AIME Exams Merged

8/10/2019 AIME Exams Merged

1/18

USA

AIME

2014

I

1 The 8 eyelets for the lace of a sneaker all lie on a rectangle, four equally spaced on each of thelonger sides. The rectangle has a width of 50 mm and a length of 80 mm. There is one eyeletat each vertex of the rectangle. The lace itself must pass between the vertex eyelets along awidth side of the rectangle and then crisscross between successive eyelets until it reaches thetwo eyelets at the other width side of the rectrangle as shown. After passing through thesefinal eyelets, each of the ends of the lace must extend at least 200 mm farther to allow a knotto be tied. Find the minimum length of the lace in millimeters.

2 An urn contains 4 green balls and 6 blue balls. A second urn contains 16 green balls andNblue balls. A single ball is drawn at random from each urn. The probability that both ballsare of the same color is 0.58. Find N.

3 Find the number of rational numbersr, 0< r


2/18

USA

AIME

2014

6 The graphs ofy = 3(x h)2 +j and y = 2(x h)2 + k have y-intercepts of 2013 and 2014,respectively, and each graph has two positive integer x-intercepts. Find h.

7 Let w and z be complex numbers such that|w| = 1 and|z| = 10. Let = arg wzz

. The

maximum possible value of tan2 can be written as pq

, where p and qare relatively primepositive integers. Findp + q. (Note that arg(w), forw= 0, denotes the measure of the anglethat the ray from 0 to w makes with the positive real axis in the complex plane.

8 The positive integers N and N2 both end in the same sequence of four digits abcd whenwritten in base 10, where digit a is not zero. Find the three-digit number abc.

9 Letx1 < x2 < x3 be three real roots of equation 2014x34029x2 +2 = 0. Find x2(x1 + x3).10 A disk with radius 1 is externally tangent to a disk with radius 5. LetA be the point where

the disks are tangent, Cbe the center of the smaller disk, and Ebe the center of the largerdisk. While the larger disk remains fixed, the smaller disk is allowed to roll along the outsideof the larger disk until the smaller disk has turned through an angle of 360. That is, if thecenter of the smaller disk has moved to the point D, and the point on the smaller disk thatbegan at A has now moved to point B , then AC is parallel to BD. Then sin2(BE A) = m

n,

where mand n are relatively prime positive integers. Find m + n.

11 A token starts at the point (0, 0) of an xy-coordinate grid and them makes a sequence of sixmoves. Each move is 1 unit in a direction parallel to one of the coordinate axes. Each moveis selected randomly from the four possible directions and independently of the other moves.The probability the token ends at a point on the graph of|y|=|x| is m

n, where m and n are

relatively prime positive integers. Findm + n.

12 LetA ={1, 2, 3, 4}, andfandg be randomly chosen (not necessarily distinct) functions fromA to A. The probability that the range offand the range of g are disjoint is m

n, where m

and n are relatively prime positive integers. Find m.

13 On square ABCD, points E, F,G, and H lie on sides AB,BC,CD, and DA, respectively,so that EG F H and EG = F H = 34. Segments EG and F H intersect at a point P,and the areas of the quadrilaterals AEPH,BFPE,CGPF, and DHPG are in the ratio269 : 275 : 405 : 411. Find the area of square ABCD.

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3/18

USA

AIME

2014

A

B C

DH

F

E

G

P

w

x

y

z

w : x : y : z = 269 : 275 : 405 : 411

14 Let m be the largest real solution to the equation

3

x

3

+ 5

x

5

+ 17

x

17

+ 19

x

19

=x2 11x 4.

There are positive integers a, b, c such that m = a +

b +

c. Find a + b + c.

15 InABC, AB = 3, BC= 4, and CA = 5. Circle intersects AB at E and B, BC at Band D, and AC at F and G. Given that EF = DF and DG

EG = 34 , length DE=

ab

c , where

a and c are relatively prime positive integers, and b is a positive integer not divisible by thesquare of any prime. Find a + b + c.


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4/18

USA

AIME

2014

II

1 Abe can paint the room in 15 hours, Bea can paint 50 percent faster than Abe, and Coe canpaint twice as fast as Abe. Abe begins to paint the room and works alone for the first hourand a half. Then Bea joins Abe, and they work together until half the room is painted. ThenCoe joins Abe and Bea, and they work together until the entire room is painted. Find thenumber of minutes after Abe begins for the three of them to finish painting the room.

