數學競賽試卷（中英雙語）
注意：
本試卷為自命題試卷，請不要在平臺上搜索考試答案或者在答題中使用任何軟件。本卷共有6道題，16道小題，滿分150分?？偞痤}時間8小時。開考前你們有10分鐘的時間瀏覽試卷。本試卷不設相應答題卡，答案請寫在桌上的A4紙上，A4紙一人5張。
1.（初等平面幾何）在平面內有正三角形ABC，D是BC上一點，E是ABD的外心，F(xiàn)是ACD的外心。請回答以下問題。
（1）做三角形DEF。求證：DEF是等邊三角形.（2分）
（2）BF、CE交于G，求證G是ABC的中心.（3分）
（3）延長BE、CF交于H，求證GH=GC.（3分）
（4）做a過E垂直于AF；b過F垂直于BE；a與b相交于K。求證：K在AB上.（8分）
命題：咲東、MOKE
審核：一信、二神
2.（代數基礎分析）已知f(x)=e^x-x^e。
（1）請計算出該函數的導數.（3分）
（2）求證：函數值恒大于等于0.（4分）
（3）請問該函數有多少個極點?（5分）
（4）請計算出該函數的零點與所有極點順次連接組成的凸多邊形的面積.（5分）
命題：MOKE、二神
審核：一信
3.（代數綜合分析）已知函數y=x*e^（x+a），e為自然底數。
（1）求該函數的極點.（5分）
（2）求證：該函數有且僅有一條漸近線，并寫出這條漸近線的類型.（8分）
（3）求證：g(x)=f(x)+e^(a-1)≥0.（10分）
（4）若x*[e*g(x)-1]≥0恒成立，請求出a的取值范圍.
命題：二神、咲東
審核：MOKE
4.（新場景應用）在3*4的網格內有一顆黑子與兩顆白子，按照如下方式移動：黑子開始時處在左上角的格子中，每次移動2格（可以橫向豎向各移動1格，禁止斜向移動）。一個白子處在黑子右下角的一格，兩個白子相距2格，每個白子每次移動一格。黑先白后。當兩個白子都挨在黑子旁邊時黑子就輸了。請問黑方是否有不輸的辦法？如果沒有，請問白方至多在多少次之后勝利？請給出證明過程.（25分）
命題：MOKE、一信
審核：咲東
5.（平面解析幾何）平面直角坐標系xOy內做一個圓，這個圓的半徑為2，圓心為坐標原點O。A（1，0），B（0，1）。C點是這個圓上的一個動點。
延長BC交x軸于D點，延長AC交y軸于E點。
（1）請求出AD*BE的值.（5分）
（2）分別做DC、EC的中垂線p、q交于K。求證：K在定直線x-y=0上.（7分）
命題：二神
審核：MOKE、咲東
（3）分別做l1、l2過A、B垂直于CD、BD交于K。求證：CK的長度、與坐標軸的夾角與C點所在位置無關，并計算出C點順時針運動360度時CK掃過部分的面積.（12分）
（4）延長KC、KB交圓于M、N，求出K的軌跡方程，并證明JG、KH、AI三線共點.（15分）
6.（數論）完全平方數可以由兩個相同的數相乘得到，它寄寓了人們對一切美好事物的無止境追求。
（1）是否存在一個八位完全平方數，使其只由1和4構成？請證明你的結論.（8分）
（2）是否存在2022位完全平方數使得它只由1、4、9、0構成，而且不以零結尾?請證明你的結論.（12分）
命題：MOKE
審核：一信、咲東
鳴謝名單
總策劃：一信
總負責人：咲東
設備：MOKE、二神
題目順序校驗：二神
題目內容校驗：MOKE（101、201～203、301～302、304、501、503、602）、咲東（102、204、303、401、502、504、601）
6.22初次擬定
6.25定稿
6.29最終審核
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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. ?
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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. ?
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(1) Make triangle DEF.? DEF is equilateral triangle. (2分) ?
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(2) BF and CE intersect with G, and it is proved that G is the center of ABC. ?
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(3) Extend BE and CF to H, and verify GH=GC. (3 marks) ?
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(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) ?
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Propositions: Sakito, MOKE ?
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Review: One faith, two gods ?
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F (x)=e^x-x^e ?
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Please calculate the derivative of this function. ?
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(2) Verify: The function value is always greater than or equal to 0. (4 marks) ?
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How many poles does this function have?? (5 points) ?
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(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. ?
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Proposition: MOKE, two gods ?
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Review: one letter ?
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Y =x*e^ (x+a), e is the natural base. ?
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(1) Find the pole of the function. ?
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(2) Verify that the function has one and only one asymptote, and write the type of this asymptote. ?
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G (x)=f(x)+e^(a-1)≥0. (10分) ?
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(4) If x*[e*g(x)-1]≥0 is always true, the value range of A is requested. ?
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Proposition: Two god, Sakito ?
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Review: MOKE ?
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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. ?
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Propositions: MOKE, a letter ?
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Review: Sakito ?
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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. ?
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Extend BC to intersect the X-axis at D and AC to intersect the Y-axis at E. ?
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(1) Select the value of AD*BE (2 分) ?
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(2) Make the perpendicular lines P and Q of DC and EC intersect K respectively.? K is on the fixed line x-y=0. ?
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Proposition: Two gods ?
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Review: MOKE, Sakito ?
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(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. ?
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(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. ?
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The perfect square number, which can be multiplied by two identical numbers, embodies the endless pursuit of all good things. ?
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(1) Is there an eight-bit perfect square number that consists of only 1 and 4?? Please prove your conclusion. ?
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(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. ?
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Proposition: MOKE ?
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Review: Ichishin, Sakito ?
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Thanks to the list ?
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Chief planner: one letter ?
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General manager: Sakito ?
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Equipment: MOKE, two gods ?
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Two gods ?
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MOKE (101, 201 ~ 203, 301 ~ 302, 304, 501, 503, 602), Sakito (102, 204, 303, 401, 502, 504, 601) ?
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6.22 Initial draft ?
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6.25 finalized ?
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6.29 Final review?
編輯人：狼人殺官服第一狼
日期：2022/7/9
