MathDB

2006 AIME Problems

Part of AIME Problems

Subcontests

(15)
15
2

Sequence

Given that a sequence satisfies x0=0x_0=0 and ∣xk∣=∣xk−1+3∣|x_k|=|x_{k-1}+3| for all integers k≥1,k\ge 1, find the minimum possible value of ∣x1+x2+⋯+x2006∣|x_1+x_2+\cdots+x_{2006}|.

Aime ii - 2006/15

Given that xx, yy, and zz are real numbers that satisfy: x=y2−116+z2−116 x=\sqrt{y^2-\frac{1}{16}}+\sqrt{z^2-\frac{1}{16}} y=z2−125+x2−125 y=\sqrt{z^2-\frac{1}{25}}+\sqrt{x^2-\frac{1}{25}} z=x2−136+y2−136 z=\sqrt{x^2-\frac{1}{36}}+\sqrt{y^2-\frac{1}{36}} and that x+y+z=mnx+y+z=\frac{m}{\sqrt{n}}, where mm and nn are positive integers and nn is not divisible by the square of any prime, find m+nm+n.
14
2

Broken Tripod

A tripod has three legs each of length 5 feet. When the tripod is set up, the angle between any pair of legs is equal to the angle between any other pair, and the top of the tripod is 4 feet from the ground. In setting up the tripod, the lower 1 foot of one leg breaks off. Let hh be the height in feet of the top of the tripod from the ground when the broken tripod is set up. Then hh can be written in the form mn,\frac m{\sqrt{n}}, where mm and nn are positive integers and nn is not divisible by the square of any prime. Find ⌊m+n⌋.\lfloor m+\sqrt{n}\rfloor. (The notation ⌊x⌋\lfloor x\rfloor denotes the greatest integer that is less than or equal to xx.)

Short and fun.

Let SnS_n be the sum of the reciprocals of the non-zero digits of the integers from 1 to 10n10^n inclusive. Find the smallest positive integer nn for which SnS_n is an integer.
13
2
12
2

Trigonometry

Find the sum of the values of xx such that cos⁡33x+cos⁡35x=8cos⁡34xcos⁡3x\cos^3 3x+ \cos^3 5x = 8 \cos^3 4x \cos^3 x, where xx is measured in degrees and 100<x<200100< x< 200.

Geometry yet again

Equilateral △ABC\triangle ABC is inscribed in a circle of radius 2. Extend AB‾\overline{AB} through BB to point DD so that AD=13AD=13, and extend AC‾\overline{AC} through CC to point EE so that AE=11AE=11. Through DD, draw a line l1l_1 parallel to AE‾\overline{AE}, and through EE, draw a line l2{l}_2 parallel to AD‾\overline{AD}. Let FF be the intersection of l1{l}_1 and l2{l}_2. Let GG be the point on the circle that is collinear with AA and FF and distinct from AA. Given that the area of △CBG\triangle CBG can be expressed in the form pqr\frac{p\sqrt{q}}{r}, where pp, qq, and rr are positive integers, pp and rr are relatively prime, and qq is not divisible by the square of any prime, find p+q+rp+q+r.
9
2

Sequence and Logs

The sequence a1,a2,…a_1, a_2, \ldots is geometric with a1=aa_1=a and common ratio rr, where aa and rr are positive integers. Given that log⁡8a1+log⁡8a2+⋯+log⁡8a12=2006,\log_8 a_1+\log_8 a_2+\cdots+\log_8 a_{12} = 2006, find the number of possible ordered pairs (a,r)(a,r).

Tangent geometry

Circles C1\mathcal{C}_1, C2\mathcal{C}_2, and C3\mathcal{C}_3 have their centers at (0,0), (12,0), and (24,0), and have radii 1, 2, and 4, respectively. Line t1t_1 is a common internal tangent to C1\mathcal{C}_1 and C2\mathcal{C}_2 and has a positive slope, and line t2t_2 is a common internal tangent to C2\mathcal{C}_2 and C3\mathcal{C}_3 and has a negative slope. Given that lines t1t_1 and t2t_2 intersect at (x,y)(x,y), and that x=p−qrx=p-q\sqrt{r}, where pp, qq, and rr are positive integers and rr is not divisible by the square of any prime, find p+q+rp+q+r.
8
2

Everybody's favorite...