2 Arnold is studying the prevalence of three health risk factors, denoted by A, B, and C. withina population of men. For each of the three factors, the probability that a randomly selectedman in the population as only this risk factor (and none of the others) is 0.1. For any twoof the three factors, the probability that a randomly selected man has exactly two of thesetwo risk factors (but not the third) is 0.14. The probability that a randomly selected manhas all three risk factors, given that he has A and B is 13 . The probability that a man hasnone of the three risk factors given that he does not have risk factor A is p

q, where p and q

are relatively prime positive integers. Find p + q.

3 A rectangle has sides of length a and 36. A hinge is installed at each vertex of the rectangleand at the midpoint of each side of length 36. The sides of length a can be pressed towardeach other keeping those two sides parallel so the rectangle becomes a convex hexagon as

shown. When the figure is a hexagon with the sides of lengtha parallel and separated by adistance of 24, the hexagon has the same area as the original rectangle. Find a2.

4 The repeating decimals 0.abababand 0.abcabcabcsatisfy

0.ababab + 0.abcabcabc=33

37,

where a, b, and c are (not necessarily distinct) digits. Find the three-digit number abc.


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5/18

USA

AIME

2014

5 Real numbers r and s are roots of p(x) = x3 +ax + b, and r+ 4 and s3 are roots ofq(x) =x3 + ax + b + 240. Find the sum of all possible values of|b|.

6 Charles has two six-sided dice. One of the dice is fair, and the other die is biased so that itcomes up six with probability 23 , and each of the other five sides has probability

115 . Charles

chooses one of the two dice at random and rolls it three times. Given that the first two rollsare both sixes, the probability that the third roll will also be a six is p

q, where p and qare

relatively prime positive integers. Findp + q.

7 Let f(x) = (x2 + 3x + 2)cos(x). Find the sum of all positive integers nfor which

n

k=1

log10 f(k)

= 1.

8 Circle Cwith radius 2 has diameter AB. Circle D is internally tangent to circle C at A.Circle Eis internally tangent to circle C,externally tangent to circle D,and tangent to AB .The radius of circle D is three times the radius of circle Eand can be written in the form

m n, where m and n are positive integers. Find m + n.9 Ten chairs are arranged in a circle. Find the number of subsets of this set of chairs that

contain at least three adjacent chairs.

10 Let z be a complex number with|z| = 2014. Let P be the polygon in the complex planewhose vertices are z and every w such that 1

z+w = 1z

+ 1w

. Then the area enclosed by P can

be written in the form n

3,where nis an integer. Find the remainder when n is divided by1000.

11 InRED,RD = 1, DRE = 75 and RED = 45. Let M be the midpoint of segmentRD. Point C lies on side ED such that RC EM. Extend segment DE through E topoint Asuch that CA = AR. Then AE= a

b

c , where a and c are relatively prime positive

integers, and b is a positive integer. Find a + b + c.

12 Suppose that the angles ofABCsatisfy cos(3A)+cos(3B)+cos(3C) = 1. Two sides of thetriangle have lengths 10 and 13. There is a positive integer m so that the maximum possiblelength for the remaining side ofABC ism. Find m.

13 Ten adults enter a room, remove their shoes, and toss their shoes into a pile. Later, a childrandomly pairs each left shoe with a right shoe without regard to which shoes belong together.The probability that for every positive integer k < 5, no collection of k pairs made by thechild contains the shoes from exactly k of the adults is m

n,wherem and n are relatively prime

positive integers. Find m + n.


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6/18

USA

AIME

2014

14 InABC, AB = 10, A= 30, and C= 45. Let H, D, and Mbe points on line B Csuchthat AH BC, BAD = CAD, and BM = CM. Point N is the midpoint of segmentHM, and point P is on ray AD such that P N B C. Then AP2 = m

n, where m and n are

relatively prime positive integers. Findm + n.

15 For any integer k 1, let p(k) be the smallest prime which does not divide k. Define theinteger functionX(k) to be the product of all primes less than p(k) ifp(k)> 2, and X(k) = 1if p(k) = 2. Let{xn} be the sequence defined by x0 = 1, and xn+1X(xn) = xnp(xn) forn0. Find the smallest positive integer, t such that xt = 2090.

The problems on this page are copyrighted by the Mathematical Association ofAmericas American Mathematics Competitions.


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7/18

USA

AIME

2012

I

1 Find the number of positive integers with three not necessarily distinct digits,abc, witha= 0,c= 0 such that both abcand cbaare divisible by 4.

