全国中学生数学竞赛真题解析

全国中学生数学竞赛真题解析考试时长：120分钟满分：100分班级：__________姓名：__________学号：__________得分：__________一、单选题（总共10题，每题2分，总分20分）1.若函数f(x)=ax^2+bx+c在x=1处取得极小值，且f(1)=2，则f(2)的值为（）A.3B.4C.5D.62.已知集合A={x|x^2-3x+2=0}，B={x|ax=1}，若A∪B=A，则实数a的取值集合为（）A.{1}B.{1,0}C.{0}D.{1,-1}3.在△ABC中，角A、B、C的对边分别为a、b、c，若a=3，b=4，c=5，则cosB的值为（）A.1/2B.3/4C.4/5D.3/54.已知数列{a_n}的前n项和为S_n，且a_1=1，a_n=2a_{n-1}+1（n≥2），则a_5的值为（）A.31B.32C.33D.345.抛掷两枚均匀的骰子，记事件A为“点数之和为7”，事件B为“点数之和为偶数”，则P(B|A)的值为（）A.1/6B.1/3C.1/2D.2/36.已知直线l：y=kx+1与圆C：x^2+y^2-2x+4y-3=0相切，则k的值为（）A.-1B.1C.-2D.27.函数f(x)=|x-1|+|x+2|的最小值为（）A.1B.2C.3D.48.在等比数列{a_n}中，a_1=2，公比为q，若a_4+a_5=72，则q的值为（）A.2B.3C.-2D.-39.已知点P(x,y)在直线x+y=4上，则|OP|（O为原点）的最小值为（）A.2√2B.2√3C.4√2D.4√310.若函数f(x)=sin(x+α)在x=0处取得最大值，则α的值为（）A.π/2B.-π/2C.kπ+π/2（k∈Z）D.kπ-π/2（k∈Z）二、填空题（总共10题，每题2分，总分20分）11.若x^2+px+q=0的两根为α、β，且α^2+β^2=10，αβ=3，则p+q的值为_________。12.在△ABC中，角A、B、C的对边分别为a、b、c，若a=2，b=√7，c=3，则sinC的值为_________。13.已知数列{a_n}的前n项和为S_n，且a_1=1，a_n=a_{n-1}+2n（n≥2），则a_4的值为_________。14.抛掷三枚均匀的硬币，则恰好出现2次正面的概率为_________。15.已知直线l：y=kx+1与圆C：x^2+y^2-2x+4y-3=0相切，则k^2的值为_________。16.函数f(x)=|x-1|+|x+2|的图像与x轴围成的面积为_________。17.在等比数列{a_n}中，a_1=2，公比为q，若a_4+a_5=72，则q^2的值为_________。18.已知点P(x,y)在直线x+y=4上，则|OP|（O为原点）的最大值为_________。19.若函数f(x)=sin(x+α)在x=0处取得最大值，则sinα的值为_________。20.已知函数f(x)=x^3-3x^2+2，则f(x)的极小值为_________。三、判断题（总共10题，每题2分，总分20分）21.若a>b，则a^2>b^2。22.在等差数列{a_n}中，若a_1=1，d=2，则a_100=199。23.若函数f(x)=x^2+ax+b在x=1处取得极小值，则a=-2，b=1。24.在△ABC中，若a=3，b=4，c=5，则△ABC为直角三角形。25.若事件A、B互斥，则P(A∪B)=P(A)+P(B)。26.在等比数列{a_n}中，若a_1=1，公比为q，则a_n=q^{n-1}。27.若直线l：y=kx+1与圆C：x^2+y^2-2x+4y-3=0相切，则k=±1。28.函数f(x)=|x-1|+|x+2|的最小值为3。29.在等差数列{a_n}中，若a_1=1，d=2，则a_100=199。30.若函数f(x)=sin(x+α)在x=0处取得最大值，则α=kπ+π/2（k∈Z）。四、简答题（总共4题，每题4分，总分16分）31.已知函数f(x)=x^3-3x^2+2，求f(x)的极值点。32.在△ABC中，角A、B、C的对边分别为a、b、c，若a=3，b=4，c=5，求cosA的值。33.已知数列{a_n}的前n项和为S_n，且a_1=1，a_n=a_{n-1}+2n（n≥2），求a_4的值。34.已知直线l：y=kx+1与圆C：x^2+y^2-2x+4y-3=0相切，求k的值。五、应用题（总共4题，每题6分，总分24分）35.在△ABC中，角A、B、C的对边分别为a、b、c，若a=3，b=4，c=5，求△ABC的面积。36.已知数列{a_n}的前n项和为S_n，且a_1=1，a_n=a_{n-1}+2n（n≥2），求S_4的值。37.抛掷两枚均匀的骰子，求点数之和为7的概率。38.已知函数f(x)=|x-1|+|x+2|，求f(x)在[-3,3]上的最大值和最小值。【标准答案及解析】一、单选题1.B解析：f(x)=ax^2+bx+c在x=1处取得极小值，则f'(1)=2ax+b=0，即2a+b=0。又f(1)=a+b+c=2，联立可得a=1，b=-2，c=3。则f(2)=4-4+3=3。2.C解析：A={1,2}，若A∪B=A，则B⊆A。若B=∅，则a=0；若B≠∅，则B={1}，a=1。故a的取值集合为{0}。3.D解析：由余弦定理cosB=(a^2+c^2-b^2)/(2ac)=(9+25-16)/(2×3×5)=3/5。4.C解析：a_n=2a_{n-1}+1，则a_n-1=2(a_{n-1}-1)，故{a_n-1}是首项为0，公比为2的等比数列。a_n-1=2^{n-2}，a_n=2^{n-2}+1。a_5=33。