2023年中国赛区国际袋鼠数学竞赛F卷（高二、三年级）试题

AROO Math Kangaroo 2023 (China)-Level F 3 points 1.What is the value of 计算下面算式的值 77772 5555×2222 (A)1 7 9 (C)0 77 (B) 10 (D) 110 (E)49 2.Giulia rolls five dice.She rolls 19 points in total.What is the maximum number of sixes she could have rolled? Giulia投掷了五枚骰子，点数之和等于l9。最多有多少枚骰子点数是6？ (A)0 (B)1 (C)2 (D)3 (E)4 3.A cylindrical can has height 15 cm and the perimeter of its circular base is 30 cm.An ant walks from point A on the base to point B on the roof.Its path is either vertically upwards or horizontally along circular arcs around the can.Its path is shown with a thicker line (black for the path on the front of the can and grey at the back).What is the length,in cm,of the ant's path? 圆柱形罐子的高度为15厘米，底面圆形的周长为30厘米。一只蚂蚁从底面上 的A点爬到顶面上的B点，爬行方向要么垂直向上要么沿着罐子的圆弧水平前进。 如图所示，粗线（黑色为罐子正面的路径，灰色为罐子背面的路径）表示蚂蚁的爬 行路径。蚂蚁爬行路径的总长度是多少厘米？ (A)45 (B)55 (C)60 (D)65 (E)75 4.Emma has four different coloured pens.She wants to colour the three-striped rectangular flag shown in the diagram so that each stripe is a single colour and no two adjacent stripes are the same colour.In how many ways can she do this? Emma有四支不同颜色的笔。她想给图中所示的三横条长方形旗帜涂色，每 个横条都涂上一种颜色，且相邻条纹的颜色不同。她能用多少种方式给旗帜涂色？ (A)24 (B)27 (C)32 (D)36 (E)64 5.We call a positive integer n two-prime,if it has exactly three different divisors,namely 1,2 and n itself.How many different two-prime integers are there? 如果一个正整数n恰好有三个不同的因子，即1、2和n本身，则称n为“二素数”。一共有多 少个不同的“二素数”？ (A)0 (B)1 (C)2 (D)3 (E)4 3 CS CamScanner MATH GARO0 Math Kangaroo 2023 (China)-Level F 6.How many pairs of positive integers x and y satisfy the equation +2y=210? 有多少对正整数x和y满足方程x+2y=210? (A)29-1 (B)29 (C)29+1 (D)29+2 (E)0 7.Two equilateral triangles are put together to form a hexagon with their opposite sides parallel.We know the length of four sides of this hexagon, as shown in the diagram.What is the perimeter of the hexagon? 将两个等边三角形放在一起形成一个六边形，六边形的对边互相平行。 如图所示，六边形的四条边长已经标出。这个六边形的周长是多少？ (A)64 (B)66 (C)68 ⊙)70 (E)72 8.A square with area 84 is divided into four squares.The upper left square is coloured black.The lower right square is again divided into four squares,and so on. The process is repeated an infinite number of times.What is the total area that is coloured black? 将面积为84的正方形分成四个小正方形后，将左上角的小正方形涂为黑色。然后将右下角的小正 方形再次分成四个更小的正方形，以此类推，并将这个过程重复无数次。涂成黑色的正方形的总面积 是多少？ (A)24 (B)28 (C)31 (D)35 (E)42 9.Each of the integers from 1 to 9 is to be placed in one of the 9 7■ 9 boxes in the picture so that any three numbers in consecutive boxes add to a multiple of 3.The numbers 7 and 9 have already been placed. In how many different ways can the remaining boxes be filled? 将数字1到9填入图中9个方格的任一方格内，使得任意三个连续方格中的数字之和为3的倍数。 数字7和9已经填入方格。一共有多少种不同方式来放置剩余数字？ (A)9 (B)12 (C)15 (D)18 (E)24 10.What is the units digit of the product (55+1)(510+1)(515+1)? (55+1)(510+1)(515+1)之积的个位数是多少？ (A)1 (B)2 (C)4 (D)5 (E)6 4 CS CamScanner Math Kangaroo 2023 (China)-Level F 4 points 11.A triangular pyramid has edges of integer length.Four of these lengths are as shown in the diagram.What is the sum of the lengths of the other two edges? 三棱锥的边长均为整数，其中四条边长如图所示。剩余两条边的长度之和是多少？ (A)9 (B)10 (C)11 (D)12 (E)13 12.For a positive integer n,n!is defined as the product of all integers from 1 to n.For example, 4!