MathDB

2005 AIME Problems

Part of AIME Problems

Subcontests

(15)
14
2
13
2
12
2

Difference of Two Series

For positive integers nn, let τ(n)\tau (n) denote the number of positive integer divisors of nn, including 11 and nn. For example, τ(1)=1\tau (1)=1 and τ(6)=4\tau(6) =4. Define S(n)S(n) by S(n)=τ(1)+τ(2)+...+τ(n).S(n)=\tau(1)+ \tau(2) + ... + \tau(n). Let aa denote the number of positive integers n≤2005n \leq 2005 with S(n)S(n) odd, and let bb denote the number of positive integers n≤2005n \leq 2005 with S(n)S(n) even. Find ∣a−b∣|a-b|.

Square on a Coordinate Plane

Square ABCDABCD has center OO, AB=900AB=900, EE and FF are on ABAB with AE<BFAE<BF and EE between AA and FF, m∠EOF=45∘m\angle EOF =45^\circ, and EF=400EF=400. Given that BF=p+qrBF=p+q\sqrt{r}, wherer p,q,p,q, and rr are positive integers and rr is not divisible by the square of any prime, find p+q+rp+q+r.
11
2
10
2
9
2
8
2

Roots of an Exponential Equation

The equation 2333x−2+2111x+2=2222x+1+12^{333x-2}+2^{111x+2}=2^{222x+1}+1 has three real roots. Given that their sum is m/nm/n where mm and nn are relatively prime positive integers, find m+nm+n.

Externally Tangent Circles

Circles C1C_1 and C2C_2 are externally tangent, and they are both internally tangent to circle C3C_3. The radii of C1C_1 and C2C_2 are 44 and 1010, respectively, and the centers of the three circles are all collinear. A chord of C3C_3 is also a common external tangent of C1C_1 and C2C_2. Given that the length of the chord is mnp\frac{m\sqrt{n}}{p} where m,n,m,n, and pp are positive integers, mm and pp are relatively prime, and nn is not divisible by the square of any prime, find m+n+pm+n+p.
7
2
6
2

A Product of Nonreal Roots

Let PP be the product of the nonreal roots of x4−4x3+6x2−4x=2005x^4-4x^3+6x^2-4x=2005. Find ⌊P⌋\lfloor P\rfloor.

Stack of Cards

The cards in a stack of 2n2n cards are numbered consecutively from 11 through 2n2n from top to bottom. The top nn cards are removed, kept in order, and form pile AA. The remaining cards form pile BB. The cards are then restacked by taking cards alternately from the tops of pile BB and AA, respectively. In this process, card number (n+1)(n+1) becomes the bottom card of the new stack, card number 11 is on top of this card, and so on, until piles AA and BB are exhausted. If, after the restacking process, at least one card from each pile occupies the same position that it occupied in the original stack, the stack is named magical. Find the number of cards in the magical stack in which card number 131131 retains its original position.
5
2
4
2
3
2
2
2

Arithmetic Series and Common Differences

For each positive integer kk, let SkS_k denote the increasing arithmetic sequence of integers whose first term is 11 and whose common difference is kk. For example, S3S_3 is the sequence 1,4,7,10,...1,4,7,10,.... For how many values of kk does SkS_k contain the term 20052005?

Breakfast at a Hotel

A hotel packed breakfast for each of three guests. Each breakfast should have consisted of three types of rolls, one each of nut, cheese, and fruit rolls. The preparer wrapped each of the nine rolls and once wrapped, the rolls were indistinguishable from one another. She then randomly put three rolls in a bag for each of the guests. Given that the probability each guest got one roll of each type is mn\frac{m}{n}, where mm and nn are relatively prime integers, find m+nm+n.
1
2
15
2