2023年第25届AMC10B国际数学竞赛（高一及以下学生）试题

2023 MAA AMC 10B 1.Mrs.Jones is pouring orange juice into four identical glasses for her four sons.She fills the first three glasses completely but runs out of juice when the fourth glass is only full.What fraction of a glass must Mrs.Jones pour from each of the first three glasses into the fourth glass so that all four glasses will have the same amount of juice? Jons女士将橙汁倒入四个相同的玻璃杯中给她的四个儿子.她将前三杯装 满后，第四杯装到3满时，果汁就没有了，要使得所有四个玻璃杯中有相同 数量的橙汁，Joes女士必须将前三个玻璃杯的每一个中橙汁的几分之几倒 入第四个玻璃杯？ 1 A)12 (B) 1 (6 D)8 2.Carlos went to a sports store to buy running shoes.Running shoes were on sale,with prices reduced by 20%on every pair of shoes.Carlos also knew that he had to pay a 7.5%sales tax on the discounted price.He had $43. What is the original(before discount)price of the most expensive shoes he could afford to buy? Carlos去一家体育用品店买跑鞋.跑鞋正在促销，每双鞋降价20%.Carlos 还知道他必须按照折扣价的7.5%缴付消费税.他有$43.问他能买得起的最 贵的跑鞋的原价（即折扣前）是多少？ (A)$46 (B)$50 (C)$48 (D)$47 (E)$49 1 2023 MAA AMC 10B 3.A 3-4-5 right triangle is inscribed in circle A,and a 5-12-13 right triangle is inscribed in circle B.What is the ratio of the area of circle A to the area of circle B? 三边长为3,4,5的直角三角形内接于圆A,三边长为5,12,13的直角三 角形内接于圆B.问圆A的面积与圆B的面积之比是多少？ 9 1 B) 1 (C 25 D)169 4 (E)25 4.Jackson's paintbrush makes a narrow strip with a width of 6.5 millimeters. Jackson has enough paint to make a strip 25 meters long.How many square centimeters of paper could Jackson cover with paint? Jackson的画笔可以画出宽度为6.5毫米的窄条.Jackson有足够的油漆来制 作25米长的窄条.问Jackson可以用颜料覆盖多少平方厘米的纸？ (A)162,500 (B)162.5 (C)1,625 (D)1,625,00 (E)16,250 2 2023 MAA AMC 10B 5.Maddy and Lara see a list of numbers written on a blackboard.Maddy adds 3 to each number in the list and finds that the sum of her new numbers is 45. Lara multiplies each number in the list by 3 and finds that the sum of her new numbers is also 45.How many numbers are written on the blackboard? Maddy和Lara看到了黑板上写着的一列数.Maddy将这列数中的每个数 加3，发现她得到的所有新数的总和为45.Lara将这列数中的每个数乘以3， 发现她得到的所有新数的总和也是45.问黑板上写了多少个数？ (A)10 (B)5 (C)6 (D)8 (E)9 6.Let L1 1,L2 =3,and Ln+2 Ln+1+Ln for n 1.How many terms in the sequence L1,L2,L3,...,L2023 are even? 令L1=1,L2=3,当n≥1时，Lm+2=Lm+1十Lm,问数列L1,L2,L3, .·,L2023中有多少项是偶数？ (A)673 (B)1011 (C)675 (D)1010 (E)674 3 2023 MAA AMC 10B 7.Square ABCDis rotated 20 clockwise about its center to obtain square EFGH, as shown below.What is the degree measure of /EAB 如下图所示，正方形ABCD绕其中心顺时针旋转20°，得到正方形EFGH. 问∠EAB的度数是多少？ F G (A)24 (B)35° (C)30° (D)32° (E)20° 8.What is the units digit of 20222023+20232022? 20222023+20232022的个位数字是多少？ (A)7 (B)1 (C)9 (D)5 (E)3 2023 MAA AMC 10B 9.The numbers 16 and 25 are a pair of consecutive positive perfect squares whose differenceis 9.How many pairs of consecutive positive perfect squares have a difference of less than or equal to 2023 数16和25是一对连续的正完全平方数，它們的差是9.问有多少对连续的 正完全平方数，它们的差小于或等于2023？ (A)674 (B)1011 (C)1010 (D)2019 (E)2017 10.You are playing a game.A 2 x 1 rectangle covers two adjacent squares(ori- ented either horizontally or vertically)of a 3 x 3 grid of squares,but you are not told which two squares are covered.Your goal is to find at least one square that is covered by the rectangle.A "turn"consists of you guessing a square,after which you are told whether that square is covered by the hid- den rectangle.What is the minimum number of turns you need to ensure that at least one of your guessed squares is covered by the rectangle? 你在玩下述的游戏.一个2×1矩形覆盖了3×3方格表中的两个相邻方格 (依水平或竖直方向)，但你不知道是哪两个方格被覆盖了，你的目标是找到 至少一个被矩形覆盖的方格.