2014 AMC 10B Problems Problem 1 Leah has 13 coins^ all of which are pennies and nickels. If 5h已 had one more nickel than she has now, then she would h^ve the same number of pennies and nickels, lr )匚ents, how much are Leah's coins worth?(A) 33 (B) 35 (C) 37 (D) 39 (E) 41Problem 223 4 2s 2-彳 + 2-^ Problem 3Randy drove the first third of his trip on a gravel road^ the next 20 miles on pavement, and ths remaining □ne -fifth on a dirt road . In miles how long was Randy's tripi?(A) 30 (B)等 (C)粤(D) 40 (E)竽Problem 4Susie pays for 4 muffins and 3 bananas. Calvin spends twice as much paying for 2 muffins and 16 bananas. A muffin is. how many times as expensive as a banana?m 吨 了(A) | (B)彩 (C) f (D) 2 (E)-Problem 5Doug constructs a square window using 8 equal-size panes of glass, as shown ・ The ratio of the height to width ft3『 each pane is 5 : 2f and the borders around and between the panes are 2 inches wide, in inches, what 谄 the sid& length of th© square wind 口w?(A) 26 (B) 28 (C) 30 (D) 32 (E) 34Problem 6Orvin went to the store with just enough rrnoney to buy 30 balloons. When h@ arrived^ he discovered that the store had a special sale on b a I loons ：: buy 1 balloon at the regular p 『i 匚已 and get a second at g off the regular price. What is the greatest number of balloons Orvin could buy?(A) 33 (B) 34 (C) 36 (D) 38 (E) 39what im(A) 16 (B) 24 (C) 32 (D) 48 (E) 64Problem 7Suppose A> U > 0 and A is 近% greater than B. what is £?⑷100(#) (B) 1 叫字)(C) 100(字)(D) 1 叫乎)(E) 100(和SolutionProblem 8A truck tr^ve>l5 f feet every t seconds. There ars 3 feet in a yard i How many yards dDss the truck travel6in 3 rninute-s?b frn 血m10i 1 口⑷廖(X (①〒⑴)石㈣〒Solutionproblem 9For real numbers w and z,丄+丄予二于=2014,ut ww + zWhat is ----------- ?w ― z、一1 J 1(A) - 2014 (B)艄(C) —(D) 1 (E) 2014Problem 10In the addition shown below A. [3. C.and D are distinct digits. How many different values are po^ible ForD?ABBCB+ BCADADI3DDD(A) 2 (E) 4 (C) 7 (D) 8 (E) 9Problem 1111. For the consume^ a single discount of n%is more adv^ntagsous than any of the following discounts:(1)two ^ucces^ive 15% discounts(2)three successive 10% discounts(3) a 25f/J discount Fellow sd by s 5xu discountWhat is the- smallest possibls positive intsgsr valus uf n?(A) 27 (B) 2S (C) 29 (D) 31 (E) 33Problem 1212. The largest divisor of 2, 014h 000* 000 is rtsslf. What is its fifth largest divisor?(A) 125t875.000 (B) 20L400. 000 (C) 251, 750.000 (D) 402： 800.000 (E) 503.500.000Problem 13Sin regular hexagons surround 召regular hexagon of ^ide length 1 as shown. What is the ar?a of A J4BC?(A) 2V5 {B) 3V3 (C) 1+3辺(D) 2 + 2“$(E) 3+2\/3Problem 14Oariica drove her new car on a trip for a whole number of hour勺averaging 55 miles per hour. At the beginning of the trip」abc miles was displayed on thm Dclannstsr^ where abc is 耳3-digit number with ti > 1 and « + 6 —c < 7. At the end of the trip，the odometer showed eba miles. What is(i~ + fc2+ c~ ?(A) 26 (B) 27 (C) 36 (D) 37 (E) 41Problem 15In rectangle 打<7 = 2GR and points E3nd F lie on AF£□ that ED and FD tnsect Z.ADCas shown, what is the ratio of the area of ADEF to the ares of rectangle ABCD?S)晋(B)^ ©瞬㈣笫Problem 16Four fair six-sided dice are rolled. What is the probability that at least three of the four dice show the sarme value? (A)吉(B)12(C) I {D)島㈣|Problem 17What is thm greatest power of 2 t hat if a fmutor of 101— 4U：，01?Problem 18(A) 21DO2(B) 21003(C) 21004(D) 21C05(E) 21D0C A list of 11 positive! integers has a mean of 10；a median of 9j and a uiniqus> mads af 8- What is the largest passible value of an integer in the list?(A) 24 {B) 30 (C) 31 (D) 33 (E) 35Problem 19Two ccrncentric circles have radii 1 and 2. Two points an the outer circle are chosen independently and uniformly at random. What iw the probabi^it/ that the chord joining the two points in tersects the inner circle?(A) g (B) i (C)匕詳(D) i (E) +Problem 20For how many integers 工is the number —51J:2 + 50 negative?