MathDB

2012 AIME Problems

Part of AIME Problems

Subcontests

(15)
8
2

Volume of a Part of a Cube

Cube ABCDEFGHABCDEFGH, labeled as shown below, has edge length 11 and is cut by a plane passing through vertex DD and the midpoints MM and NN of AB‾\overline{AB} and CG‾\overline{CG} respectively. The plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form pq\frac{p}{q}, where pp and qq are relatively prime positive integers. Find p+qp+q. [asy]
draw((0,0)--(10,0)--(10,10)--(0,10)--cycle); draw((0,10)--(4,13)--(14,13)--(10,10)); draw((10,0)--(14,3)--(14,13)); draw((0,0)--(4,3)--(4,13), dashed); draw((4,3)--(14,3), dashed); dot((0,0)); dot((0,10)); dot((10,10)); dot((10,0)); dot((4,3)); dot((14,3)); dot((14,13)); dot((4,13)); dot((14,8)); dot((5,0)); label("A", (0,0), SW); label("B", (10,0), S); label("C", (14,3), E); label("D", (4,3), NW); label("E", (0,10), W); label("F", (10,10), SE); label("G", (14,13), E); label("H", (4,13), NW); label("M", (5,0), S); label("N", (14,8), E); [/asy]

Complex Product Nonlinear Equation System

The complex numbers zz and ww satisfy the system \begin{align*}z+\frac{20i}{w}&=5+i,\\w+\frac{12i}{z}&=-4+10i.\end{align*} Find the smallest possible value of ∣zw∣2|zw|^2.
15
1
14
2
13
2

Cocentric Circles of radii 3,4,5

Three concentric circles have radii 33, 44, and 55. An equilateral triangle with one vertex on each circle has side length ss. The largest possible area of the triangle can be written as a+bcda+\frac{b}{c}\sqrt{d}, where a,b,ca,b,c and dd are positive integers, bb and cc are relatively prime, and dd is not divisible by the square of any prime. Find a+b+c+da+b+c+d.

Five Congruent Equilateral Triangles

Equilateral △ABC\triangle ABC has side length 111\sqrt{111}. There are four distinct triangles AD1E1AD_1E_1, AD1E2AD_1E_2, AD2E3AD_2E_3, and AD2E4AD_2E_4, each congruent to △ABC\triangle ABC, with BD1=BD2=11BD_1 = BD_2=\sqrt{11}. Find ∑k=14(CEk)2\sum^4_{k=1}(CE_k)^2.
12
2

Steiner's Theorem Returns

Let △ABC\triangle ABC be a right triangle with right angle at CC. Let DD and EE be points on AB‾\overline{AB} with DD between AA and EE such that CD‾\overline{CD} and CE‾\overline{CE} trisect ∠C\angle C. If DEBE=815\frac{DE}{BE} = \frac{8}{15}, then tan⁡B\tan B can be written as mpn\frac{m\sqrt{p}}{n}, where mm and nn are relatively prime positive integers, and pp is a positive integer not divisible by the square of any prime. Find m+n+pm+n+p.

P-Safe Numbers

For a positive integer pp, define the positive integer nn to be pp-safe if nn differs in absolute value by more than 22 from all multiples of pp. For example, the set of 1010-safe numbers is {3,4,5,6,7,13,14,15,16,17,23,…}\{3, 4, 5, 6, 7, 13, 14, 15, 16, 17,23, \ldots \}. Find the number of positive integers less than or equal to 10,00010,000 which are simultaneously 77-safe, 1111-safe, and 1313-safe.·
11
2

Froggy

A frog begins at P0=(0,0)P_0 = (0,0) and makes a sequence of jumps according to the following rule: from Pn=(xn,yn)P_n=(x_n,y_n), the frog jumps to Pn+1P_{n+1}, which may be any of the points (xn+7,yn+2)(x_n+7, y_n+2), (xn+2,yn+7)(x_n+2,y_n+7), (xn−5,yn−10)(x_n-5, y_n-10), or (xn−10,yn−5)(x_n-10,y_n-5). There are MM points (x,y)(x,y) with ∣x∣+∣y∣≤100|x|+|y| \le 100 that can be reached by a sequence of such jumps. Find the remainder when MM is divided by 10001000.

Recursively Defined Functions

Let f1(x)=23−33x+1f_1(x) = \frac{2}{3}-\frac{3}{3x+1}, and for n≥2n \ge 2, define fn(x)=f1(fn−1(x))f_n(x) = f_1(f_{n-1} (x)). The value of x that satisfies f1001(x)=x−3f_{1001}(x) = x - 3 can be expressed in the form mn\frac{m}{n}, where mm and nn are relatively prime positive integers. Find m+nm + n.
9
2

Logarithms in x,y,z

Let xx, yy, and zz be positive real numbers that satisfy 2log⁡x(2y)=2log⁡2x(4z)=log⁡2x4(8yz)≠0. 2\log_x(2y) = 2\log_{2x}(4z) = \log_{2x^4}(8yz) \neq 0. The value of xy5zxy^5z can be expressed in the form 12p/q\frac{1}{2^{p/q}}, where pp and qq are relatively prime integers. Find p+qp+q.

Double Trigonometric Ratios

Let xx and yy be real numbers such that sin⁡xsin⁡y=3\frac{\sin{x}}{\sin{y}} = 3 and cos⁡xcos⁡y=12\frac{\cos{x}}{\cos{y}} = \frac{1}{2}. The value of sin⁡2xsin⁡2y+cos⁡2xcos⁡2y\frac{\sin{2x}}{\sin{2y}} + \frac{\cos{2x}}{\cos{2y}} can be expressed in the form pq\frac{p}{q}, where pp and qq are relatively prime positive integers. Find p+qp + q.
10
2

Sets of perfect squares ending in 256

Let S\mathcal{S} be the set of all perfect squares whose rightmost three digits in base 1010 are 256256. Let T\mathcal{T} be the set of all numbers of the form x−2561000\frac{x-256}{1000}, where xx is in S\mathcal{S}. In other words, T\mathcal{T} is the set of numbers that result when the last three digits of each number in S\mathcal{S} are truncated. Find the remainder when the tenth smallest element of T\mathcal{T} is divided by 10001000.