2 The terms of an arithmetic sequence add to 715. The first term of the sequence is increasedby 1, the second term is increased by 3, the third term is increased by 5, and in general, thekth term is increased by the kth odd positive integer. The terms of the new sequence add to836. Find the sum of the first, last, and middle terms of the original sequence.

3 Nine people sit down for dinner where there are three choices of meals. Three people order

the beef meal, three order the chicken meal, and three order the fish meal. The waiter servesthe nine meals in random order. Find the number of ways in which the waiter could serve themeal types to the nine people such that exactly one person receives the type of meal orderedby that person.

4 Butch and Sundance need to get out of Dodge. To travel as quickly as possible, each alternateswalking and riding their only horse, Sparky, as follows. Butch begins walking as Sundancerides. When Sundance reaches the first of their hitching posts that are conveniently locatedat one-mile intervals along their route, he ties Sparky to the post and begins walking. WhenButch reaches Sparky, he rides until he passes Sundance, then leaves Sparky at the nexthitching post and resumes walking, and they continue in this manner. Sparky, Butch, and

Sundance walk at 6, 4, and 2.5 miles per hour, respectively. The first time Butch and Sundancemeet at a milepost, they aren miles from Dodge, and have been traveling for t minutes. Findn + t.

5 Let B be the set of all binary integers that can be written using exactly 5 zeros and 8 oneswhere leading zeros are allowed. If all possible subtractions are performed in which oneelement ofB is subtracted from another, find the number of times the answer 1 is obtained.

6 The complex numbers z and w satisfy z13 = w, w11 = z, and the imaginary part of z issin

mn

for relatively prime positive integers mand n with m < n. Find n.

7 At each of the sixteen circles in the network below stands a student. A total of 3360 coins aredistributed among the sixteen students. All at once, all students give away all their coins bypassing an equal number of coins to each of their neighbors in the network. After the trade,all students have the same number of coins as they started with. Find the number of coinsthe student standing at the center circle had originally.


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8/18

USA

AIME

2012

8 CubeABCDEFGH, labeled as shown below, has edge length 1 and is cut by a plane passingthrough vertexD and the midpointsMandNofAB and C Grespectively. The plane dividesthe cube into two solids. The volume of the larger of the two solids can be written in theform p

q, where p and qare relatively prime positive integers. Find p + q.

A B

CD

EF

GH

M

N

9 Let x, y, and z be positive real numbers that satisfy

2logx

(2y) = 2 log2x

(4z) = log2x4(8yz)

= 0.

The value of xy5z can be expressed in the form 12p/q

, where p and q are relatively primeintegers. Find p + q.

10 LetSbe the set of all perfect squares whose rightmost three digits in base 10 are 256. LetTbe the set of all numbers of the form x256

1000 , where x is inS. In other words,T is the set of

numbers that result when the last three digits of each number inS are truncated. Find theremainder when the tenth smallest element ofT is divided by 1000.


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9/18

USA

AIME

2012

11 A frog begins at P0 = (0, 0) and makes a sequence of jumps according to the following rule:from Pn= (xn, yn), the frog jumps to Pn+1, which may be any of the points (xn+ 7, yn+ 2),(xn + 2, yn + 7), (xn5, yn10), or (xn10, yn5). There are M points (x, y) with|x| + |y| 100 that can be reached by a sequence of such jumps. Find the remainder whenM is divided by 1000.

12 LetABCbe a right triangle with right angle at C. Let D and Ebe points on AB with DbetweenAand Esuch that CD and CEtrisect C. If DE

BE = 8

15, then tan B can be written

as m

p

n , where mand nare relatively prime positive integers, and p is a positive integer not

divisible by the square of any prime. Find m + n +p.

13 Three concentric circles have radii 3, 4, and 5. An equilateral triangle with one vertex on each

circle has side length s. The largest possible area of the triangle can be written as a + bc

d,

wherea, b, cand d are positive integers, b and c are relatively prime, and d is not divisible bythe square of any prime. Find a + b + c + d.

14 Complex numbersa,b and c are the zeros of a polynomial P(z) =z3 + qz + r, and|a|2 + |b|2 +|c|2 = 250. The points corresponding to a, b, and c in the complex plane are the vertices of aright triangle with hypotenuse h. Find h2.

15 There are n mathematicians seated around a circular table with n seats numbered 1, 2, 3, , nin clockwise order. After a break they again sit around the table. The mathematicians note

that there is a positive integer a such that(1) for eachk, the mathematician who was seated in seatk before the break is seated in seat kaafter the break (where seat i +nis seati); (2) for every pair of mathematicians, the number ofmathematicians sitting between them after the break, counting in both the clockwise and thecounterclockwise directions, is different from either of the number of mathematicians sittingbetween them before the break.