Hexagon ABCDEFABCDEF is divided into four rhombuses, P,Q,R,S,\mathcal{P, Q, R, S,} and T,\mathcal{T,} as shown. Rhombuses P,Q,R,\mathcal{P, Q, R,} and S\mathcal{S} are congruent, and each has area 2006\sqrt{2006}. Let KK be the area of rhombus T\mathcal{T}. Given that KK is a positive integer, find the number of possible values for KK.
[asy] size(150);defaultpen(linewidth(0.7)+fontsize(10)); draw(rotate(45)*polygon(4)); pair F=(1+sqrt(2))*dir(180), C=(1+sqrt(2))*dir(0), A=F+sqrt(2)*dir(45), E=F+sqrt(2)*dir(-45), B=C+sqrt(2)*dir(180-45), D=C+sqrt(2)*dir(45-180); draw(F--(-1,0)^^C--(1,0)^^A--B--C--D--E--F--cycle); pair point=origin; 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("P\mathcal{P}", intersectionpoint( A--(-1,0), F--(0,1) )); label("S\mathcal{S}", intersectionpoint( E--(-1,0), F--(0,-1) )); label("R\mathcal{R}", intersectionpoint( D--(1,0), C--(0,-1) )); label("Q\mathcal{Q}", intersectionpoint( B--(1,0), C--(0,1) )); label("T\mathcal{T}", point); dot(A^^B^^C^^D^^E^^F);[/asy]

Triangle confetti!

There is an unlimited supply of congruent equilateral triangles made of colored paper. Each triangle is a solid color with the same color on both sides of the paper. A large equilateral triangle is constructed from four of these paper triangles. Two large triangles are considered distinguishable if it is not possible to place one on the other, using translations, rotations, and/or reflections, so that their corresponding small triangles are of the same color. Given that there are six different colors of triangles from which to choose, how many distinguishable large equilateral triangles may be formed?
7
2

Parallel Lines

An angle is drawn on a set of equally spaced parallel lines as shown. The ratio of the area of shaded region C\mathcal{C} to the area of shaded region B\mathcal{B} is 11/511/5. Find the ratio of shaded region D\mathcal{D} to the area of shaded region A\mathcal{A}.
[asy] defaultpen(linewidth(0.7)+fontsize(10)); for(int i=0; i<4; i=i+1) { fill((2*i,0)--(2*i+1,0)--(2*i+1,6)--(2*i,6)--cycle, mediumgray); } pair A=(1/3,4), B=A+7.5*dir(-17), C=A+7*dir(10); draw(B--A--C); fill((7.3,0)--(7.8,0)--(7.8,6)--(7.3,6)--cycle, white); clip(B--A--C--cycle); for(int i=0; i<9; i=i+1) { draw((i,1)--(i,6)); } label("A\mathcal{A}", A+0.2*dir(-17), S); label("B\mathcal{B}", A+2.3*dir(-17), S); label("C\mathcal{C}", A+4.4*dir(-17), S); label("D\mathcal{D}", A+6.5*dir(-17), S);[/asy]

Sum 1000

Find the number of ordered pairs of positive integers (a,b)(a,b) such that a+b=1000a+b=1000 and neither aa nor bb has a zero digit.
6
2

Sum of Repeating Decimals

Let S\mathcal{S} be the set of real numbers that can be represented as repeating decimals of the form 0.abc‾0.\overline{abc} where a,b,ca, b, c are distinct digits. Find the sum of the elements of S\mathcal{S}.