一个“回合”是指，你猜测是某个方格，然后被 告知这个方格是否被矩形覆盖.为确保至少猜中一个被矩形覆盖的方格，你 最少需要多少个回合？ (A)3 (B)5 (C)4 (D)8 (E)6 5 2023 MAA AMC 10B 11.Suzanne went to the bank and withdrew $800.The teller gave her this amount using $20 bills,$50 bills,and $100 bills,with at least one of each denomination.How many different collections of bills could Suzanne have received? S1 zanne去银行提取800美元.出纳员用20美元纸币、50美元纸币、100美 元纸币给了她这笔钱，每种面值的纸币至少有一张.问Suzanne得到的纸币 有多少种不同的可能组合？ (A)45 (B)21 (C)36 (D)28 (E)32 12.When the roots of the below polynomial are removed from the real number line,what remains is the union of 11 disjoint open intervals.On how many of these intervals is P(x)positive? 当把下面的多项式的根从实数轴上移除后，剩下的是11个不相交的开区间 的并集.问在其中的多少个区间上，P(x)为正值？ P(x)=(x-1)1(x-2)2(x-3)3.·(x-10)10 (A)3 (B)7 (C)6 (D)4 (E)5 2023 MAA AMC 10B 13.What is the area of the region in the coordinate plane defined by the below inequality? 坐标平面中由下列不等式定义的区域的面积是多少？ lx-1+ly-1≤1？ (A)2 (B)8 (C)4 (D)15 (E)12 14.How many ordered pairs of integers(m,n)satisfy the equation m2+mn+ n2=m2n2? 满足方程m2+mn+n2=m2n2的有序整数对(m,n)有多少个？ (A)7 (B)1 (C)3 (D)6 (E)5 15.What is the least positive integer m such that m.2!.3!.4!.5!...16!is a perfect square? 使得m·2!·3!.4!.5!…16!是完全平方数的最小正整数m是多少？ (A)30 (B)30,030 (C)70 (D)1430 (E)1001 7 2023 MAA AMC 10B 16.Define an upno to be a positive integer of 2 or more digits where the digits are strictly increasing moving left to right.Similarly,define a doronno to be a positive integer of 2 or more digits where the digits are strictly decreasing moving left to right.For instance,the number 258 is an upno and 8620 is a downno.Let U equal the total number of upnos and D equal the total number of downnos.What is U-D? 对于一个有2位或者更多位的正整数，若它的数字从左到右严格递增，则定 义其为“上升数”类似地，对于一个有2位或者更多位的正整数，若它的数字 从左到右严格递减，则定义其为“下降数”.例如，数258是上升数，8620则 是下降数.令U为上升数的总个数，D为下降数的总个数.问U一D是多 少？ (A)512 (B)10 (C)0 (D)9 (E)511 17.A rectangular box P has distinct edge lengths a,b,and c.The sum of the lengths of all 12 edges of p is 13,the sum of the areas of all 6 faces of p 11 isand the volme ofPisWhat is the ength of the interio diagonal connecting two vertices of P? 一个长方体盒子P的三条不同的棱长为？，b,C.P的所有12条棱的长度之 11 和为13，P的所有6个面的面积之和为2，并且P的体积为2：问连接P 的两个顶点，且在其内部的对角线最长是多少？ (A)2 (B) 9 (C (D) (E) 3-2 8 2023 MAA AMC 10B 18.Suppose that a,b,and c are positive integers such that ab c 14+15=210 Which of the following statements are necessarily true? I.If gcd(a,14)=1 or gcd(b,15)=1 or both,then gcd(c,210)=1. II.If gcd(c,210)=1,then gcd(a,14)=1 or gcd(b,15)=1 or both. III.gcd(c,210)=1 if and only if gcd(a,14)=1 and gcd(b,15)=1. 假设a,b,c是正整数，满足 b 14+15=210· 问以下哪些陈述一定是正确的？ I.如果gcd(a,14)=1,或者gcd(b,15)=1,或者两式都成立，那么 gcd(c,210)=1. I.如果gcd(c,210)=1,那么gcd(a,14)=1,或者gcd(b,15)=1,或者 两式都成立 IΠ.gcd(c,210)=1,当且仅当gcd(a,14)=1,并且gcd(b,15)=1. 这里gcd表示最大公约数 (A)L,IL,andI[I,IⅡ和IⅢ (B)I only|只有I (C)I and II only|只有I和IⅡ (D)III only|只有III (E)IⅡand III only|只有IⅡ和II 9 2023 MAA AMC 10B 19.Sonya the frog chooses a point uniformly at random lying within the square [0,6]x [0,6]in the coordinate plane and hops to that point.She then chooses a distance uniformly at random in the interval [0,1]and a direction uni- formly at random from [north,east,south,west).All her choices are inde- pendent.She now hops the chosen distance in the chosen direction.What is the probability that she lands outside the square? 青蛙Sonya在坐标平面中由[0,6]×[0,6]形成的正方形内均匀随机的选择一 个，点，并且跳到该点.然后，她在区间[0,1]中均匀随机的选择一个距离，并 从{东、南、西、北}中均匀随机的选择一个方向，她的所有选择都是独立的 现在，她朝着选定的方向跳跃选定的距离，问她落在正方形外的概率是多少？ (A) 6 1 B)12 (C D)0 (E) 10