Find the number of possible values ofn with 1< n


10/18

USA

AIME

2012

II

1 Find the number of ordered pairs of positive integer solutions (m, n) to the equation 20m+12n= 2012.

2 Two geometric sequences a1, a2, a3, . . . and b1, b2, b3 . . .have the same common ratio, witha1= 27,b1= 99, and a15 = b11. Find a9.

3 At a certain university, the division of mathematical sciences consists of the departments of

mathematics, statistics, and computer science. There are two male and two female professorsin each department. A committee of six professors is to contain three men and three womenand must also contain two professors from each of the three departments. Find the numberof possible comittees that can be formed subject to these requirements.

4 Ana, Bob, and Cao bike at constant rates of 8.6 meters per second, 6.2 meters per second, and5 meters per second, respectively. They all begin biking at the same time from the northeastcorner of a rectangular field whose longer side runs due west. Ana starts biking along theedge of the field, initially heading west, Bob starts biking along the edge of the field, initiallyheading south, and Cao bikes in a straight line across the field to a point D on the south edgeof the field. Cao arrives at point D at the same time that Ana and Bob arrive at D for the

first time. The ratio of the fields length to the fields width to the distance from point D tothe southeast corner of the field can be represented as p : q: r, where p, q, and r are positiveintegers with p and q relatively prime. Find p + q+ r.

5 In the accompanying figure, the outer square has side length 40. A second square S of sidelength 15 is constructed inside S with the same center as S and with sides parallel to thoseof S. From each midpoint of a side of S, segments are drawn to the two closest vertices of S.The result is a four-pointed starlike figure inscribed in S. The star figure is cut out and thenfolded to form a pyramid with base S. Find the volume of this pyramid.


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11/18

USA

AIME

2012

6 Letz= a + bibe the complex number with|z|= 5 and b >0 such that the distance between(1 + 2i)z3 and z5 is maximized, and let z4 =c + di. Find c + d.

7 LetSbe the increasing sequence of positive integers whose binary representation has exactly8 ones. Let Nbe the 1000th number in S. Find the remainder when N is divided by 1000.

8 The complex numbersz and w satisfy the system z+ 20iw

= 5 + i

w+ 12i

z

=

4 + 10i Find the smallest possible value of|

zw

|2.

9 Letxand y be real numbers such that sinxsin y

= 3 and cosxcos y

= 12

. The value of sin 2xsin 2y

+ cos 2xcos 2y

can

be expressed in the form pq

, where p and qare relatively prime positive integers. Find p + q.

10 Find the number of positive integers n less than 1000 for which there exists a positive realnumber xsuch that n= xx. Note:x is the greatest integer less than or equal to x.

11 Letf1(x) = 2

3 3

3x+1, and for n2, define fn(x) =f1(fn1(x)). The value of x that satisfies

f1001(x) =x 3 can be expressed in the form mn, wherem and n are relatively prime positiveintegers. Find m + n.

12 For a positive integer p, define the positive integer n to be p-safe if n differs in absolute

value by more than 2 from all multiples of p. For example, the set of 10-safe numbers is3, 4, 5, 6, 7, 13, 14, 15, 16, 17, 23,.... Find the number of positive integers less than or equal to10, 000 which are simultaneously 7-safe, 11-safe, and 13-safe.

13 EquilateralABChas side length 111. There are four distinct triangles AD1E1, AD1E2,AD2E3, and AD2E4, each congruent to ABC, with BD1= BD2=

11. Find

4

k=1(CEk)2.




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USA

AIME

2012

14 In a group of nine people each person shakes hands with exactly two of the other peoplefrom the group. Let N be the number of ways this handshaking can occur. Consider twohandshaking arrangements different if and only if at least two people who shake hands underone arrangement do not shake hands under the other arrangement. Find the remainder whenN is divided by 1000.

15 Triangle ABC is inscribed in circle with AB = 5, BC = 7, and AC = 3. The bisectorof angle A meets side BC at D and circle at a second point E. Let be the circle withdiameter DE. Circles and meet at Eand a second point F. Then AF2 = m

n, where m

and n are relatively prime positive integers. Find m + n.