(A) 8 (B) 10 (C) 12 (D) 14(E) 16Problem 21Trapezoid ABCD has parallel sides AB af length 33 and CD of length 21. The other two sides are of lengths 1(' and 14. The angles at A and B are acute. What js the Imn gth of the shorter diagonal of ABCD?{A) 10?6 (B) 25 (C) S\/10 (D) 18“(E) 26SolutiariProblem 22Eight semicircles line the insid目af a ^qu^re with 吕id日length 2 as shown. What is the radium Q F the circle tangent tc ell of these semicircles?A sphere is inscribed in a truncated right circular cons as shown. The volume of thm truncated cone is twice that of the sphere. What is the ratro of the radius af the bottam base of the truncated cone to the radius of the tap base of the truncated cone?Problem 23(A) | ©近(D)2 (E)^^ Problem 24The numbers 1, 2} 3}斗』5 are to be arranged in a circle. An arrangement rs bad if rt is not true that fcr every n from 1 to 15 one can find a subset of the numberm that appear consecutively on the circle that sum to J ?. Arrangements that differ only by a rotation or a reflection mne considered the same. Haw many different bad arrangements are there?(A) 1 (B) 2 (C) 3 (D) 4 (E) 5 Problem 25In a small pond there are eleven lily pads in a row labeled 0 through 10, A frog is sitting on pad 1. When the frog is on pad N } 0 < N < 10, it 呷illju 叩 to pad N-l with probability — and to pad N +1 with probability 1 - Each jump is independent of the previous jumps. If the frog reaches pad 0 it will be eaten by a patiently waiting snake, If the frog reaches pad 10 it will exit the pond> never to return, what is the probability that the frog will escape being eaten by the snake?| (B)曙(C)磊(D)纟(E) \答案：1. C2. E3. E4. B5.A6. C7. A8. E9. A10. C11. C12. C13. B14. D15. A16. B17. D18. E19. D20. C21. B22. B23. E24. B25. Cwwt 冷10诲+ * +矿叮(A) 3 (B) & (C) y (D)爭 (E) 170Problem 2Roy's 亡吕t m吕t 百 of a can cf eat food ev&ry morning and y of a ean of eat food every evening. Before 怡目ding hiis 匚 a )t □“ Mlonday morn in Rciy ope me d a bck containing 6 cans of eat feotl*. On what day of the wisek dbd the 匚已t finish eating all the cat food in the box? (A) Tuesday (U) Wednesday (C) Thur 吕d 暫(U) Friday (E) Saturday Problem 3 Bridget bakes 48 loaves of bread for her bakery. She sells half of them in the morning for 32,501 each. In the aftemaon she sells two thirds of what she has left 3 and bscauss they are not freshshe charges only half priice. Tn the late aftern 口口n shethe r^mainiiing loaves -at B dollar leach. Each loaf 匚口百上鼻S (J_75 farher to make, In dollarSj what is her profit for the day? (A) 24 (B) 36 (C) 44 (D) 4S (E) 52Problem 4Walking down Jane Street, Ralph passed four houses in 吕 row, each painted a different 匚olcr- He passed th® orange house before the red housSi and the passed the blue house before ths yellow house. Ths blue house was not n&^t to t.h& ysillow house. How many orderings of the colored fhouses 囂「白 possible?(A) 2 (B) 3 (C) 4 (D) 5 (E) 6Problem 5On an 3l^eb rs quiz d 10% of Elis students scored 70 point 勺 35% scored 80 points^ 30% scored 90 pointSj and the rest scored 100 points. What is the differsincc between ths mean and median score cf ths students' scares on this quis?(A) 1 (B) 2 (C) 3 (D) 4 (E) 5Problerri 6SuppoBis that o 匚OWE ; give 6- gallons of milk in n days. At thus rate, how marTpr gallanE 口f nnilk will d cows give in c days?Probfem 7IN 口 r^zSrd 『自 £l nurfib&f£ ir. 些、ci f and b S-Stiisfy J : < LL <L b. Hdw marhy Gf th€i Follow in g id 白口u£ lit i 自弓 mustba true?(I) H + 空 V ci + b(II) Ji — y < a — 6(III) xy < ab(A) 0 (B) 1 (C) 2 (D) 3 (E) 4(C)cibdc (D) bcdeProblem 8 Which of the following niumbers Is a perfect square?Problem 9(A) 14115!~^T~(D)17fl81~2~18?193~2~Tfie two legs of a right trimngl已which are altitudes, ha\/e lengths 2\/3 and 6. How long is the third altitude of the triangte?(A) 1 (B) 2 (C) 3 (D) 4 (E) 5Problem 10Five positive consecutive integers starting with a have average b. what is the average of 5 consecutive integers that start with b?