速度沿着正方形 BCCB 的边界以方向 BCCBB运动。点 X、Y 在同一时刻分别从点 A、B开始运动。求线断 XY 的中点的轨迹。 4. 解方程 cos2x + cos22x + cos23x = 1。 5. 在圆 K 上有三个不同的点 A、B、C。试在 K 上再作出一点 D 使得这四点所形成的四According to Office Party of mass line education practice activities led group of unified arrangements, units main led to for cadres workers

46、 Shang a times party lecture, due to himself level limited, only put this stage concentrated learning of experience today and everyone common with learning exchange, purpose is let we further deep understanding mass line of connotation, insisted party of mass line, do masses work, following main fro

47、m four a aspects told: a, and full awareness party of mass line education practice activities of major meaning Partys 18 major distinct proposed to focus on the partys advanced nature and purity, in the party, with “pragmatic and clean for the people“ as the main content of the partys mass line and

48、educational practice. This is comrade XI Jinping as party General Secretary and adhere to strictly administering the party, strengthen the partys major decision is in line with the public expectations and strengthening the construction of learning-oriented service innovation of Marxist partys major

49、deployment, is the important measure to advance the great cause of socialism with Chinese characteristics. First, carry out the partys mass line in education practice, was carrying 18 of the party spirit, summon a powerful force for realizing the great rejuvenation of the Chinese nation the Chinese dream of urgent needs. Partys 18 to promote strategic deployment to building socialism with Chinese characteristics, presents two 100-year goal, namely, by 2020 Chinas gross domestic product and per capita income of urban

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