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13/18

USA

AIME

2013

I

1 The AIME Triathlon consists of a half-mile swim, a 30-mile bicycle, and an eight-mile run.Tom swims, bicycles, and runs at constant rates. He runs five times as fast as he swims, andhe bicycles twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarterhours. How many minutes does he spend bicycling?

2 Find the number of five-digit positive integers,n, that satisfy the following conditions:

(a) the number n is divisible by 5, (b) the first and last digits ofn are equal, and (c) the sumof the digits ofn is divisible by 5.

3 Let ABCD be a square, and let E and Fbe points on AB and BC, respectively. The linethrough E parallel to BCand the line through F parallel to AB divide ABCD into twosquares and two non square rectangles. The sum of the areas of the two squares is 9

10of the

area of square ABCD. Find AEEB

+ EBAE

.

4 In the array of 13 squares shown below, 8 squares are colored red, and the remaining 5 squaresare colored blue. If one of all possible such colorings is chosen at random, the probabilitythat the chosen colored array appears the same when rotated 90 around the central squareis 1

n, where n is a positive integer. Find n.

5 The real root of the equation 8x3 3x2 3x 1 = 0 can be written in the form 3a+

3b+1

c ,

where a, b, and c are positive integers. Find a + b + c.

6 Melinda has three empty boxes and 12 textbooks, three of which are mathematics textbooks.One box will hold any three of her textbooks, one will hold any four of her textbooks, andone will hold any five of her textbooks. If Melinda packs her textbooks into these boxes inrandom order, the probability that all three mathematics textbooks end up in the same boxcan be written as m

n, where m and n Are relatively prime positive integers. Find m + n.


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14/18

USA

AIME

2013

7 A rectangular box has width 12 inches, length 16 inches, and height mn

inches, where m andn are relatively prime positive integers. Three faces of the box meet at a corner of the box.The center points of those three faces are the vertices of a triangle with an area of 30 squareinches. Find m + n.

8 The domain of the functionf(x) = arcsin(logm(nx)) is a closed interval of length 1

2013, where

mand n are positive integers andm >1. Find the remainder when the smallest possible summ + n is divided by 1000.

9 A paper equilateral triangle ABC has side length 12. The paper triangle is folded so thatvertex A touches a point on side BC a distance 9 from point B. The length of the line

segment along which the triangle is folded can be written as mp

n , where m, n, and p are

positive integers, m and n are relatively prime, and p is not divisible by the square of anyprime. Find m + n +p.

BA

CB

A

C

10 There are nonzero integersa, b, r , ands such that the complex number r + siis a zero of thepolynomial P(x) =x3 ax2 + bx 65. For each possible combination ofa and b, let pa,b bethe sum of the zeroes ofP(x). Find the sum of thepa,bs for all possible combinations ofaand b.

11 Ms. Maths kindergarten class has 16 registered students. The classroom has a very large

number, N, of play blocks which satisfies the conditions:

(a) If 16, 15, or 14 students are present, then in each case all the blocks can be distributedin equal numbers to each student, and (b) There are three integers 0 < x < y < z < 14 suchthat when x, y, or z students are present and the blocks are distributed in equal numbers toeach student, there are exactly three blocks left over.

Find the sum of the distinct prime divisors of the least possible value of N satisfying theabove conditions.


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15/18

USA

AIME

2013

12 LetP QRbe a triangle with P= 75 and Q= 60. A regular hexagon ABCDEF withside length 1 is drawn insideP QR so that side AB lies on P Q, side CD lies on QR, andone of the remaining vertices lies on RP. There are positive integers a, b, c, and d such that

the area ofP QR can be expressed in the form a+bc

d , wherea and d are relatively prime

and c is not divisible by the square of any prime. Find a + b + c + d.

13 Triangle AB0C0 has side lengths AB0 = 12, B0C0 = 17, and C0A = 25. For each positiveinteger n, points Bn and Cn are located on ABn1 and ACn1, respectively, creating threesimilar trianglesABnCn Bn1CnCn1 ABn1Cn1. The area of the union of alltriangles Bn1CnBn for n

1 can be expressed as p

q, where p and qare relatively prime

positive integers. Find q.

14 For


16/18

USA

AIME

2013

II

1 Suppose that the measurement of time during the day is converted to the metric system sothat each day has 10 metric hours, and each metric hour has 100 metric minutes. Digitalclocks would then be produced that would read 9:99 just before midnight, 0:00 at midnight,1:25 at the former 3:00 am, and 7:50 at the former 6:00 pm. After the conversion, a personwho wanted to wake up at the equivalent of the former 6:36 am would have to set his newdigital alarm clock for A:BC, where A, B, and C are digits. Find 100A + 10B + C.

