2024年中国赛区国际袋鼠数学竞赛F卷（高二、三年级）试题

考式说明/Instruction Time allowed:75 minutes 考试时长：75分钟 Language:English Chinese 试卷语言：英文和中文 ..、0000000900900X990000 00000000000000000000000000000000000.' This is a 30-question multiple-choice competition. 考试包含30道单项选择题。 Problems 1 to 10 are worth 3 points each. 1-10题每题3分 Problems 11 to 20 are worth 4 points each. 11-20题每题4分 Problems 21 to 30 are worth 5 points each. 21-30题每题5分 1 point will be deducted for an incorrect answer and no point for a blank answer. 答错扣1分，不答不扣分。 The full score is 150(Starting point is 30) 总分为150分（起始分数为30分） 000000 Do not open this booklet until you are told to do so. 收到监考老师指令后，方可打开试卷册。 Mark your answers clearly on the answer sheet using a 2B pencil. 务必使用2B铅笔在答题卡上正确填涂自己的答案。 Calculator cannot be used during the competition. 考试期间禁止使用计算器。 DO NOT discuss problems and solutions in any online or public forum until April 30. 4月30日前请勿以任何形式公开讨论试题和答案。 只 扫描全能王创建 MATH KANGAROO 2024 (CHINA)-LEVEL F 3 points 1.A pattern is made of equal pentagons.Which of the tiles below,when placed in the contral hole,will form a self-intersecting loop? 右侧图形由相同的五边形瓷砖组成。请问将下列哪块瓷砖放入图 形的中心处可以形成一条自相交义的环线？ (B (E 2.Which of these integers is two less than a multiple of ten,two more than a square,and two times a prime? 请问下列哪一个整数比10的倍数少2，比一个平方数多2，并且是一个质数的两倍？ (A)78 (B)58 (C)38 (D)18 (E)6 3.A young kangaroo cut a pizza into six equal slices.After eating one slice,he arranged the remaining slices with equal gaps between slices.What size is the angle of each gap? 一只小袋鼠将披萨分为6等份。小袋鼠吃了一份之后，将剩余的披萨摆放整 齐，使得每份披萨之间的夹角相等。请问相邻披萨之间的夹角为多少度？ (A)5° (B)8° (C)9° (D)10° (E)12° 4.Juuso has an unusual habit of drawing the xy-plane with the positive coordinate axes pointing left and down.What would the graph of the equation y=+1 look like in a coordinate system drawn by Juuso? Juuso画x平面直角坐标系时，习惯将向左和向下的方向定为坐标轴的正方向。请问下列哪一项 是方程y=x+1在Juuso画的xy平面直角坐标系上的图像？ 寸六 扫描全能王创建 MATH KANGAROO 2024 (CHINA)-LEVEI Kaito has maniputeddie.The pobbit of og,5hbuthe probability of rolling a 6 is twice the probability of rolling a 1.What is the probability of rolling a 6? Kato操控一枚般子，使得点数2、3、4、5中各个点数朝上的概率始终为后'但是点数6朝上的 概率是点数1朝上的概率的两倍。请问点数6期上的概率是多少？ 国月 (C)38 (D) 6.Which of the expressions below has the same value as 1615+1615+1615+1615? 请问下列哪个表达式与165+1615+1616+165的值相等？ (A)1619 (B)431 (C)460 (D)1660 (E)4122 7.Beaver wishes to color the squares and triangles of the following figure so that no two neighbouring figures,even those sharing a single vertex,are the same color. Beaver想为下图中的正方形和三角形上色，使得任意两个相邻图形不同色，包括共用一个顶点的 两个图形也不同色。 What is the least number of colors needed? 请问Beaver至少需要几种颜色？ (A)3 (B)4 (C)5 (D)6 (E)7 8.There are 6 glasses on a table with their open ends up.In any one move,we turn over exactly 4 of them.What is the least number of moves required to have all glasses upside down? 桌上有6个开口朝上的玻璃杯。每一轮只翻转4个玻璃杯，请问至少需要几轮才能使得所有玻璃 杯开口朝下？ (A)2 (B)3 (C)4 (D)5 (E)6 9.A student started with the number 1 and multiplied it by either 6 or 10.He then multiplied the result by either 6 or 10,and continued this procedure many times.Which of the following cannot be one of the numbers he obtained? 一名学生从数字1开始，将其乘6或10，然后再将得到的结果乘6或10，以此类推，多次重 复该过程。请问这名学生无法得到下列哪一个数字？ (A)2100320580 (B)290320580 (C)290320570 (D)2110380530 (E)250550 扫描全能王创建 可只 v然8ao加 MATH KANGAROO 2024 (CHINA)-LEVEL F 10.A black trail and a grey trail cross a park,as shown.Each trail divides 、A the park into two regions of equal area.Which of the following must be true B about the areas A,B and C? 