MathDB

2014 AIME Problems

Part of AIME Problems

Subcontests

(15)
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Convoluted Shoelaces

The 88 eyelets for the lace of a sneaker all lie on a rectangle, four equally spaced on each of the longer sides. The rectangle has a width of 5050 mm and a length of 8080 mm. There is one eyelet at each vertex of the rectangle. The lace itself must pass between the vertex eyelets along a width side of the rectangle and then crisscross between successive eyelets until it reaches the two eyelets at the other width side of the rectrangle as shown. After passing through these final eyelets, each of the ends of the lace must extend at least 200200 mm farther to allow a knot to be tied. Find the minimum length of the lace in millimeters.
[asy] size(200); defaultpen(linewidth(0.7)); path laceL=(-20,-30)..tension 0.75 ..(-90,-135)..(-102,-147)..(-152,-150)..tension 2 ..(-155,-140)..(-135,-40)..(-50,-4)..tension 0.8 ..origin; path laceR=reflect((75,0),(75,-240))*laceL; draw(origin--(0,-240)--(150,-240)--(150,0)--cycle,gray); for(int i=0;i<=3;i=i+1) { path circ1=circle((0,-80*i),5),circ2=circle((150,-80*i),5); unfill(circ1); draw(circ1); unfill(circ2); draw(circ2); } draw(laceL--(150,-80)--(0,-160)--(150,-240)--(0,-240)--(150,-160)--(0,-80)--(150,0)^^laceR,linewidth(1));[/asy]

Painting Rooms

Abe can paint the room in 15 hours, Bea can paint 50 percent faster than Abe, and Coe can paint twice as fast as Abe. Abe begins to paint the room and works alone for the first hour and a half. Then Bea joins Abe, and they work together until half the room is painted. Then Coe joins Abe and Bea, and they work together until the entire room is painted. Find the number of minutes after Abe begins for the three of them to finish painting the room.
13
2

Splitting up a Square

On square ABCD,ABCD, points E,F,G,E,F,G, and HH lie on sides AB‾,BC‾,CD‾,\overline{AB},\overline{BC},\overline{CD}, and DA‾,\overline{DA}, respectively, so that EG‾⊥FH‾\overline{EG} \perp \overline{FH} and EG=FH=34.EG=FH = 34. Segments EG‾\overline{EG} and FH‾\overline{FH} intersect at a point P,P, and the areas of the quadrilaterals AEPH,BFPE,CGPF,AEPH, BFPE, CGPF, and DHPGDHPG are in the ratio 269:275:405:411.269:275:405:411. Find the area of square ABCDABCD.
[asy] size(200); defaultpen(linewidth(0.8)+fontsize(10.6)); pair A = (0,sqrt(850)); pair B = (0,0); pair C = (sqrt(850),0); pair D = (sqrt(850),sqrt(850)); draw(A--B--C--D--cycle); dotfactor = 3; dot("AA",A,dir(135)); dot("BB",B,dir(215)); dot("CC",C,dir(305)); dot("DD",D,dir(45)); pair H = ((2sqrt(850)-sqrt(120))/6,sqrt(850)); pair F = ((2sqrt(850)+sqrt(306)+7)/6,0); dot("HH",H,dir(90)); dot("FF",F,dir(270)); draw(H--F); pair E = (0,(sqrt(850)-6)/2); pair G = (sqrt(850),(sqrt(850)+sqrt(100))/2); dot("EE",E,dir(180)); dot("GG",G,dir(0)); draw(E--G); pair P = extension(H,F,E,G); dot("PP",P,dir(60)); label("ww", (H+E)/2,fontsize(15)); label("xx", (E+F)/2,fontsize(15)); label("yy", (G+F)/2,fontsize(15)); label("zz", (H+G)/2,fontsize(15)); label("w:x:y:z=269:275:405:411w:x:y:z=269:275:405:411",(sqrt(850)/2,-4.5),fontsize(11)); [/asy]

Bad Shoe Placement

Ten adults enter a room, remove their shoes, and toss their shoes into a pile. Later, a child randomly pairs each left shoe with a right shoe without regard to which shoes belong together. The probability that for every positive integer k<5,k<5, no collection of kk pairs made by the child contains the shoes from exactly kk of the adults is mn,\tfrac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n.
3
2

Rational numbers, sum to 1000

Find the number of rational numbers rr, 0<r<10<r<1, such that when rr is written as a fraction in lowest terms, the numerator and denominator have a sum of 10001000.