(A) d + 3 (B) ti + 4 (C) a 4^5 (D) u 4- 6 {E) <i + 7Problem 11A cusEomer who intends to purchase art appliance has three coupon5^ only ore of which may be uw总d;Coupon 1: .10/( off the listed price if the listed price rs at least S50Coupon 2. $20 off the listed price if the listed pnce is at Itfast 1100coupon 3： 18*X off the amount by which the listed price e^teeds $100For which of the following listed prices will coupari 1 offer a greater price reduction than either coupon 2 or coupon 3?(A) S179.95 (B) S199.95 (C) S219.95 (D) $239.95 (E) $259.95Problem 12A regular hexagon has side length 6. Ccngruerit arcs with radius 3 are drawn with the center at each of the vertical, creating circular sac tors 启弓shown. The region inside the ha^agon but outside the sectors is shaded as shown Whe t is thm^rea of the shaded regian?(A)幫価—跖(B) 27V3-&r (C) 54?5-18TT(D) &4^3-12^(E) 伽Problem 13Equilateral A>1R(7 hms side length 1』and squmrms AB DE, BOH I, CAFG bs outbids the triangle. What is the area of hexagon DEFGH I?Problem 14Ths y-intencBptSj P目nd Q, of two perp&ndicjldr lines intersecting mt the point j4(6h 8)have m sunn of zero. What is the area of AAPQ?(A) 45 (B) 4S (c) 54 (D) M (E) 72Problem 15Oavidl drives from his home to the airport to cat匚h a flight. He drives 35 milES in th^ first hour, but realizes that he will be 1 hour late if he continues at this speed. He increases his speed by 15 miles per hour for the rest of the way to the airport and arrives 30 minutes early. How many miles is the airport from his home?(A) 140 (B) 175 (C) 210 (D) 245 (E) 280Problem 16Fn rectangle A拜C0 A/J =L OC7 = 2』and points E, F、and <7 are midpoints of f 门』-and respectively・Point H is the midpoint of GE. What is the arsa of the shaded region?12 + 3>/3-4-(C) 3 + v5 (D)(E) 6/ Z?⑷吉㈣兽©害 (D)卷(E) 111 / 12Problem 17Three fair six-sided dn^e are rolled. What 祐 the prob ability that the values shown on two of the dice sum to the value shown an the rematning die?(A) j (B)第(C)籟(D)备(E) | Problem 18A square in the coordinate plane has vertices whose y-co ordinates are 0』1』4, and 5. What is the area of thm square?(A) 16 (B) 17 (C) 25 (D) 26 (E) 27 Problem 19Four cubes with edge lengths 1, 2, 3, and 4 are stacked as shown. What is the length of the portion of XV contdin^d in th© cub@ with edge length 3?(A)警 (B) 2^/3 (C)警 (D) 4 ㈣ 3闪Problem 20The product (8)(888 . .. 8), where ttie second factor has k digits, is an integer whose digits have a sum of 1000. What is fc?(A) 901 (B) 911 (C) 919 (D) 991 (E) 999Problem 21Positive integers a and b are such that the grsplis of = EAX + & and y = 驻+ b intersect this ir-a^is at ths same point. What is the sum erf all possible jr-coondinat&s of ths SB points of intersection?⑷-20 (B) -18 (C) 一苗(D) -12 (E) —8Problem 22[n r直亡tanglm AnC'D r AB= 20 and BG =10. Let £? be a point Ort CD such that Z.CBE= 15°. What is AE?(A) (B) IO A/3(C) 18 (D) 11 辺(E) 20Problem 23A rectmngubr piece of paper whose length is V3 times the width has area A, The paper is divided into three equal sections along the opposite I eng th Sj artd then a dotted fine is drawn from the first divider to the 亡end divider cn the opposite side shown. The paper 左then fbldEd flat along thiim dotted line to cnemtm a. new whmp日with area /?. What is the ratio J? : A?(A) 1 : 2 (B) 3 : &Problem 24A ssquanc^ cf natural nuimber^ i£cori£trut?ted by listing the first 4H then skipping one^ fci sting the newt 5, skipping 2. listing 6』skipping 3d andj on the nth itEration, listing n + 3 and skipping n. The sequence bsgins L 2h 3h 4, 6, 7. 8. 9.10. 13. What is ths 500- QOOth number in the sequence?(A) 996,506 (B) 996507 (C) 996508 (D) 996509 (E) 996510Problem 25The number 5B67is between 22011 and 2Z014r now many pairs or integers ^re there such thmt(C) 2 : 3 (D) 3 : 4 (E) 4 : 5Problem 201 < m < 2012 and(A) 278 (B) 279 (C) 280 (D) 281 (E) 282答案：1.C 2.D 3.D 4.B 5.E 6.B 7.B 8.B 9.D 10.C 11.C 12.C 13.C 14.C 15.B 16.B 17.D 18.E 19.D 20.B 21.C 22.A 23.C 24.B 25.A。