如图所示，一条黑色小道和一条灰色小道横穿公园。每一条小道都将公园划 分为两个面积相同的区域。请问关于面积A、B、C的说法中，下列哪项是正 确的？ (A)A=C (B)B=A+C (C)B= (A+@) D)B=号A+O) 2 3 (®)B=A+C) 4 points 11.Exactly one of these statements about a certain positive integer n is true.Which statement is true? 下列关于某个正整数几的表述中恰好只有一项正确，请问是哪一项？ (A)n is divisible by3n可以被3整除 (B)m is divisible by6n可以被6整除 (C)n is odd n是奇数 (D)n=2 (E)n is prime n是质数 12.A triangular pyramid ABCD has sides of length 5,6,7,8,9 and 10.The points M,N,P,Q,R and S are the midpoints of the edges of the pyramid,as shown. 三棱锥ABCD的各边长为5、6、7、8、9、10。点M、N、P、Q、R、S是三棱锥各边上的中 点，如图所示。 M R What is the perimeter of the closed hexagonal line MNPQRSM? 请问闭合的六边图形MNPQRSM的周长是多少？ (A)19 (B)20 (C)21 (D)22 (E)23 3 扫描全能王创建 MATH MATH KANGAROO 2024 (CHINA)-LEVEL F 13.A quadrilateral ABCD has two right angles at B and C,where AB =4,BC 8 and CD 2.Point X lies on BC.What is the minimum value of AX+DX? 四边形ABCD有两个直角位于B和C处，且AB=4,BC=8, CD=2。点X位于BC边上。请问AX+DX的最小值是多少？ X (A)9V2 (B)12 (C)13 (D)10 (E)None of the previous以上都不正确 14.John has a number of all black or all white unit cubes and wants to build a 3x 3x 3 cube using 27 of them.He wants the surface to be exactly half black and half white.What is the smallest number of black cubes he can use? Jon有很多全黑和全白的单位立方体。他想用其中27个单位立方体构建一个3×3×3的大立 方体。他希望这个大立方体的表面上，恰好一半为黑色，一半为白色，请问John最少需要用多少个全 黑单位立方体？ (A)14 (B)13 (C)12 (D)11 (E)None of the previous以上都不正确 15.A diagonal,a semicircle and a quadrant are drawn in a square of side 6 cm. What is the area,in cm2,of the shaded part? 在一个边长为6cm的正方形中画一条对角线、一个半圆和一个四分之一圆。 请问阴影部分的面积为多少cm2? (A)9 (B)3π (C)6m-9 (D)10m/3 (E)12 16.The figure shows four squares.The smaller ones have side lengths a,b and c.The vertices A and C of two of the smaller squares coincide with two diagonally opposite vertices of the large square.The vertex B of the third small square is on the side of the large one.Which of the following expressions represents the side length of the largest square? 右图中有4个正方形。三个小正方形的边长分别为a、b、c。其中两个小正方形的顶点A和C恰 好是最大正方形的两个对角顶点，第三个小正方形的顶点B落在最大正方形的边上。请问下列哪一项 表示最大正方形的边长？ (A)j(a+6+c) (B)Va2+62+c2 (C)V(a+b)2+c2 (D)√6-a2+c2 (E)Va2+ab+62+c2 扫描全能王创建 MATH MATH KANGAROO 2024 (CHINA)-LEVEL F 17.Wo have two positivo numbers p and d,with p<q.Which of these expressions is the largest? p和q为两个正数，且p<?。请问下列哪个表达式的值最大？ (A)P+3 (B)D+2 (C)+q (D)+q 4 3 2 3 18.How many three-digit numbers are there that contain at least one of the digits 1,2 or 3? 请问有多少个3位数，其数位上至少包含1、2、3这三个数学中的一个？ (A)27 (B)147 (C)441 (D)557 (E)606 19.I write down a 4-digit non-zero number N=pgra.When I place a decimal point between the g and the r,I find that the resulting number po.rs is the average of the two-digit numbers pq and 7s. What is the sum of the digits of N? 写下一个非零的四位数N=9。在g和？之间加入一个小数点，得到的数字四T是两位数 西和的平均值。请问数字N的各位数字之和为多少？ (A)14 (B)18 (C)21 (D)25 (E)27 20.Two candles of equal length start burning at the same time.One of the candles will burn down in 4 hours,the other in 5 hours,each at their own constant rate.How many hours will they have to burn before one candle is 3 times the length of the other? 两支等长的蜡烛同时开始燃烧。每支蜡烛的燃烧速度都是恒定的，其中一支蜡烛4小时后燃尽，另 外一支5小时后燃尽。请问两支蜡烛必须燃烧多少小时之后，其中一支蜡烛的长度是另一支蜡烛长度 的3倍？ ω智 45 B) 63 7 (D)3 (®)14 5 points 21.Andre has six cards with one number written on each side of each card.The pairs of numbers on the cards are (5,12),(3,11),(0,16),(7,8),(4,14)and (9,10).The cards can be placed in any order in the blank spaces of the figure. Andre有6张卡片，每张卡片的正反两面上各写有一个数字。6张卡片上的数对分别为(5,12)、 (3,11、(0,16)、(7,8)、(4,14)、(9,10)。将这6张卡片以任意顺序放入下方空格中。 =? What is the smallest result he can get? 请问Andre可以得到的最小值是多少？ (A)-23 (B)-24 (C)-25 (D)-26 (E)-27 5 扫描全能王创建 Aao加 MATH KANGAROO 2024 (CHINA)-LEVEL F 22.Kangaroo solves the equation ax2+ba+c=0,and Beaver solves the equation bx2+ax+c=0, where a,b,c are pairwise distinct non-zero integers.It turns out that the equations share a solution. Which of the following must be true? Kangaroo解方程a2+b+c=0,Beaver解方程bz2+ax+c=0,其中a、b、c是互不相同 的非零整数，且两个方程有一个相同的解。请问下列哪项的说法一定是正确的？ (A)The common solution must be 0. 两个方程的公共解一定是0。 (B)The quadratic equation ax2+bx+c=0 has exactly one real solution. 二次方程ax2+bx+c=0只有一个实数解。 (C)a>0 (D)b<0 (E)a+b+c=0 23.I have a strip of paper that is 12cm long and 2cm wide.I 459 make a crease across it at 45 and then fold it,so that the two parts of the strip are aligned in a right angle,as shown. What is the smallest possible length,in cm,of XY? 一张纸条，长12cm,宽2cm。在纸条的45°处做一条折痕， 然后沿着折痕进行折叠，此时纸条的两个部分呈直角，如图所示。 请问XY的长度最短可能为多少cm? (A)6v2 (B)7V2 (C)10 (D)8 (E)6+√ 24.Rasika has several unbiased 12-sided dice,each with faces labelled 1 to 12.When rolling all the dice at once,the probability of rolling a 12 exactly once is equal to the probability of rolling no 12s. How many dice does Rasika have? Rasika有几枚质地均匀的12面骰子，每枚骰子上的各个面从1到l2进行编号。同时投掷全部骰 子，编号12恰好出现一次的概率等于编号12完全没有出现的概率。请问Rasika一共有几枚骰子？ (A)8 (B)9 (C)10 (D)11 (E)12 25.A polynomial p(x)satisfies the relation p(x+1)=x2-x+2p(6)for every real x.What is the sum of the coefficients of p? 对于每个实数x,多项式p(x)满足p(a+1)=x2-x+2(6)。请问p的系数之和是多少？ (A)-40 (B)-6 (C)12 (D)40 (E)None of the previous以上都不正确 扫描全能王创建 SAROO MATH KANGAROO 2024 (CHINA)-LEVEL F 26.The values of ,y and z satisfy 2=3,2v =7 and 6*=7.Which of the following gives the relationship between a,y and 2? x、y、之的值满足2=3,2”=7,6=7。请问下列哪个选项表示、、名之间的关系？ )=辛 ®)=号+1@：=是- 1 (D)名=二 但)=y-是 27.A strip of paper consists of eight squares.Initially each square contains the number 0.In every move,we chose 4 consecutive squares and add one to each of the numbers in those squares.The figure shows the outcome after a number of moves but unfortunately some ink is covering some of the squares.What number is written on the square with the question mark? 一张纸条由8个方格组成。最初，每个方格中都包含数字0。每一轮选择4个连续方格，并将所 选方格中的每个数字加1。图中所示为多轮操作后的结果，但其中一些方格被墨水遮盖。请问图中问 号所在方格上的数字为多少？ 42 (A)24 (B)30 (C)36 (D)48 (E)None of the previous以上都不正确 28.A function f:R->R satisfies f(20-x)=f(22+x)for all real x.It is known that f has exactly two roots.What is the sum of these two roots? 对于所有实数x,函数f:R→R满足f(20-x)=f(22+x)。已知f恰好有两个根。请问这 两个根的和是多少？ (A)-1 (B)20 (C)21 (D)22 (E)None of the previous以上都不正确 29.Twelve points are equally spaced on a circle.How many triangles containing a 45 angle can be formed by choosing three of these points? 一个圆上有12个等距分布的点。选取其中三个点，请问可以组成多少个含一个45°角的三角形？ (A)48 (B)60 (C)72 (D)84 (E)96 30.A special four-digit number abcd satisfies the equation abcd aa+6+c+dd.What is the value of a? 一个特殊的4位数abcd满足方程abcd=aa+b+ce+d4。请问a的值是多少？ (A)2 (B)3 (C)4 (D)5 (E)6 扫描全能王创建 题号 答案 题号 答案 题号 答案 题号 答案 题号 答案 1 A 2 C 3 飞 4 D 5 D 6 B 7 C 8 B 9 B 10 B 11 C 12 C 13 0 14 E 15 A 16 C 17 A 18 E 19 B 20 A 21 C 22 E 23 B 24 D 25 A 26 A 27 A 28 E 29 C 30 B