Squishing a Rectangle

A rectangle has sides of length aa and 3636. A hinge is installed at each vertex of the rectangle and at the midpoint of each side of length 3636. The sides of length aa can be pressed toward each 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 length aa parallel and separated by a distance of 24,24, the hexagon has the same area as the original rectangle. Find a2a^2.
[asy] pair A,B,C,D,E,F,R,S,T,X,Y,Z; dotfactor = 2; unitsize(.1cm); A = (0,0); B = (0,18); C = (0,36);
// don't look here
D = (12*2.236, 36); E = (12*2.236, 18); F = (12*2.236, 0); draw(A--B--C--D--E--F--cycle); dot(" ",A,NW); dot(" ",B,NW); dot(" ",C,NW); dot(" ",D,NW); dot(" ",E,NW); dot(" ",F,NW);
//don't look here
R = (12*2.236 +22,0); S = (12*2.236 + 22 - 13.4164,12); T = (12*2.236 + 22,24); X = (12*4.472+ 22,24); Y = (12*4.472+ 22 + 13.4164,12); Z = (12*4.472+ 22,0); draw(R--S--T--X--Y--Z--cycle); dot(" ",R,NW); dot(" ",S,NW); dot(" ",T,NW); dot(" ",X,NW); dot(" ",Y,NW); dot(" ",Z,NW);
// sqrt180 = 13.4164 // sqrt5 = 2.236
[/asy]
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11
2

Wandering Token

A token starts at the point (0,0)(0,0) of an xyxy-coordinate grid and them makes a sequence of six moves. Each move is 11 unit in a direction parallel to one of the coordinate axes. Each move is 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∣|y|=|x| is mn\tfrac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n.

Nice Angles

In △RED,RD=1,∠DRE=75∘\triangle RED, RD =1, \angle DRE = 75^\circ and ∠RED=45∘\angle RED = 45^\circ. Let MM be the midpoint of segment RD‾\overline{RD}. Point CC lies on side ED‾\overline{ED} such that RC‾⊥EM‾\overline{RC} \perp \overline{EM}. Extend segment DE‾\overline{DE} through EE to point AA such that CA=ARCA = AR. Then AE=a−bc,AE = \tfrac{a-\sqrt{b}}{c}, where aa and cc are relatively prime positive integers, and bb is a positive integer. Find a+b+ca+b+c.
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2

Heated Argument

Let ww and zz be complex numbers such that ∣w∣=1|w| = 1 and ∣z∣=10|z| = 10. Let θ=arg⁡(w−zz)\theta = \arg\left(\tfrac{w-z}{z}\right). The maximum possible value of tan⁡2θ\tan^2 \theta can be written as pq\tfrac{p}{q}, where pp and qq are relatively prime positive integers. Find p+qp+q. (Note that arg⁡(w)\arg(w), for w≠0w \neq 0, denotes the measure of the angle that the ray from 00 to ww makes with the positive real axis in the complex plane.

Scary Sums and Logs

Let f(x)=(x2+3x+2)cos⁡(πx)f(x) = (x^2+3x+2)^{\cos(\pi x)}. Find the sum of all positive integers nn for which ∣∑k=1nlog⁡10f(k)∣=1.\left| \sum_{k=1}^n \log_{10} f(k) \right| = 1.
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2

Circle revolving around circle

A disk with radius 11 is externally tangent to a disk with radius 55. Let AA be the point where the disks are tangent, CC be the center of the smaller disk, and EE be the center of the larger disk. While the larger disk remains fixed, the smaller disk is allowed to roll along the outside of the larger disk until the smaller disk has turned through an angle of 360∘360^\circ. That is, if the center of the smaller disk has moved to the point DD, and the point on the smaller disk that began at AA has now moved to point BB, then AC‾\overline{AC} is parallel to BD‾\overline{BD}. Then sin⁡2(∠BEA)=mn\sin^2(\angle BEA)=\tfrac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n.

