2020年美國數學競賽(AMC10A)的試題與解答

廣東省廣州市華南師范大學(510631) 李湖南

1.What value ofxsatisfies

譯文方程的解x是多少?

解化簡可得,故(E)正確.

2.The numbers 3,5,7,a,andbhave an average(arithmetic mean)of 15.What is the average ofaandb?

A.0 B.15 C.30 D.45 E.60

譯文數字3,5,7,a和b的平均值(算術平均值)是15.則a和b的平均值是多少?

解依題意有3+5+7+a+b=15×5,則a+b=60,平均值為30,故(C)正確.

3.Assuming34 and5,what is the value in simplest form of the following expression?

A.?1 B.1 C.D.E.

譯文設則表達式的最簡形式是什么?

解約分即得故(A)正確.

4.A driver travels for 2 hours at 60 miles per hour,during which her car gets 30 miles per gallon of gasoline.She is paid$0.50 per mile,and her only expense is gasoline at$2.00 per gallon.What is her net rate of pay,in dollars per hour,after this expense?

A.20 B.22 C.24 D.25 E.26

譯文一位司機以60 英里/小時的速度駕車2 小時,她的車每跑30 英里需要消耗1 加侖汽油.她能獲得0.50 美元/英里的報酬,唯一的花費就是2 美元/加侖的汽油.問她每小時除去消耗之后的凈收益是多少美元?

解1 個小時她能跑60 英里,獲得60×0.50=30 美元,汽油費為60÷30×2 = 4 美元,故凈收益為26 美元,(E)正確.

5.What is the sum of all real numbersxfor which

A.12 B.15 C.18 D.21 E.25

譯文滿足方程的所有實數x之和是多少?

解分別解方程x2?12x+34=2 和x2?12x+34=?2,可得x1,2=4,8 和x3,4=6,故和為18,(C)正確.

6.How many 4-digit positive integers(that is,integers between 1000 and 9999,inclusive)having only even digits are divisible by 5?

A.80 B.100 C.125 D.200 E.500

譯文有多少個四位的正整數(也就是在1000 和9999之間的整數)能被5 整除且所有數字均為偶數?

解依題意,符合條件的四位數的個位數只能是0,十位數和百位數可以是0,2,4,6,8,千位數只能是2,4,6,8,共有1×5×5×4=100 種選擇,故(B)正確.

7.The 25 integers from?10 to 14,inclusive,can be arranged to form a 5-by-5 square in which the sum of the numbers in each row,the sum of the numbers in each column,the sum of the numbers along each of the main diagonals are all the same.What is the value of this common sum?

A.2 B.5 C.10 D.25 E.50

譯文將25 個整數分別是從?10 到14,放入5×5 的格子中,使得格子里的每行、每列和兩條對角線的數字和均相等.問這個數字和是多少?

解這是一個5 階幻方問題,25 個數字之和是(?10)+(?9)+···+13+14 = 50,分別放入5 行,故每行的數字和是10,(C)正確.

8.What is the value of 1+2+3?4+5+6+7?8+···+197+198+199?200?

A.9800 B.9900 C.10000 D.10100 E.10200

譯文1+2+3?4+5+6+7?8+···+197+198+199?200的值是多少?

解原式= 1 +(2 +3?4)+5 +(6 +7?8)+···+197+(198+199?200)= 2×(1+5+···+197)=,故(B)正確.

9.A single bench section at a school event can hold either 7 adults or 11 children.WhenNbench sections are connected end to end,an equal number of adults and children seated together will occupy all the bench space.What is the least possible positive integer value ofN?

A.9 B.18 C.27 D.36 E.77

譯文在某學校的活動中,一條長凳可以坐7 個成人或者11 個兒童.當N條長凳首尾相接的時候,剛好坐滿了相同數量的成人和兒童.問N的最小正整數值是多少?

解設有x條長凳坐了兒童,則有N ?x條長凳坐了成人,依題意有11x= 7(N ?x),因而故Nmin=18,(B)正確.

10.Seven cubes,whose volumes are 1,8,27,64,125,216,and 343 cubic units,are stacked vertically to form a tower in which the volumes of the cubes decrease from bottom to top.Except for the bottom cube,the bottom face of each cube lies completely on top of the cube below it.What is the total surface area of the tower(including the bottom)in square units?

A.644 B.658 C.664 D.720 E.749

譯文七個立方體,體積分……