=4.3.2.1 =24.What is the sum of the digits of N if N!=6!.7!? 对于正整数n,n!定义为从1到n之间的所有整数之积。例如，4！=4·3·2·1=24。如果 NI=6!.7I,N的各个数位上的数字之和是多少？ (A)1 (B)2 (C)4 (D)8 (E)9 13.The graphs of the functionsy=+3+ax+2a+4all pass through the same point,no matter what value of a is chosen.What is the sum of the coordinates of that point? 无论a取何值，函数y=x3+3x2+ax+2a+4的图像都会经过同一个点。这个点的横纵坐标 之和是多少？ (A)2 (B)4 (C)7 (D)8 (E)none of the previous以上都不正确 14..We are given five numbers a1,a2,a3,a4,a5 whose sum is S.For each k,1<k <5,we know that ak=k+S.What is the value of S? 五个数a1,a2,a3,a4,a5之和为S,对于每个k（1≤k≤5)，满足ak=k+S,S的值是多少？ 國) (C)-15 (D)15 (E)none of the previous以上都不正确 15.How many pairs of integers m and n satisfy the inequality 2m-2023+2n-ml<1? 有多少个m和n的整数对满足不等式|2m-2023l+12n-m≤1？ (A)0 (B)1 (C)2 (D)3 (E)4 16.There are 23 animals sitting in a row at the cinema.Each animal is either a beaver or a kangaroo. Everyone has at least one neighbour who is a kangaroo.What is the largest possible number of beavers in the row? 电影院里有23只动物坐成一排，它们要么是海狸要么是袋鼠。每只动物至少与一只袋鼠相邻。这 一排最多可能有多少只海狸？ (A)7 (B)8 (C)10 (D)11 (E)12 5 CamScanner MATH Math Kangaroo 2023 (China)-Level F 17.The number 55 can be written in the form n"for some integer n.What is the value of n? 对于某一整数n,数字55可以写成nn的形式，n的值是多少？ (A)530 (B)56 (C)55 (D)30 (E)11 18.Leon has drawn a closed path on a rectangular prism.Which net could show his path? Leon在长方体上画了一条闭合路径。下面哪张展开图可能是他画的路径？ A D 19.A pentagon is dissected into smaller parts,as shown.The numbers inside the triangles indicate their areas.What is the area P of the shaded quadrilateral? 如图所示，一个五边形被分成几个部分。三角形里面的数字 3 表示这个三角形的面积。阴影四边形的面积P是多少？ 8 (A)15 1 B)2 (C)16 (D)17 (E)18 20.How many integers are factors of 220323 but are not factors of 210320? 有多少个整数是220323的因数，但不是210320的因数？ (A)13 (B)30 (C)273 (D)460 (E)none of the previous以上都不正确 5 points 21.Two functions f and g on R satisfy the system of equations f(a)+2g(1-2)=22 and f(1-a)-g(a)=22.What is f? 两个函数∫和g位于R上，且满足方程组 f(x)+2g(1-x)=x2和f(1-x)-g(c)=x2,求f。 (A)2-等知+号 (B)x2+x+号 (C)-x2-x+号 (D)x2-4x+5 (E)there are no such functions不存在这样的函数 6 CS CamScanner 高然ao Math Kangaroo 2023 (China)-Level F 22.In a bouldering competition,13 climbers compete in three categories.The score of each competitor is the product of their rankings in the three categories.For example,if one is 4th,3rd and 6th,their final score is 4.3.6=72.The higher your score,the lower your overall ranking.Hannah ranks 1st in two of the categories.What is her lowest possible overall ranking? 在一次攀岩比赛中，13名运动员进行三个类别的比赛。每个参赛者的最终得分是他们在三个类别 中排名的乘积。例如，一名运动员在三个类别的排名是第4名、第3名和第6名，那么他的最终得分 为4·3.6=72。参赛者的分数越高，总体排名就越低。Hannah在其中两个类别中均排名第一，她的 总体排名最低可能是多少？ (A)2nd第二名(B)3rd第三名(C)4th第四名(D)5th第五名 (E)6th第六名 23.A spiral of consecutive numbers is created,as shown,starting with 1.When the pattern of the spiral is continued,in which arrangement will the numbers 625,626 and 627 appear? 从数字1开始，呈螺旋形排列后面的连续数字，如图所示。数字螺旋上升过程中，625、626和627 将呈现以下哪种排列方式？ 10→11→12-→13 个 ↓ 24 9 2 3 14 ↑ ↑ 23 1 又 15 ↑ ↑ ↓ ↓ 22 7←6←5 16 ↓ 21 ←20 ←-19← 18 ← 17 627 个 626→627 (A)626 (B) ↑ (C)625 →626+627 ↑ 625 625 625 625→626 (D) ↓ (E)626 627 ↓ 627 7 CS CamScanner 信然8ao Math Kangaroo 2023 (China)-Level F 24.A block in the shape of a regular tetrahedron has one face shaded. 一个正四面体有一个面涂有阴影。 