2 Positive integers a and b satisfy the condition

log2(log2a(log2b(21000))) = 0.

Find the sum of all possible values ofa + b.

3 A large candle is 119 centimeters tall. It is designed to burn down more quickly when it is firstlit and more slowly as it approaches its bottom. Specifically, the candle takes 10 seconds toburn down the first centimeter from the top, 20 seconds to burn down the second centimeter,and 10k seconds to burn down the k-th centimeter. Suppose it takesTseconds for the candleto burn down completely. Then T

2 seconds after it is lit, the candles height in centimeters

will be h. Find 10h.4 In the Cartesian plane let A = (1, 0) and B =

2, 2

3

. Equilateral triangle ABC is con-structed so that C lies in the first quadrant. Let P = (x, y) be the center ofABC. Thenx y can be written as p

q

r , where p and r are relatively prime positive integers and q is an

integer that is not divisible by the square of any prime. Find p + q+ r.

5 In equilateralABClet points D and Etrisect B C. Then sin(DAE) can be expressed inthe form a

b

c , where aand care relatively prime positive integers, and bis an integer that is

not divisible by the square of any prime. Find a + b + c.

6 Find the least positive integer N such that the set of 1000 consecutive integers beginning

with 1000 Ncontains no square of an integer.7 A group of clerks is assigned the task of sorting 1775 files. Each clerk sorts at a constant

rate of 30 files per hour. At the end of the first hour, some of the clerks are reassigned toanother task; at the end of the second hour, the same number of the remaining clerks are alsoreassigned to another task, and a similar reassignment occurs at the end of the third hour.The group finishes the sorting in 3 hours and 10 minutes. Find the number of files sortedduring the first one and a half hours of sorting.


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17/18

USA

AIME

2013

8 A hexagon that is inscribed in a circle has side lengths 22, 22, 20, 22, 22, and 20 in that order.The radius of the circle can be written as p+

q, where p and qare positive integers. Find

p + q.

9 A 7 1 board is completely covered by m 1 tiles without overlap; each tile may cover anynumber of consecutive squares, and each tile lies completely on the board. Each tile is eitherred, blue, or green. Let Nbe the number of tilings of the 7 1 board in which all three colorsare used at least once. For example, a 1 1 red tile followed by a 2 1 green tile, a 1 1green tile, a 2 1 blue tile, and a 1 1 green tile is a valid tiling. Note that if the 2 1 bluetile is replaced by two 1 1 blue tiles, this results in a different tiling. Find the remainderwhen N is divided by 1000.

10 Given a circle of radius

13, let A be a point at a distance 4 +

13 from the center O of thecircle. Let B be the point on the circle nearest to point A. A line passing through the pointA intersects the circle at points K and L. The maximum possible area forBK L can bewritten in the form ab

c

d , where a, b, c, and d are positive integers, a and d are relatively

prime, and c is not divisible by the square of any prime. Find a + b + c + d.

11 Let A={1, 2, 3, 4, 5, 6, 7} and let Nbe the number of functions f from set A to set A suchthat f(f(x)) is a constant function. Find the remainder when N is divided by 1000.

12 LetSbe the set of all polynomials of the form z3 + az2 + bz + c, wherea,b, andc are integers.Find the number of polynomials in Ssuch that each of its roots z satisfies either

|z

|= 20 or

|z| = 13.13 InABC, AC=B C, and point D is on BC so that CD= 3 BD. Let Ebe the midpoint

ofAD. Given that CE=

7 and BE= 3, the area ofABCcan be expressed in the formm

n, where mand n are positive integers and n is not divisible by the square of any prime.

Find m + n.

14 For positive integers n and k, let f(n, k) be the remainder when n is divided by k, and for

n >1 let F(n) = max1kn

2

f(n, k). Find the remainder when100

n=20

F(n) is divided by 1000.

15 Let A, B , C be angles of an acute triangle with

cos2 A + cos2 B+ 2 sin A sin B cos C=15

8 and

cos2 B+ cos2 C+ 2 sin B sin Ccos A=14

9 .

There are positive integers p, q, r, and s for which

cos2 C+ cos2 A + 2 sin Csin A cos B= p qr

s ,




18/18

USA

AIME

2013

wherep + qands are relatively prime and r is not divisible by the square of any prime. Findp + q+ r+ s.


Thi fil d l d d f th A PS M th Ol i d R P P 6
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