MathDB

2008 AIME Problems

Part of AIME Problems

Subcontests

(15)
6
2

Triangular Array

A triangular array of numbers has a first row consisting of the odd integers 1,3,5,…,99 1,3,5,\ldots,99 in increasing order. Each row below the first has one fewer entry than the row above it, and the bottom row has a single entry. Each entry in any row after the top row equals the sum of the two entries diagonally above it in the row immediately above it. How many entries in the array are multiples of 67 67? [asy]size(200); defaultpen(fontsize(10)); label("1", origin); label("3", (2,0)); label("5", (4,0)); label("⋯\cdots", (6,0)); label("97", (8,0)); label("99", (10,0));
label("4", (1,-1)); label("8", (3,-1)); label("12", (5,-1)); label("196", (9,-1)); label(rotate(90)*"⋯\cdots", (6,-2));[/asy]

Two Sequences

The sequence {an} \{a_n\} is defined by a_0 \equal{} 1,a_1 \equal{} 1, \text{ and } a_n \equal{} a_{n \minus{} 1} \plus{} \frac {a_{n \minus{} 1}^2}{a_{n \minus{} 2}}\text{ for }n\ge2. The sequence {bn} \{b_n\} is defined by b_0 \equal{} 1,b_1 \equal{} 3, \text{ and } b_n \equal{} b_{n \minus{} 1} \plus{} \frac {b_{n \minus{} 1}^2}{b_{n \minus{} 2}}\text{ for }n\ge2. Find b32a32 \frac {b_{32}}{a_{32}}.
13
2

Polynomial in Two Variables

Let p(x,y) \equal{} a_0 \plus{} a_1x \plus{} a_2y \plus{} a_3x^2 \plus{} a_4xy \plus{} a_5y^2 \plus{} a_6x^3 \plus{} a_7x^2y \plus{} a_8xy^2 \plus{} a_9y^3. Suppose that \begin{align*}p(0,0) &\equal{} p(1,0) \equal{} p( \minus{} 1,0) \equal{} p(0,1) \equal{} p(0, \minus{} 1) \\&\equal{} p(1,1) \equal{} p(1, \minus{} 1) \equal{} p(2,2) \equal{} 0.\end{align*} There is a point (ac,bc) \left(\tfrac {a}{c},\tfrac {b}{c}\right) for which p\left(\tfrac {a}{c},\tfrac {b}{c}\right) \equal{} 0 for all such polynomials, where a a, b b, and c c are positive integers, a a and c c are relatively prime, and c>1 c > 1. Find a \plus{} b \plus{} c.

Complex Hexagon

A regular hexagon with center at the origin in the complex plane has opposite pairs of sides one unit apart. One pair of sides is parallel to the imaginary axis. Let R R be the region outside the hexagon, and let S\equal{}\{\frac{1}{z}|z\in R\}. Then the area of S S has the form a\pi\plus{}\sqrt{b}, where a a and b b are positive integers. Find a\plus{}b.
12
2

Cars on Highway

On a long straight stretch of one-way single-lane highway, cars all travel at the same speed and all obey the safety rule: the distance from the back of the car ahead to the front of the car behind is exactly one car length for each 15 kilometers per hour of speed or fraction thereof (Thus the front of a car traveling 52 kilometers per hour will be four car lengths behind the back of the car in front of it.) A photoelectric eye by the side of the road counts the number of cars that pass in one hour. Assuming that each car is 4 meters long and that the cars can travel at any speed, let M M be the maximum whole number of cars that can pass the photoelectric eye in one hour. Find the quotient when M M is divided by 10.

