1、2017 年數(shù)學(xué)競(jìng)賽（中國(guó)區(qū)）選拔賽試題主辦青少年國(guó)際競(jìng)賽與交流中心數(shù)學(xué)競(jìng)賽中國(guó)規(guī)則1) 比賽時(shí)間：10:0011:00, 1 小時(shí)。不許提前交卷2) 共 8 道題，滿分 100 分。請(qǐng)?jiān)诖痤}紙上作答，寫(xiě)在試卷或草稿紙上的無(wú)效。3) 請(qǐng)?jiān)诖痤}紙規(guī)定位置工整填寫(xiě)學(xué)校、及頁(yè)碼。4）答題紙（一）僅填寫(xiě)最后結(jié)果。答題紙（二）需寫(xiě)出推導(dǎo)、演算過(guò)程，僅填寫(xiě)無(wú)法得分。5) 答題紙（二）可按需加紙(A4)，請(qǐng)務(wù)必在每頁(yè)注明學(xué)校、題號(hào)，頁(yè)碼以免丟失或錯(cuò)放。不允許使用計(jì)算器。6)可使用鉛筆作圖，但不要使用鉛筆作答。比賽結(jié)束后，請(qǐng)將答題紙及試題卷整理好并交給監(jiān)考老師。9) 嚴(yán)禁。評(píng)卷過(guò)程中，如發(fā)現(xiàn)同校卷，成績(jī)一律無(wú)

2、效。歡迎參加題。信息高中數(shù)學(xué)競(jìng)賽（中國(guó)區(qū)）選拔賽。希望您能夠花 30 秒鐘的時(shí)間，完成下列這道問(wèn)賽事研究使用，十分感謝！您是通過(guò)什么途徑知道高中數(shù)學(xué)競(jìng)賽的？青少年國(guó)際競(jìng)賽與交流中心 ITCCC青少年國(guó)際競(jìng)賽與交流中心 ITCCCCEAIE-ITCCCC. 青少年國(guó)際競(jìng)賽與交流中心 ITCCC 其它競(jìng)賽項(xiàng)目: IJSO 青少年科學(xué)競(jìng)賽、RoboRAVE 機(jī) 器賽、Botball 機(jī)器賽、太空城市設(shè)計(jì)大賽、未來(lái)之城大賽、數(shù)學(xué)競(jìng)賽、iGEM工程機(jī)器、國(guó)際青少年微大賽等。D. 校內(nèi)宣傳(校內(nèi)海報(bào)、同學(xué)介紹)E. 學(xué)校組織或老師介紹F. 機(jī)構(gòu)（請(qǐng)注明機(jī)構(gòu)名稱）。F其他（請(qǐng)注明）。祝比賽順利！CEAIE

3、ITCCC，更快了解賽事進(jìn)程。關(guān)注青少年國(guó)際競(jìng)賽與交流中心1.On the side there is a simple octagon (t is,here the sides dontersect) where 4 of theernalanglearked in blue) are acute. Given any simple octagon, whats theumsible number of acuteernal angles?t AB k EF, AE EF, with D the2.ABEF is a trzoid suchersection of the Diagonals

4、, and C thependicular projection of D on AE. The lengths of the bases are given by|AB| = 8 and|EF| = 17. Find theheight EC3. Let 2k be the largester of 2 dividing 30!. Find k.4. Let d(n) be the total number of digits needed to write all the numbers from 1 to n in base 10; for exd(5) = 5 and d(20) =

5、31. Find d(2012).le5. Call an ordered triple of sets (A,B,C) nice if |AB| = |BC| = |C A| = 2 and |ABC| = 0. How many ordered triples of subsets of 1,2,3.,9 are nice? (Note: |S| denotes the number of elements of S, and S Tdenotes theersection of S and T.)6. Brets an nnn cube (where n is aneger) which

6、 he dipso blue pa. He then cuts the cubeo abunch of 111 cubes, and notit the number of un-paed cubes (which isitive) evenly divides thenumber of paed cubes. What is the largestsible side length of Bretts original cube.7. Given a2017 + b2017= 2017, nd theum value of a + b8. If the sum of the reciprocals of the roots of the quadraticis , what is k?答題紙（一）Name and School:QuestionAnswer12345678Answer the questionshe answer sheet, Sho