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2005 AIME I Problems

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2005 AIME I (Answer Key)
Printable version: AoPS Resources • PDF | MathLinks Resources • PDF

Instructions

  1. This is a 15-question, 3-hour examination. All answers are integers ranging from 000 to 999, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.
  2. No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, calculators are not permitted.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

Contents

Problem 1

Six circles form a ring with with each circle externally tangent to two circles adjacent to it. All circles are internally tangent to a circle C with radius 30. Let K be the area of the region inside circle C and outside of the six circles in the ring. Find \lfloor K \rfloor.

Solution

Problem 2

For each positive integer k, let S_k denote the increasing arithmetic sequence of integers whose first term is 1 and whose common difference is k. For example, S_3 is the sequence 1,4,7,10,\ldots. For how many values of k does S_k contain the term 2005?

Solution

Problem 3

How many positive integers have exactly three proper divisors, each of which is less than 50?

Solution

Problem 4

The director of a marching band wishes to place the members into a formation that includes all of them and has no unfilled positions. If they are arranged in a square formation, there are 5 members left over. The director realizes that if he arranges the group in a formation with 7 more rows than columns, there are no members left over. Find the maximum number of members this band can have.

Solution

Problem 5

Robert has 4 indistinguishable gold coins and 4 indistinguishable silver coins. Each coin has an engraving of one face on one side, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the number of possible distunguishable arrangements of the 8 coins.

Solution

Problem 6

Let P be the product of the nonreal roots of x^4-4x^3+6x^2-4x=2005. Find \lfloor P\rfloor.

Solution

Problem 7

In quadrilateral ABCD, BC=8, CD=12, AD=10, and m\angle A= m\angle B = 60^\circ. Given that AB = p + \sqrt{q}, where p and q are positive integers, find p+q.

Solution

Problem 8

The equation 2^{333x-2} + 2^{111x+2} = 2^{222x+1} + 1 has three real roots. Given that their sum is \frac mn where m and n are relatively prime positive integers, find m+n.

Solution

Problem 9

Twenty seven unit cubes are painted orange on a set of four faces so that two non-painted faces share an edge. The 27 cubes are randomly arranged to form a 3\times 3 \times 3 cube. Given the probability of the entire surface area of the larger cube is orange is \frac{p^a}{q^br^c}, where p,q, and r are distinct primes and a,b, and c are positive integers, find a+b+c+p+q+r.

Solution

Problem 10

Triangle ABC lies in the Cartesian Plane and has an area of 70. The coordinates of B and C are (12,19) and (23,20), respectively, and the coordinates of A are (p,q). The line containing the median to side BC has slope -5. Find the largest possible value of p+q.

Solution

Problem 11

A semicircle with diameter d is contained in a square whose sides have length 8. Given the maximum value of d is m - \sqrt{n}, find m+n.

Solution

Problem 12

For positive integers n, let \tau (n) denote the number of positive integer divisors of n, including 1 and n. For example, \tau (1)=1 and \tau(6) =4. Define S(n) by S(n)=\tau(1)+ \tau(2) + \cdots + \tau(n). Let a denote the number of positive integers n \leq 2005 with S(n) odd, and let b denote the number of positive integers n \leq 2005 with S(n) even. Find |a-b|.

Solution

Problem 13

A particle moves in the Cartesian Plane according to the following rules:

  1. From any lattice point (a,b), the particle may only move to (a+1,b), (a,b+1), or (a+1,b+1).
  2. There are no right angle turns in the particle's path.

How many different paths can the particle take from (0,0) to (5,5)?

Solution

Problem 14

Consider the points A(0,12), B(10,9), C(8,0), and D(-4,7). There is a unique square S such that each of the four points is on a different side of S. Let K be the area of S. Find the remainder when 10K is divided by 1000.

Solution

Problem 15

Triangle ABC has BC=20. The incircle of the triangle evenly trisects the median AD. If the area of the triangle is m \sqrt{n} where m and n are integers and n is not divisible by the square of a prime, find m+n.

Solution

See Also

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