Blue/Green Flags

There are two distinguishable flagpoles, and there are 19 19 flags, of which 10 10 are identical blue flags, and 9 9 are identical green flags. Let N N be the number of distinguishable arrangements using all of the flags in which each flagpole has at least one flag and no two green flags on either pole are adjacent. Find the remainder when N N is divided by 1000 1000.
9
2

Stack of Crates

Ten identical crates each of dimensions 3 3 ft × \times 4 4 ft × \times 6 6 ft. The first crate is placed flat on the floor. Each of the remaining nine crates is placed, in turn, flat on top of the previous crate, and the orientation of each crate is chosen at random. Let mn \frac{m}{n} be the probability that the stack of crates is exactly 41 41 ft tall, where m m and n n are relatively prime positive integers. Find m m.

Particle Moves

A particle is located on the coordinate plane at (5,0) (5,0). Define a move for the particle as a counterclockwise rotation of π/4 \pi/4 radians about the origin followed by a translation of 10 10 units in the positive x x-direction. Given that the particle's position after 150 150 moves is (p,q) (p,q), find the greatest integer less than or equal to |p|\plus{}|q|.
10
2

Isosceles Trapezoid

Let ABCD ABCD be an isosceles trapezoid with AD‾∥BC‾ \overline{AD}\parallel{}\overline{BC} whose angle at the longer base AD‾ \overline{AD} is π3 \dfrac{\pi}{3}. The diagonals have length 1021 10\sqrt {21}, and point E E is at distances 107 10\sqrt {7} and 307 30\sqrt {7} from vertices A A and D D, respectively. Let F F be the foot of the altitude from C C to AD‾ \overline{AD}. The distance EF EF can be expressed in the form mn m\sqrt {n}, where m m and n n are positive integers and n n is not divisible by the square of any prime. Find m \plus{} n.

Growing Path

The diagram below shows a 4×4 4\times4 rectangular array of points, each of which is 1 1 unit away from its nearest neighbors. [asy]unitsize(0.25inch); defaultpen(linewidth(0.7));
int i, j; for(i = 0; i < 4; ++i) for(j = 0; j < 4; ++j) dot(((real)i, (real)j));[/asy]Define a growing path to be a sequence of distinct points of the array with the property that the distance between consecutive points of the sequence is strictly increasing. Let m m be the maximum possible number of points in a growing path, and let r r be the number of growing paths consisting of exactly m m points. Find mr mr.
11
2

Binary Sequences

Consider sequences that consist entirely of A A's and B B's and that have the property that every run of consecutive A A's has even length, and every run of consecutive B B's has odd length. Examples of such sequences are AA AA, B B, and AABAA AABAA, while BBAB BBAB is not such a sequence. How many such sequences have length 14?

Tangent Circles

In triangle ABC ABC, AB \equal{} AC \equal{} 100, and BC \equal{} 56. Circle P P has radius 16 16 and is tangent to AC‾ \overline{AC} and BC‾ \overline{BC}. Circle Q Q is externally tangent to P P and is tangent to AB‾ \overline{AB} and BC‾ \overline{BC}. No point of circle Q Q lies outside of △ABC \triangle ABC. The radius of circle Q Q can be expressed in the form m \minus{} n\sqrt {k}, where m m, n n, and k k are positive integers and k k is the product of distinct primes. Find m \plus{} nk.
8
2
15
2