Product of Numbers and Their Floors

Find the number of positive integers nn less than 10001000 for which there exists a positive real number xx such that n=x⌊x⌋n = x \lfloor x \rfloor. Note: ⌊x⌋\lfloor x \rfloor is the greatest integer less than or equal to xx.
7
2

Trading coins

At each of the sixteen circles in the network below stands a student. A total of 3360 coins are distributed among the sixteen students. All at once, all students give away all their coins by passing an equal number of coins to each of their neighbors in the network. After the trade, all students have the same number of coins as they started with. Find the number of coins the student standing at the center circle had originally.
[asy] import graph;
unitsize(1 cm);
pair[] O;
O[1] = (0,0); O[2] = 0.6*dir(270); O[3] = 0.6*dir(270 + 360/5); O[4] = 0.6*dir(270 + 2*360/5); O[5] = 0.6*dir(270 + 3*360/5); O[6] = 0.6*dir(270 + 4*360/5); O[7] = 1.2*dir(90); O[8] = 1.2*dir(90 + 360/5); O[9] = 1.2*dir(90 + 2*360/5); O[10] = 1.2*dir(90 + 3*360/5); O[11] = 1.2*dir(90 + 4*360/5); O[12] = 2*dir(270); O[13] = 2*dir(270 + 360/5); O[14] = 2*dir(270 + 2*360/5); O[15] = 2*dir(270 + 3*360/5); O[16] = 2*dir(270 + 4*360/5);
draw(O[1]--O[2]); draw(O[1]--O[3]); draw(O[1]--O[4]); draw(O[1]--O[5]); draw(O[1]--O[6]); draw(O[7]--O[5]--O[8]--O[6]--O[9]--O[2]--O[10]--O[3]--O[11]--O[4]--cycle); draw(O[12]--O[10]--O[13]--O[11]--O[14]--O[7]--O[15]--O[8]--O[16]--O[9]--cycle); draw(O[12]--O[13]--O[14]--O[15]--O[16]--cycle);
for(int i = 1; i <= 16; ++i) { filldraw(Circle(O,0.2),white,black); } [/asy]

Eight-One Binary Sequences

Let SS be the increasing sequence of positive integers whose binary representation has exactly 88 ones. Let NN be the 1000th1000^{th} number in SS. Find the remainder when NN is divided by 10001000.
6
2
5
2

Binary integers whose difference is 1

Let BB be the set of all binary integers that can be written using exactly 5 zeros and 8 ones where leading zeros are allowed. If all possible subtractions are performed in which one element of BB is subtracted from another, find the number of times the answer 1 is obtained.

Pyramid from Star Folding

In the accompanying figure, the outer square has side length 40. A second square S' of side length 15 is constructed inside S with the same center as S and with sides parallel to those of S. From each midpoint of a side of S, segments are drawn to the two closest vertices of S'. The result is a four-pointed starlike figure inscribed in S. The star figure is cut out and then folded to form a pyramid with base S'. Find the volume of this pyramid. [asy] draw((0,0)--(8,0)--(8,8)--(0,8)--(0,0)); draw((2.5,2.5)--(4,0)--(5.5,2.5)--(8,4)--(5.5,5.5)--(4,8)--(2.5,5.5)--(0,4)--(2.5,2.5)--(5.5,2.5)--(5.5,5.5)--(2.5,5.5)--(2.5,2.5)); [/asy]
4
2

Running away from a place

Butch and Sundance need to get out of Dodge. To travel as quickly as possible, each alternates walking and riding their only horse, Sparky, as follows. Butch begins walking as Sundance rides. When Sundance reaches the first of their hitching posts that are conveniently located at one-mile intervals along their route, he ties Sparky to the post and begins walking. When Butch reaches Sparky, he rides until he passes Sundance, then leaves Sparky at the next hitching post and resumes walking, and they continue in this manner. Sparky, Butch, and Sundance walk at 6, 4, and 2.5 miles per hour, respectively. The first time Butch and Sundance meet at a milepost, they are nn miles from Dodge, and have been traveling for tt minutes. Find n+tn + t.

Biking Across the Field

Ana, Bob, and Cao bike at constant rates of 8.68.6 meters per second, 6.26.2 meters per second, and 55 meters per second, respectively. They all begin biking at the same time from the northeast corner of a rectangular field whose longer side runs due west. Ana starts biking along the edge of the field, initially heading west, Bob starts biking along the edge of the field, initially heading south, and Cao bikes in a straight line across the field to a point D on the south edge of the field. Cao arrives at point D at the same time that Ana and Bob arrive at D for the first time. The ratio of the field's length to the field's width to the distance from point D to the southeast corner of the field can be represented as p:q:rp : q : r, where pp, qq, and rr are positive integers with p and q relatively prime. Find p+q+rp + q + r.
3
2

Ordering and serving meals randomly

Nine people sit down for dinner where there are three choices of meals. Three people order the beef meal, three order the chicken meal, and three order the fish meal. The waiter serves the nine meals in random order. Find the number of ways in which the waiter could serve the meal types to the nine people such that exactly one person receives the type of meal ordered by that person.

Gender Roles in Math Committees

At a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women and must also contain two professors from each of the three departments. Find the number of possible comittees that can be formed subject to these requirements.
2
2
1
2