Fake Complex Numbers

Let zz be a complex number with ∣z∣=2014|z| = 2014. Let PP be the polygon in the complex plane whose vertices are zz and every ww such that 1z+w=1z+1w\tfrac{1}{z+w} = \tfrac{1}{z} + \tfrac{1}{w}. Then the area enclosed by PP can be written in the form n3,n\sqrt{3}, where nn is an integer. Find the remainder when nn is divided by 10001000.
14
2

Degree Six Polynomial's Roots

Let mm be the largest real solution to the equation 3x−3+5x−5+17x−17+19x−19=x2−11x−4.\frac{3}{x-3}+\frac{5}{x-5}+\frac{17}{x-17}+\frac{19}{x-19}= x^2-11x-4. There are positive integers a,b,ca,b,c such that m=a+b+cm = a + \sqrt{b+\sqrt{c}}. Find a+b+ca+b+c.

Find a Length in a Rigid Triangle

In △ABC\triangle ABC, AB=10AB=10, ∠A=30∘\angle A=30^\circ, and ∠C=45∘\angle C=45^\circ. Let H,DH,D, and MM be points on line BC‾\overline{BC} such that AH‾⊥BC‾\overline{AH}\perp\overline{BC}, ∠BAD=∠CAD\angle BAD=\angle CAD, and BM=CMBM=CM. Point NN is the midpoint of segment HM‾\overline{HM}, and point PP is on ray ADAD such that PN‾⊥BC‾\overline{PN}\perp\overline{BC}. Then AP2=mnAP^2=\tfrac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm+n.
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2

Balls and Urns

An urn contains 44 green balls and 66 blue balls. A second urn contains 1616 green balls and NN blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is 0.580.58. Find NN.

Health Risk Factors Probability Problem

Arnold is studying the prevalence of three health risk factors, denoted by A, B, and C. within a population of men. For each of the three factors, the probability that a randomly selected man in the population as only this risk factor (and none of the others) is 0.1. For any two of the three factors, the probability that a randomly selected man has exactly two of these two risk factors (but not the third) is 0.14. The probability that a randomly selected man has all three risk factors, given that he has A and B is 13\tfrac{1}{3}. The probability that a man has none of the three risk factors given that he does not have risk factor A is pq\tfrac{p}{q}, where pp and qq are relatively prime positive integers. Find p+qp+q.
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15
2

Circle and 3-4-5

In △ABC \triangle ABC , AB=3 AB = 3 , BC=4 BC = 4 , and CA=5 CA = 5 . Circle ω\omega intersects AB‾\overline{AB} at EE and BB, BC‾\overline{BC} at BB and DD, and AC‾\overline{AC} at FF and GG. Given that EF=DFEF=DF and DGEG=34\tfrac{DG}{EG} = \tfrac{3}{4}, length DE=abcDE=\tfrac{a\sqrt{b}}{c}, where aa and cc are relatively prime positive integers, and bb is a positive integer not divisible by the square of any prime. Find a+b+ca+b+c.

Function with Primes

For any integer k≥1k\ge1, let p(k)p(k) be the smallest prime which does not divide kk. Define the integer function X(k)X(k) to be the product of all primes less than p(k)p(k) if p(k)>2p(k)>2, and X(k)=1X(k)=1 if p(k)=2p(k)=2. Let {xn}\{x_n\} be the sequence defined by x0=1x_0=1, and xn+1X(xn)=xnp(xn)x_{n+1}X(x_n)=x_np(x_n) for n≥0n\ge0. Find the smallest positive integer, tt such that xt=2090x_t=2090.
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2

Bicycles and trains

Jon and Steve ride their bicycles on a path that parallels two side-by-side train tracks running in the east/west direction. Jon rides east at 20 miles per hour, and Steve rides west at 20 miles per hour. Two trains of equal length traveling in opposite directions at constant but different speeds, each pass the two riders. Each train takes exactly 1 minute to go past Jon. The westbound train takes 10 times as long as the eastbound train to go past Steve. The length of each train is mn\tfrac mn, where mm and nn are relatively prime positive integers. Find m+nm+n.

Repeating Decimal Addition

The repeating decimals 0.ababab‾0.abab\overline{ab} and 0.abcabcabc‾0.abcabc\overline{abc} satisfy 0.ababab‾+0.abcabcabc‾=3337,0.abab\overline{ab}+0.abcabc\overline{abc}=\frac{33}{37}, where a,ba,b, and cc are (not necessarily distinct) digits. Find the three-digit number abcabc.
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