Square Paper Folding

A square piece of paper has sides of length 100 100. From each corner a wedge is cut in the following manner: at each corner, the two cuts for the wedge each start at distance 17 \sqrt {17} from the corner, and they meet on the diagonal at an angle of 60∘ 60^\circ (see the figure below). The paper is then folded up along the lines joining the vertices of adjacent cuts. When the two edges of a cut meet, they are taped together. The result is a paper tray whose sides are not at right angles to the base. The height of the tray, that is, the perpendicular distance between the plane of the base and the plane formed by the upper edges, can be written in the form mn \sqrt [n]{m}, where m m and n n are positive integers, m<1000 m < 1000, and m m is not divisible by the n nth power of any prime. Find m \plus{} n. [asy]import math; unitsize(5mm); defaultpen(fontsize(9pt)+Helvetica()+linewidth(0.7));
pair O=(0,0); pair A=(0,sqrt(17)); pair B=(sqrt(17),0); pair C=shift(sqrt(17),0)*(sqrt(34)*dir(75)); pair D=(xpart(C),8); pair E=(8,ypart(C));
draw(O--(0,8)); draw(O--(8,0)); draw(O--C); draw(A--C--B); draw(D--C--E);
label("17\sqrt{17}",(0,2),W); label("17\sqrt{17}",(2,0),S); label("cut",midpoint(A--C),NNW); label("cut",midpoint(B--C),ESE); label("fold",midpoint(C--D),W); label("fold",midpoint(C--E),S); label("30∘30^\circ",shift(-0.6,-0.6)*C,WSW); label("30∘30^\circ",shift(-1.2,-1.2)*C,SSE);[/asy]

Conditions for Integer

Find the largest integer n n satisfying the following conditions: (i) n2 n^2 can be expressed as the difference of two consecutive cubes; (ii) 2n\plus{}79 is a perfect square.
7
2
14
2

Geometric Maximization

Let AB‾ \overline{AB} be a diameter of circle ω \omega. Extend AB‾ \overline{AB} through A A to C C. Point T T lies on ω \omega so that line CT CT is tangent to ω \omega. Point P P is the foot of the perpendicular from A A to line CT CT. Suppose AB \equal{} 18, and let m m denote the maximum possible length of segment BP BP. Find m2 m^{2}.

System

Let a a and b b be positive real numbers with a≥b a\ge b. Let ρ \rho be the maximum possible value of ab \frac{a}{b} for which the system of equations a^2\plus{}y^2\equal{}b^2\plus{}x^2\equal{}(a\minus{}x)^2\plus{}(b\minus{}y)^2has a solution in (x,y) (x,y) satisfying 0≤x<a 0\le x<a and 0≤y<b 0\le y<b. Then ρ2 \rho^2 can be expressed as a fraction mn \frac{m}{n}, where m m and n n are relatively prime positive integers. Find m\plus{}n.
5
2

Cone

A right circular cone has base radius r r and height h h. The cone lies on its side on a flat table. As the cone rolls on the surface of the table without slipping, the point where the cone's base meets the table traces a circular arc centered at the point where the vertex touches the table. The cone first returns to its original position on the table after making 17 17 complete rotations. The value of h/r h/r can be written in the form mn m\sqrt {n}, where m m and n n are positive integers and n n is not divisible by the square of any prime. Find m \plus{} n.

Trapezoid Midpoints

In trapezoid ABCD ABCD with BC‾∥AD‾ \overline{BC}\parallel\overline{AD}, let BC\equal{}1000 and AD\equal{}2008. Let \angle A\equal{}37^\circ, \angle D\equal{}53^\circ, and m m and n n be the midpoints of BC‾ \overline{BC} and AD‾ \overline{AD}, respectively. Find the length MN MN.
3
2

Ed and Sue

Ed and Sue bike at equal and constant rates. Similarly, they jog at equal and constant rates, and they swim at equal and constant rates. Ed covers 74 74 kilometers after biking for 2 2 hours, jogging for 3 3 hours, and swimming for 4 4 hours, while Sue covers 91 91 kilometers after jogging for 2 2 hours, swimming for 3 3 hours, and biking for 4 4 hours. Their biking, jogging, and swimming rates are all whole numbers of kilometers per hour. Find the sum of the squares of Ed's biking, jogging, and swimming rates.

Cheese-Cutting

A block of cheese in the shape of a rectangular solid measures 10 10 cm by 13 13 cm by 14 14 cm. Ten slices are cut from the cheese. Each slice has a width of 1 1 cm and is cut parallel to one face of the cheese. The individual slices are not necessarily parallel to each other. What is the maximum possible volume in cubic cm of the remaining block of cheese after ten slices have been cut off?
2
2
4
2
1
2