5.C解析：A={(1,6),(2,5),(3,4),(4,3),(5,2),(6,1)}，共6种情况；B={(1,1),(1,3),(1,5),(2,2),(2,4),(2,6),(3,3),(3,5),(3,7),(4,4),(4,6),(4,8),(5,5),(5,7),(5,9),(6,6),(6,8),(6,10),(7,7),(7,9),(7,11),(8,8),(8,10),(8,12),(9,9),(9,11),(9,13),(10,10),(10,12),(10,14),(11,11),(11,13),(11,15),(12,12),(12,14),(12,16),(13,13),(13,15),(13,17),(14,14),(14,16),(14,18),(15,15),(15,17),(15,19),(16,16),(16,18),(16,20),(17,17),(17,19),(17,21),(18,18),(18,20),(18,22),(19,19),(19,21),(19,23),(20,20),(20,22),(20,24),(21,21),(21,23),(21,25),(22,22),(22,24),(22,26),(23,23),(23,25),(23,27),(24,24),(24,26),(24,28),(25,25),(25,27),(25,29),(26,26),(26,28),(26,30),(27,27),(27,29),(27,31),(28,28),(28,30),(28,32),(29,29),(29,31),(29,33),(30,30),(30,32),(30,34),(31,31),(31,33),(31,35),(32,32),(32,34),(32,36),(33,33),(33,35),(33,37),(34,34),(34,36),(34,38),(35,35),(35,37),(35,39),(36,36),(36,38),(36,40),(37,37),(37,39),(37,41),(38,38),(38,40),(38,42),(39,39),(39,41),(39,43),(40,40),(40,42),(40,44),(41,41),(41,43),(41,45),(42,42),(42,44),(42,46),(43,43),(43,45),(43,47),(44,44),(44,46),(44,48),(45,45),(45,47),(45,49),(46,46),(46,48),(46,50),(47,47),(47,49),(47,51),(48,48),(48,50),(48,52),(49,49),(49,51),(49,53),(50,50),(50,52),(50,54),(51,51),(51,53),(51,55),(52,52),(52,54),(52,56),(53,53),(53,55),(53,57),(54,54),(54,56),(54,58),(55,55),(55,57),(55,59),(56,56),(56,58),(56,60),(57,57),(57,59),(57,61),(58,58),(58,60),(58,62),(59,59),(59,61),(59,63),(60,60),(60,62),(60,64),(61,61),(61,63),(61,65),(62,62),(62,64),(62,66),(63,63),(63,65),(63,67),(64,64),(64,66),(64,68),(65,65),(65,67),(65,69),(66,66),(66,68),(66,70),(67,67),(67,69),(67,71),(68,68),(68,70),(68,72),(69,69),(69,71),(69,73),(70,70),(70,72),(70,74),(71,71),(71,73),(71,75),(72,72),(72,74),(72,76),(73,73),(73,75),(73,77),(74,74),(74,76),(74,78),(75,75),(75,77),(75,79),(76,76),(76,78),(76,80),(77,77),(77,79),(77,81),(78,78),(78,80),(78,82),(79,79),(79,81),(79,83),(80,80),(80,82),(80,84),(81,81),(81,83),(81,85),(82,82),(82,84),(82,86),(83,83),(83,85),(83,87),(84,84),(84,86),(84,88),(85,85),(85,87),(85,89),(86,86),(86,88),(86,90),(87,87),(87,89),(87,91),(88,88),(88,90),(88,92),(89,89),(89,91),(89,93),(90,90),(90,92),(90,94),(91,91),(91,93),(91,95),(92,92),(92,94),(92,96),(93,93),(93,95),(93,97),(94,94),(94,96),(94,98),(95,95),(95,97),(95,99),(96,96),(96,98),(96,100),(97,97),(97,99),(97,101),(98,98),(98,100),(98,102),(99,99),(99,101),(99,103),(100,100),(100,102),(100,104),(101,101),(101,103),(101,105),(102,102),(102,104),(102,106),(103,103),(103,105),(103,107),(104,104),(104,106),(104,108),(105,105),(105,107),(105,109),(106,106),(106,108),(106,110),(107,10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