数学竞赛试卷（中英双语）

注意：

本试卷为自命题试卷，请不要在平台上搜索考试答案或者在答题中使用任何软件。本卷共有6道题，16道小题，满分150分。总答题时间8小时。开考前你们有10分钟的时间浏览试卷。本试卷不设相应答题卡，答案请写在桌上的A4纸上，A4纸一人5张。

1.（初等平面几何）在平面内有正三角形ABC，D是BC上一点，E是ABD的外心，F是ACD的外心。请回答以下问题。

（1）做三角形DEF。求证：DEF是等边三角形.（2分）

（2）BF、CE交于G，求证G是ABC的中心.（3分）

（3）延长BE、CF交于H，求证GH=GC.（3分）

（4）做a过E垂直于AF；b过F垂直于BE；a与b相交于K。求证：K在AB上.（8分）

命题：咲东、MOKE

审核：一信、二神

2.（代数基础分析）已知f(x)=e^x-x^e。

（1）请计算出该函数的导数.（3分）

（2）求证：函数值恒大于等于0.（4分）

（3）请问该函数有多少个极点?（5分）

（4）请计算出该函数的零点与所有极点顺次连接组成的凸多边形的面积.（5分）

命题：MOKE、二神

审核：一信

3.（代数综合分析）已知函数y=x*e^（x+a），e为自然底数。

（1）求该函数的极点.（5分）

（2）求证：该函数有且仅有一条渐近线，并写出这条渐近线的类型.（8分）

（3）求证：g(x)=f(x)+e^(a-1)≥0.（10分）

（4）若x*[e*g(x)-1]≥0恒成立，请求出a的取值范围.

命题：二神、咲东

审核：MOKE

4.（新场景应用）在3*4的网格内有一颗黑子与两颗白子，按照如下方式移动：黑子开始时处在左上角的格子中，每次移动2格（可以横向竖向各移动1格，禁止斜向移动）。一个白子处在黑子右下角的一格，两个白子相距2格，每个白子每次移动一格。黑先白后。当两个白子都挨在黑子旁边时黑子就输了。请问黑方是否有不输的办法？如果没有，请问白方至多在多少次之后胜利？请给出证明过程.（25分）

命题：MOKE、一信

审核：咲东

5.（平面解析几何）平面直角坐标系xOy内做一个圆，这个圆的半径为2，圆心为坐标原点O。A（1，0），B（0，1）。C点是这个圆上的一个动点。

延长BC交x轴于D点，延长AC交y轴于E点。

（1）请求出AD*BE的值.（5分）

（2）分别做DC、EC的中垂线p、q交于K。求证：K在定直线x-y=0上.（7分）

命题：二神

审核：MOKE、咲东

（3）分别做l1、l2过A、B垂直于CD、BD交于K。求证：CK的长度、与坐标轴的夹角与C点所在位置无关，并计算出C点顺时针运动360度时CK扫过部分的面积.（12分）

（4）延长KC、KB交圆于M、N，求出K的轨迹方程，并证明JG、KH、AI三线共点.（15分）

6.（数论）完全平方数可以由两个相同的数相乘得到，它寄寓了人们对一切美好事物的无止境追求。

（1）是否存在一个八位完全平方数，使其只由1和4构成？请证明你的结论.（8分）

（2）是否存在2022位完全平方数使得它只由1、4、9、0构成，而且不以零结尾?请证明你的结论.（12分）

命题：MOKE

审核：一信、咲东

鸣谢名单

总策划：一信

总负责人：咲东

设备：MOKE、二神

题目顺序校验：二神

题目内容校验：MOKE（101、201～203、301～302、304、501、503、602）、咲东（102、204、303、401、502、504、601）

6.22初次拟定

6.25定稿

6.29最终审核

Note:  

 

 

 

Please do not search for answers on the platform or use any software in answering questions.  There are 6 questions and 16 short questions, with a full mark of 150 points.  The total answer time is 8 hours.  You have ten minutes to look over the papers before the test begins.  Please write your answers on A4 paper on the desk. Each person has 5 A4 papers.  

 

 

 

1. (Elementary plane geometry) In the plane is the regular triangle ABC, D is a point on BC, E is the outer center of ABD, F is the outer center of ACD.  Please answer the following questions.  