Another geometry with equilateral triangle

Square ABCDABCD has sides of length 1. Points EE and FF are on BC‾\overline{BC} and CD‾\overline{CD}, respectively, so that △AEF\triangle AEF is equilateral. A square with vertex BB has sides that are parallel to those of ABCDABCD and a vertex on AE‾\overline{AE}. The length of a side of this smaller square is a−bc\displaystyle \frac{a-\sqrt{b}}{c}, where aa, bb, and cc are positive integers and bb is not divisible by the square of any prime. Find a+b+ca+b+c.
5
2

Square Roots

The number 1046+46810+14415+2006 \sqrt{104\sqrt{6}+468\sqrt{10}+144\sqrt{15}+2006} can be written as a2+b3+c5,a\sqrt{2}+b\sqrt{3}+c\sqrt{5}, where a,b,a, b, and cc are positive integers. Find a⋅b⋅ca\cdot b\cdot c.

Dice rolling

When rolling a certain unfair six-sided die with faces numbered 1,2,3,4,51, 2, 3, 4, 5, and 66, the probability of obtaining face FF is greater than 16\frac{1}{6}, the probability of obtaining the face opposite is less than 16\frac{1}{6}, the probability of obtaining any one of the other four faces is 16\frac{1}{6}, and the sum of the numbers on opposite faces is 77. When two such dice are rolled, the probability of obtaining a sum of 77 is 47288\frac{47}{288}. Given that the probability of obtaining face FF is mn\frac{m}{n}, where mm and nn are relatively prime positive integers, find m+nm+n.
4
2

Factorial Frenzy

Let NN be the number of consecutive 0's at the right end of the decimal representation of the product 1!×2!×3!×4!⋯99!×100!.1!\times2!\times3!\times4!\cdots99!\times100!. Find the remainder when NN is divided by 1000.

Permutations of numbers

Let (a1,a2,a3,...,a12)(a_1,a_2,a_3,...,a_{12}) be a permutation of (1,2,3,...,12)(1,2,3,...,12) for which a1>a2>a3>a4>a5>a6 and a6<a7<a8<a9<a10<a11<a12. a_1>a_2>a_3>a_4>a_5>a_6 \text{ and } a_6<a_7<a_8<a_9<a_{10}<a_{11}<a_{12}. An example of such a permutation is (6,5,4,3,2,1,7,8,9,10,11,12)(6,5,4,3,2,1,7,8,9,10,11,12). Find the number of such permutations.
2
2
1
2
10
2

Stacked Circles

This is the one with the 8 circles? I made each circle into the square in which the circle is inscribed, then calculated it with that. It got the right answer but I don't think that my method is truly valid...

Soccer

Seven teams play a soccer tournament in which each team plays every other team exactly once. No ties occur, each team has a 50%50\% chance of winning each game it plays, and the outcomes of the games are independent. In each game, the winner is awarded a point and the loser gets 0 points. The total points are accumilated to decide the ranks of the teams. In the first game of the tournament, team AA beats team BB. The probability that team AA finishes with more points than team BB is m/nm/n, where mm and nn are relatively prime positive integers. Find m+nm+n.
3
2
11
2

Tower of cubes

A collection of 8 cubes consists of one cube with edge-length kk for each integer k, 1≤k≤8.k,\thinspace 1 \le k \le 8. A tower is to be built using all 8 cubes according to the rules: ∙\bullet Any cube may be the bottom cube in the tower. ∙\bullet The cube immediately on top of a cube with edge-length kk must have edge-length at most k+2.k+2. Let TT be the number of different towers than can be constructed. What is the remainder when TT is divided by 1000?

Sequence

A sequence is defined as follows a1=a2=a3=1a_1=a_2=a_3=1, and, for all positive integers nn, an+3=an+2+an+1+ana_{n+3}=a_{n+2}+a_{n+1}+a_n. Given that a28=6090307a_{28}=6090307, a29=11201821a_{29}=11201821, and a30=20603361a_{30}=20603361, find the remainder when ∑k=128ak\displaystyle \sum^{28}_{k=1} a_k is divided by 1000.