MathDB

2004 AIME Problems

Part of AIME Problems

Subcontests

(15)
15
2

A hard function/sequence

For all positive integers x x, let f(x) \equal{} \begin{cases}1 & \text{if }x \equal{} 1 \\ \frac x{10} & \text{if }x\text{ is divisible by 10} \\ x \plus{} 1 & \text{otherwise}\end{cases}and define a sequence as follows: x_1 \equal{} x and x_{n \plus{} 1} \equal{} f(x_n) for all positive integers n n. Let d(x) d(x) be the smallest n n such that x_n \equal{} 1. (For example, d(100) \equal{} 3 and d(87) \equal{} 7.) Let m m be the number of positive integers x x such that d(x) \equal{} 20. Find the sum of the distinct prime factors of m m.

More Paper Folding

A long thin strip of paper is 1024 units in length, 1 unit in width, and is divided into 1024 unit squares. The paper is folded in half repeatedly. For the first fold, the right end of the paper is folded over to coincide with and lie on top of the left end. The result is a 512 by 1 strip of double thickness. Next, the right end of this strip is folded over to coincide with and lie on top of the left end, resulting in a 256 by 1 strip of quadruple thickness. This process is repeated 8 more times. After the last fold, the strip has become a stack of 1024 unit squares. How many of these squares lie below the square that was originally the 942nd square counting from the left?
14
2

Unicorns

A unicorn is tethered by a 20-foot silver rope to the base of a magician's cylindrical tower whose radius is 8 feet. The rope is attached to the tower at ground level and to the unicorn at a height of 4 feet. The unicorn has pulled the rope taut, the end of the rope is 4 feet from the nearest point on the tower, and the length of the rope that is touching the tower is a−bc\frac{a-\sqrt{b}}c feet, where a,b,a, b, and cc are positive integers, and cc is prime. Find a+b+ca+b+c.

Lucky Sevens

Consider a string of nn 7's, 7777⋯777777\cdots77, into which ++ signs are inserted to produce an arithmetic expression. For example, 7+77+777+7+7=8757+77+777+7+7=875 could be obtained from eight 7's in this way. For how many values of nn is it possible to insert ++ signs so that the resulting expression has value 7000?
13
2

Complex roots.

The polynomial P(x)=(1+x+x2+⋯+x17)2−x17P(x)=(1+x+x^2+\cdots+x^{17})^2-x^{17} has 34 complex roots of the form zk=rk[cos⁡(2πak)+isin⁡(2πak)],k=1,2,3,…,34z_k=r_k[\cos(2\pi a_k)+i\sin(2\pi a_k)], k=1, 2, 3,\ldots, 34, with 0<a1≤a2≤a3≤⋯≤a34<10<a_1\le a_2\le a_3\le\cdots\le a_{34}<1 and rk>0r_k>0. Given that a1+a2+a3+a4+a5=m/na_1+a_2+a_3+a_4+a_5=m/n, where mm and nn are relatively prime positive integers, find m+nm+n.

Pentagon Problem

Let ABCDEABCDE be a convex pentagon with AB∥CEAB\parallel CE, BC∥ADBC\parallel AD, AC∥DEAC\parallel DE, ∠ABC=120∘\angle ABC=120^\circ, AB=3AB=3, BC=5BC=5, and DE=15DE=15. Given that the ratio between the area of triangle ABCABC and the area of triangle EBDEBD is m/nm/n, where mm and nn are relatively prime positive integers, find m+nm+n.
11
2

FRUSTRAted ;)

A solid in the shape of a right circular cone is 4 inches tall and its base has a 3-inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid CC and a frustum-shaped solid FF, in such a way that the ratio between the areas of the painted surfaces of CC and FF and the ratio between the volumes of CC and FF are both equal to kk. Given that k=m/nk=m/n, where mm and nn are relatively prime positive integers, find m+nm+n.

Minimum Distance Travelled

A right circular cone has a base with radius 600 and height 2007200\sqrt{7}. A fly starts at a point on the surface of the cone whose distance from the vertex of the cone is 125, and crawls along the surface of the cone to a point on the exact opposite side of the cone whose distance from the vertex is 3752375\sqrt{2}. Find the least distance that the fly could have crawled.
9
2

Similar figures.

Let ABCABC be a triangle with sides 3, 4, and 5, and DEFGDEFG be a 6-by-7 rectangle. A segment is drawn to divide triangle ABCABC into a triangle U1U_1 and a trapezoid V1V_1 and another segment is drawn to divide rectangle DEFGDEFG into a triangle U2U_2 and a trapezoid V2V_2 such that U1U_1 is similar to U2U_2 and V1V_1 is similar to V2V_2. The minimum value of the area of U1U_1 can be written in the form m/nm/n, where mm and nn are relatively prime positive integers. Find m+nm+n.

Arithmetic and Geometric

A sequence of positive integers with a1=1a_1=1 and a9+a10=646a_9+a_{10}=646 is formed so that the first three terms are in geometric progression, the second, third, and fourth terms are in arithmetic progression, and, in general, for all n≥1n\ge1, the terms a2n−1a_{2n-1}, a2na_{2n}, a2n+1a_{2n+1} are in geometric progression, and the terms a2na_{2n}, a2n+1a_{2n+1}, and a2n+2a_{2n+2} are in arithmetic progression. Let ana_n be the greatest term in this sequence that is less than 1000. Find n+ann+a_n.
6
2

Snakelike

An integer is called snakelike if its decimal representation a1a2a3⋯aka_1a_2a_3\cdots a_k satisfies ai<ai+1a_i<a_{i+1} if ii is odd and ai>ai+1a_i>a_{i+1} if ii is even. How many snakelike integers between 1000 and 9999 have four distinct digits?

