MathDB

2010 AIME Problems

Part of AIME Problems

Subcontests

(15)
1
2
2
2
3
2
4
2

Jackie and Phil's Coins

Jackie and Phil have two fair coins and a third coin that comes up heads with probability 47 \frac47. Jackie flips the three coins, and then Phil flips the three coins. Let mn \frac{m}{n} be the probability that Jackie gets the same number of heads as Phil, where m m and n n are relatively prime positive integers. Find m \plus{} n.

Dave's Airport Snafu

Dave arrives at an airport which has twelve gates arranged in a straight line with exactly 100 100 feet between adjacent gates. His departure gate is assigned at random. After waiting at that gate, Dave is told the departure gate has been changed to a different gate, again at random. Let the probability that Dave walks 400 400 feet or less to the new gate be a fraction mn \frac{m}{n}, where m m and n n are relatively prime positive integers. Find m\plus{}n.
5
2
6
2
7
2

Minimally Intersecting

Define an ordered triple (A,B,C) (A, B, C) of sets to be minimally intersecting if |A \cap B| \equal{} |B \cap C| \equal{} |C \cap A| \equal{} 1 and A \cap B \cap C \equal{} \emptyset. For example, ({1,2},{2,3},{1,3,4}) (\{1,2\},\{2,3\},\{1,3,4\}) is a minimally intersecting triple. Let N N be the number of minimally intersecting ordered triples of sets for which each set is a subset of {1,2,3,4,5,6,7} \{1,2,3,4,5,6,7\}. Find the remainder when N N is divided by 1000 1000. Note: ∣S∣ |S| represents the number of elements in the set S S.

Imaginary roots

Let P(z) \equal{} z^3 \plus{} az^2 \plus{} bz \plus{} c, where a a, b b, and c c are real. There exists a complex number w w such that the three roots of P(z) P(z) are w \plus{} 3i, w \plus{} 9i, and 2w \minus{} 4, where i^2 \equal{} \minus{} 1. Find |a \plus{} b \plus{} c|.
8
2

Floor Region

For a real number a a, let ⌊a⌋ \lfloor a \rfloor denominate the greatest integer less than or equal to a a. Let R \mathcal{R} denote the region in the coordinate plane consisting of points (x,y) (x,y) such that \lfloor x \rfloor ^2 \plus{} \lfloor y \rfloor ^2 \equal{} 25. The region R \mathcal{R} is completely contained in a disk of radius r r (a disk is the union of a circle and its interior). The minimum value of r r can be written as mn \tfrac {\sqrt {m}}{n}, where m m and n n are integers and m m is not divisible by the square of any prime. Find m \plus{} n.

Two sets

Let N N be the number of ordered pairs of nonempty sets A \mathcal{A} and B \mathcal{B} that have the following properties: • \mathcal{A} \cup \mathcal{B} \equal{} \{1,2,3,4,5,6,7,8,9,10,11,12\}, • \mathcal{A} \cap \mathcal{B} \equal{} \emptyset, • The number of elements of A \mathcal{A} is not an element of A \mathcal{A}, • The number of elements of B \mathcal{B} is not an element of B \mathcal{B}. Find N N.
9
2

Nonlinear System

Let (a,b,c) (a,b,c) be the real solution of the system of equations x^3 \minus{} xyz \equal{} 2, y^3 \minus{} xyz \equal{} 6, z^3 \minus{} xyz \equal{} 20. The greatest possible value of a^3 \plus{} b^3 \plus{} c^3 can be written in the form mn \frac{m}{n}, where m m and n n are relatively prime positive integers. Find m \plus{} n.

Hexagon within the hexagon

Let ABCDEF ABCDEF be a regular hexagon. Let G G, H H, I I, J J, K K, and L L be the midpoints of sides AB AB, BC BC, CD CD, DE DE, EF EF, and AF AF, respectively. The segments AH AH, BI BI, CJ CJ, DK DK, EL EL, and FG FG bound a smaller regular hexagon. Let the ratio of the area of the smaller hexagon to the area of ABCDEF ABCDEF be expressed as a fraction mn \frac {m}{n} where m m and n n are relatively prime positive integers. Find m \plus{} n.
10
2
11
2

Revolving a Region

Let R \mathcal{R} be the region consisting of the set of points in the coordinate plane that satisfy both |8 \minus{} x| \plus{} y \le 10 and 3y \minus{} x \ge 15. When R \mathcal{R} is revolved around the line whose equation is 3y \minus{} x \equal{} 15, the volume of the resulting solid is mπnp \frac {m\pi}{n\sqrt {p}}, where m m, n n, and p p are positive integers, m m and n n are relatively prime, and p p is not divisible by the square of any prime. Find m \plus{} n \plus{} p.

T-grid problem

Define a T-grid to be a 3×3 3\times3 matrix which satisfies the following two properties: (1) Exactly five of the entries are 1 1's, and the remaining four entries are 0 0's. (2) Among the eight rows, columns, and long diagonals (the long diagonals are {a13,a22,a31} \{a_{13},a_{22},a_{31}\} and {a11,a22,a33} \{a_{11},a_{22},a_{33}\}, no more than one of the eight has all three entries equal. Find the number of distinct T-grids.
12
2
13
2

Regional Area

Rectangle ABCD ABCD and a semicircle with diameter AB AB are coplanar and have nonoverlapping interiors. Let R \mathcal{R} denote the region enclosed by the semicircle and the rectangle. Line ℓ \ell meets the semicircle, segment AB AB, and segment CD CD at distinct points N N, U U, and T T, respectively. Line ℓ \ell divides region R \mathcal{R} into two regions with areas in the ratio 1:2 1: 2. Suppose that AU \equal{} 84, AN \equal{} 126, and UB \equal{} 168. Then DA DA can be represented as 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.

Teams in Card Game

The 52 52 cards in a deck are numbered 1,2,…,52 1, 2, \ldots, 52. Alex, Blair, Corey, and Dylan each picks a card from the deck without replacement and with each card being equally likely to be picked, The two persons with lower numbered cards from a team, and the two persons with higher numbered cards form another team. Let p(a) p(a) be the probability that Alex and Dylan are on the same team, given that Alex picks one of the cards a a and a\plus{}9, and Dylan picks the other of these two cards. The minimum value of p(a) p(a) for which p(a)≥12 p(a)\ge\frac12 can be written as mn \frac{m}{n}. where m m and n n are relatively prime positive integers. Find m\plus{}n.
14
2
15
2

Congruent Incircles

In △ABC \triangle{ABC} with AB=12 AB = 12, BC=13 BC = 13, and AC=15 AC = 15, let M M be a point on AC‾ \overline{AC} such that the incircles of △ABM \triangle{ABM} and △BCM \triangle{BCM} have equal radii. Let p p and q q be positive relatively prime integers such that AMCM=pq \tfrac{AM}{CM} = \tfrac{p}{q}. Find p+q p + q.

Two Circumcircles

In triangle ABC ABC, AC \equal{} 13, BC \equal{} 14, and AB\equal{}15. Points M M and D D lie on AC AC with AM\equal{}MC and \angle ABD \equal{} \angle DBC. Points N N and E E lie on AB AB with AN\equal{}NB and \angle ACE \equal{} \angle ECB. Let P P be the point, other than A A, of intersection of the circumcircles of △AMN \triangle AMN and △ADE \triangle ADE. Ray AP AP meets BC BC at Q Q. The ratio BQCQ \frac{BQ}{CQ} can be written in the form mn \frac{m}{n}, where m m and n n are relatively prime positive integers. Find m\minus{}n.