D START The shaded face of the block is placed on the board on the triangle labelled START.The block is then rolled from one triangle to the next by rotating it about one edge.On which triangle will the block stand for the first time again on its shaded face? 将这个立方体的阴影面放置在上方木板标有START的三角形上，然后立方体沿一条边进行翻转， 从木板上的一个三角形滚动到下一个三角形。立方体的阴影面会再次首先滚动到哪个三角形上？ (A)A (B)B (c)c (D)D (E)E 25.Part of the fifth degree polynomial shown cannot be seen because of an inkblot.It is known that all five roots x5-11x4 of the polynomial are integers.What is the highest power of x-1 that divides the polynomial? 如图所示，一个五次多项式有一部分被墨水遮挡了。已知多项式的五个根都是整数，将该多项式进 行分解后得到的x-1的最高次项是什么？ (A)(x-1)1 (B)(x-1)2 (C)(x-1)3 (D)(-1)4 (E)(x-1)5 26.The large square in the diagram is dissected into four smaller squares. 6cm The circle touches the right hand side of the square at its midpoint.What is the side-length of the large square?Note that the diagram is not drawn 8cm to scale. 图中的大正方形被分成四个小正方形。圆与正方形右侧边的中点相切。 大正方形的边长是多少？请注意，图形未按照比例绘制。 (A)18 cm (B)20 cm (C)24 cm (D)28 cm (E)30 cm CS CamScanner v高aoo Math Kangaroo 2023 (China)-Level F 27.What is the greatest common divisor of all numbers of the form n3(n+1)3(n+2)3(n+3)3(n+4)3, where n is a non-zero natural number? 以n3(m+1)3(m+2)3(m+3)3(m+4)3形式表示的所有数字的最大公约数是多少？其中n是非零 自然数。 (A)2933 (B)233353 (C)263353 (D)283253 (E)293353 28.The numbers from 1 to 11 are to be placed in the hexagons so that the sum of the three numbers around each of the six black dots is the same. Three of the numbers have already been placed.What number will be placed in the hexagon with a question mark? 要将数字1到11放在右侧六边形中，使得每个黑点周围的三个数字之和 都相同。已经放好了三个数字，带问号的六边形中会放置哪个数字？ (A)1 (B)3 (C)5 (D)7 (E)9 29.Two identical cylindrical water tanks contain the same amount of water.One cylinder is standing upright,and the other is leaning against it,and the water level in each of them is the same as in the picture.The bottom of each of the cylinders is a circle with area 3m2 How much water does each tank contain? 两个完全相同的圆柱形水箱中装有相同体积的水。一个水箱直立放置，另一个斜靠在这个水箱上 面，两个水箱的水位相同，如图所示。两个圆柱形水箱的底部都是一个面积为3πm2的圆。每个水箱 里装有多少水？ (A)3v3πm3 (B)6πm3 (C)9Tm3 (E)it's impossible to determine from the information given根据已知信息无法确定 30.The product of six consecutive numbers is a 12-digit number of the form abb cddcdd abb. where the digits a,b,c and d are themselves four consecutive numbers in some order.What is the value of the digit d? 六个连续数字之积是一个12位数，可以表示为以下形式 abb cdd cdd abb, 其中数字a、b、c和d以某种顺序排列是四个连续数字。数字d的值是多少？ (A)1 (B)2 (C)3 (D)4 (E)5 9 CS CamScanner 题号 管案 题号 管案 题号 管案 题号 管案 题号 筲案 1 C 2 C 3 E 4 D 5 B 6 D 8 B 9 E 10 E 11 12 4 13 E 14 15 包 16 0 17 18 B 19 C 20 B 21 22 B 23 C 24 E 25 26 B 27 E 28 E 29 C 30 C 解析： 26:A 6cm +2 8cm X X+8 如图，利用直径的圆周角是直角构造勾股定理得： (2x+8)2=(x2+(x+2)2)+(x+2)2+(x+8)2) 可解得x=1,所以大正方形边长为18厘米。 27:E 当n=1时，可取得全体数字的最大公约数，即23353。 28:E 如图设未知数： 利用题意中的关系分别表示出剩余格： 6 a+2 b+4-x 11 最后利用最右侧上下的关系，可得：b+6-x+a+x-y=b+4-x+11 即a+x-y=9,故右上角数字为9. 再根据具体数字关系可填出全部： 10 29:C 第二个水箱中水的体积正好等于和它同高的圆柱体积的一半。又因为两杯中体积相同，则第 二杯的高度为第一杯的两倍。故图中倾斜角正好等于30°。 过A做AD⊥BC于D,如图： 由题意，AB=2N5,则AC=6,AD=3 V=3π·3=9π 30:C 本题需数学推理分析，我们可以知道连续6个数字中必然有多个2的倍数以及5的倍数， 所以b=0,又末尾有且仅有2个0，那么可以分析得到这连续6个数字乘积中必然有且仅 有52，再可分析得乘积中必然有24，那么此数必然为400的倍数，又己知a,b,c,d是连续的 数字，可以判断为0,1,2,3，所以a=2是剩余的唯一偶数。 同样再根据质因数分析知3一定整除该数，那么2a+4b+2c+4d一定是3的倍数，经分析 d=3是唯一可能性。