Banana!

Three clever monkeys divide a pile of bananas. The first monkey takes some bananas from the pile, keeps three-fourths of them, and divides the rest equally between the other two. The second monkey takes some bananas from the pile, keeps one-fourth of them, and divides the rest equally between the other two. The third monkey takes the remaining bananas from the pile, keeps one-twelfth of them, and divides the rest equally between the other two. Given that each monkey receives a whole number of bananas whenever the bananas are divided, and the numbers of bananas the first, second, and third monkeys have at the end of the process are in the ratio 3:2:13: 2: 1, what is the least possible total for the number of bananas?
3
2
2
2
1
2
12
2

Log area.

Let SS be the set of ordered pairs (x,y)(x, y) such that 0<x≤10<x\le 1, 0<y≤10<y\le 1, and [log⁡2(1x)]\left[\log_2{\left(\frac 1x\right)}\right] and [log⁡5(1y)]\left[\log_5{\left(\frac 1y\right)}\right] are both even. Given that the area of the graph of SS is m/nm/n, where mm and nn are relatively prime positive integers, find m+nm+n. The notation [z][z] denotes the greatest integer that is less than or equal to zz.

Isosceles Trapezoid

Let ABCDABCD be an isosceles trapezoid, whose dimensions are AB=6AB = 6, BC=5=DABC=5=DA, and CD=4CD=4. Draw circles of radius 3 centered at AA and BB, and circles of radius 2 centered at CC and DD. A circle contained within the trapezoid is tangent to all four of these circles. Its radius is −k+mnp\frac{-k+m\sqrt{n}}p, where kk, mm, nn, and pp are positive integers, nn is not divisible by the square of any prime, and kk and pp are relatively prime. Find k+m+n+pk+m+n+p.
5
2

Alpha vs. Beta

Alpha and Beta both took part in a two-day problem-solving competition. At the end of the second day, each had attempted questions worth a total of 500 points. Alpha scored 160 points out of 300 points attempted on the first day, and scored 140 points out of 200 points attempted on the second day. Beta who did not attempt 300 points on the first day, had a positive integer score on each of the two days, and Beta's daily success rate (points scored divided by points attempted) on each day was less than Alpha's on that day. Alpha's two-day success ratio was 300/500 = 3/5. The largest possible two-day success ratio that Beta could achieve is m/nm/n, where mm and nn are relatively prime positive integers. What is m+nm+n?

Employment's Going up!

In order to complete a large job, 1000 workers were hired, just enough to complete the job on schedule. All the workers stayed on the job while the first quarter of the work was done, so the first quarter of the work was completed on schedule. Then 100 workers were laid off, so the second quarter of the work was completed behind schedule. Then an additional 100 workers were laid off, so the third quarter of the work was completed still further behind schedule. Given that all workers work at the same rate, what is the minimum number of additional workers, beyond the 800 workers still on the job at the end of the third quarter, that must be hired after three-quarters of the work has been completed so that the entire project can be completed on schedule or before?
4
2
10
2
7
2

Aime 2004 #7 - coefficient in expansion

Let CC be the coefficient of x2x^2 in the expansion of the product (1−x)(1+2x)(1−3x)⋯(1+14x)(1−15x).(1 - x)(1 + 2x)(1 - 3x)\cdots(1 + 14x)(1 - 15x). Find ∣C∣|C|.

Paper Folding

ABCDABCD is a rectangular sheet of paper that has been folded so that corner BB is matched with point B′B' on edge ADAD. The crease is EFEF, where EE is on ABAB and FFis on CDCD. The dimensions AE=8AE=8, BE=17BE=17, and CF=3CF=3 are given. The perimeter of rectangle ABCDABCD is m/nm/n, where mm and nn are relatively prime positive integers. Find m+nm+n. [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, B=(25,0), C=(25,70/3), D=(0,70/3), E=(8,0), F=(22,70/3), Bp=reflect(E,F)*B, Cp=reflect(E,F)*C; draw(F--D--A--E); draw(E--B--C--F, linetype("4 4")); filldraw(E--F--Cp--Bp--cycle, white, black); pair point=( 12.5, 35/3 ); label("AA", A, dir(point--A)); label("BB", B, dir(point--B)); label("CC", C, dir(point--C)); label("DD", D, dir(point--D)); label("EE", E, dir(point--E)); label("FF", F, dir(point--F)); label("B′B^\prime", Bp, dir(point--Bp)); label("C′C^\prime", Cp, dir(point--Cp));[/asy]
8
2

Stars

Define a regular nn-pointed star to be the union of nn line segments P1P2,P2P3,…,PnP1P_1P_2, P_2P_3,\ldots, P_nP_1 such that ∙\bullet the points P1,P2,…,PnP_1, P_2,\ldots, P_n are coplanar and no three of them are collinear, ∙\bullet each of the nn line segments intersects at least one of the other line segments at a point other than an endpoint, ∙\bullet all of the angles at P1,P2,…,PnP_1, P_2,\ldots, P_n are congruent, ∙\bullet all of the nn line segments P2P3,…,PnP1P_2P_3,\ldots, P_nP_1 are congruent, and ∙\bullet the path P1P2,P2P3,…,PnP1P_1P_2, P_2P_3,\ldots, P_nP_1 turns counterclockwise at an angle of less than 180 degrees at each vertex. There are no regular 3-pointed, 4-pointed, or 6-pointed stars. All regular 5-pointed stars are similar, but there are two non-similar regular 7-pointed stars. How many non-similar regular 1000-pointed stars are there?

2004^2004

How many positive integer divisors of 200420042004^{2004} are divisible by exactly 20042004 positive integers?