 

 

 

(1) Make triangle DEF.  DEF is equilateral triangle. (2分)  

 

 

 

(2) BF and CE intersect with G, and it is proved that G is the center of ABC.  

 

 

 

(3) Extend BE and CF to H, and verify GH=GC. (3 marks)  

 

 

 

(4) Do a over E perpendicular to AF;  B over F is perpendicular to BE;  A intersects B at K.  K is on AB. (8分)  

 

 

 

Propositions: Sakito, MOKE  

 

 

 

Review: One faith, two gods  

 

 

 

F (x)=e^x-x^e  

 

 

 

Please calculate the derivative of this function.  

 

 

 

(2) Verify: The function value is always greater than or equal to 0. (4 marks)  

 

 

 

How many poles does this function have?  (5 points)  

 

 

 

(4) Please calculate the area of the convex polygon formed by the sequential connection of the zero point of the function with all the poles.  

 

 

 

Proposition: MOKE, two gods  

 

 

 

Review: one letter  

 

 

 

Y =x*e^ (x+a), e is the natural base.  

 

 

 

(1) Find the pole of the function.  

 

 

 

(2) Verify that the function has one and only one asymptote, and write the type of this asymptote.  

 

 

 

G (x)=f(x)+e^(a-1)≥0. (10分)  

 

 

 

(4) If x*[e*g(x)-1]≥0 is always true, the value range of A is requested.  

 

 

 

Proposition: Two god, Sakito  

 

 

 

Review: MOKE  

 

 

 

4. (New scene application) Move a black spot and two white spots in a 3*4 grid as follows: The spots start in the upper left corner of the grid and move 2 squares at a time (1 square can be moved horizontally and 1 square can be moved vertically, oblique movement is prohibited).  A white son is in the lower right corner of the black one square, two white son is 2 square apart, each white son moves one square at a time.  Black comes before white.  When both white pieces are next to black pieces, black pieces lose.  Does black have a way not to lose?  If not, how many times can white win?  Please give the proof process.  

 

 

 

Propositions: MOKE, a letter  

 

 

 

Review: Sakito  

 

 

 

5. (plane analytic geometry) Make a circle in the plane rectangular coordinate system xOy, the radius of the circle is 2, the center of the circle is the coordinate origin O.  A (1,0), B (0,1).  C is a moving point on this circle.  

 

 

 

Extend BC to intersect the X-axis at D and AC to intersect the Y-axis at E.  

 

 

 

(1) Select the value of AD*BE (2 分)  

 

 

 

(2) Make the perpendicular lines P and Q of DC and EC intersect K respectively.  K is on the fixed line x-y=0.  

 

 

 

Proposition: Two gods  

 

 

 

Review: MOKE, Sakito  

 

 

 

(3) L1 and L2 intersect with K through A and B perpendicular to CD and BD, respectively.  Verify that the length of CK and the included Angle with the coordinate axis are independent of the position of point C, and calculate the area of the part swept by CK when point C moves 360 degrees clockwise.  

 

 

 

(4) Extend the intersection circle of KC and KB to M and N, work out the trajectory equation of K, and prove that the three lines JG, KH and AI have common points.  

 

 

 

The perfect square number, which can be multiplied by two identical numbers, embodies the endless pursuit of all good things.  

 

 

 

(1) Is there an eight-bit perfect square number that consists of only 1 and 4?  Please prove your conclusion.  

 

 

 

(2) Is there a perfect square number with 2022 bits such that it consists only of 1, 4, 9 and 0 and does not end in zero?  Please prove your conclusion.  

 

 

 

Proposition: MOKE  

 

 

 

Review: Ichishin, Sakito  

 

 

 

Thanks to the list  

 

 

 

Chief planner: one letter  

 

 

 

General manager: Sakito  

 

 

 

Equipment: MOKE, two gods  

 

 

 

Two gods  

 

 

 

MOKE (101, 201 ~ 203, 301 ~ 302, 304, 501, 503, 602), Sakito (102, 204, 303, 401, 502, 504, 601)  

 

 

 

6.22 Initial draft  

 

6.25 finalized  

 

6.29 Final review 

编辑人：狼人杀官服第一狼

日期：2022/7/9