MathDB

2007 AIME Problems

Part of AIME Problems

Subcontests

(15)
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An Equilateral Triangle

Let ABCABC be an equilateral triangle, and let DD and FF be points on sides BCBC and ABAB, respectively, with FA=5FA=5 and CD=2CD=2. Point EE lies on side CACA such that ∠DEF=60∘\angle DEF = 60^\circ. The area of triangle DEFDEF is 14314\sqrt{3}. The two possible values of the length of side ABAB are p±qrp \pm q\sqrt{r}, where pp and qq are rational, and rr is an integer not divisible by the square of a prime. Find rr.

Tangent Circles

Four circles ω,\omega, ωA,\omega_{A}, ωB,\omega_{B}, and ωC\omega_{C} with the same radius are drawn in the interior of triangle ABCABC such that ωA\omega_{A} is tangent to sides ABAB and ACAC, ωB\omega_{B} to BCBC and BABA, ωC\omega_{C} to CACA and CBCB, and ω\omega is externally tangent to ωA,\omega_{A}, ωB,\omega_{B}, and ωC\omega_{C}. If the sides of triangle ABCABC are 13,13, 14,14, and 15,15, the radius of ω\omega can be represented in the form mn\frac{m}{n}, where mm and nn are relatively prime positive integers. Find m+n.m+n.
14
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13
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Intersection of Plane and Pyramid

A square pyramid with base ABCDABCD and vertex EE has eight edges of length 4. A plane passes through the midpoints of AE‾\overline{AE}, BC‾\overline{BC}, and CD‾\overline{CD}. The plane's intersection with the pyramid has an area that can be expressed as p\sqrt{p}. Find pp.

Triangular Array

A triangular array of squares has one square in the first row, two in the second, and in general, kk squares in the kkth row for 1≤k≤11.1 \leq k \leq 11. With the exception of the bottom row, each square rests on two squares in the row immediately below (illustrated in given diagram). In each square of the eleventh row, a 00 or a 11 is placed. Numbers are then placed into the other squares, with the entry for each square being the sum of the entries in the two squares below it. For how many initial distributions of 00's and 11's in the bottom row is the number in the top square a multiple of 33? [asy] defaultpen(linewidth(0.7)); path p=origin--(1,0)--(1,1)--(0,1)--cycle; int i,j; for(i=0; i<12; i=i+1) { for(j=0; j<11-i; j=j+1) { draw(shift(i/2+j,i)*p); }}[/asy]
12
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Rotating Isosceles Triangle

In isosceles triangle ABCABC, AA is located at the origin and BB is located at (20,0)(20, 0). Point CC is in the first quadrant with AC=BCAC = BC and ∠BAC=75∘\angle BAC = 75^\circ. If △ABC\triangle ABC is rotated counterclockwise about point AA until the image of CC lies on the positive y-axis, the area of the region common to the original and the rotated triangle is in the form p2+q3+r6+sp\sqrt{2}+q\sqrt{3}+r\sqrt{6}+s where pp, qq, rr, ss are integers. Find (p−q+r−s)/2(p-q+r-s)/2.

Log of Geometric Sequence

The increasing geometric sequence x0,x1,x2,…x_{0},x_{1},x_{2},\ldots consists entirely of integral powers of 3.3. Given that ∑n=07log⁡3(xn)=308and56≤log⁡3(∑n=07xn)≤57,\sum_{n=0}^{7}\log_{3}(x_{n}) = 308\qquad\text{and}\qquad 56 \leq \log_{3}\left ( \sum_{n=0}^{7}x_{n}\right ) \leq 57, find log⁡3(x14).\log_{3}(x_{14}).
11
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Summation Pattern Problem

For each positive integer pp, let b(p)b(p) denote the unique positive integer kk such that ∣k−p∣<12|k-\sqrt{p}|<\frac{1}{2}. For example, b(6)=2b(6) = 2 and b(23)=5b(23)=5. If S=∑p=12007b(p)S = \textstyle\sum_{p=1}^{2007}b(p), find the remainder when S is divided by 1000.

Rolling tube

Two long cylindrical tubes of the same length but different diameters lie parallel to each other on a flat surface. The larger tube has radius 7272 and rolls along the surface toward the smaller tube, which has radius 2424. It rolls over the smaller tube and continues rolling along the flat surface until it comes to rest on the same point of its circumference as it started, having made one complete revolution. If the smaller tube never moves, and the rolling occurs with no slipping, the larger tube ends up a distance xx from where it starts. The distance xx can be expressed in the form aπ+bc,a\pi+b\sqrt{c}, where a,a, b,b, and cc are integers and cc is not divisible by the square of any prime. Find a+b+c.a+b+c.
10
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Grid Combinatorics

In the 6×4 6\times4 grid shown, 12 12 of the 24 24 squares are to be shaded so that there are two shaded squares in each row and three shaded squares in each column. Let N N be the number of shadings with this property. Find the remainder when N N is divided by 1000 1000. [asy]size(100); defaultpen(linewidth(0.7)); int i; for(i=0; i<5; ++i) { draw((i,0)--(i,6)); } for(i=0; i<7; ++i) { draw((0,i)--(4,i)); }[/asy]

Probability of sets

Let SS be a set with six elements. Let PP be the set of all subsets of S.S. Subsets AA and BB of SS, not necessarily distinct, are chosen independently and at random from PP. the probability that BB is contained in at least one of AA or S−AS-A is mnr,\frac{m}{n^{r}}, where mm, nn, and rr are positive integers, nn is prime, and mm and nn are relatively prime. Find m+n+r.m+n+r. (The set S−AS-A is the set of all elements of SS which are not in A.A.)
9
2

Triangle and Tangent Circles

In right triangle ABCABC with right angle CC, CA=30CA=30 and CB=16CB=16. Its legs CA‾\overline{CA} and CB‾\overline{CB} are extended beyond AA and BB. Points O1O_{1} and O2O_{2} lie in the exterior of the triangle and are the centers of two circles with equal radii. The circle with center O1O_{1} is tangent to the hypotenuse and to the extension of leg CA, the circle with center O2O_{2} is tangent to the hypotenuse and to the extension of leg CB, and the circles are externally tangent to each other. The length of the radius of either circle can be expressed as p/qp/q, where pp and qq are relatively prime positive integers. Find p+qp+q.

Inscribed circles in rectangle

Rectangle ABCDABCD is given with AB=63AB=63 and BC=448.BC=448. Points EE and FF lie on ADAD and BCBC respectively, such that AE=CF=84.AE=CF=84. The inscribed circle of triangle BEFBEF is tangent to EFEF at point P,P, and the inscribed circle of triangle DEFDEF is tangent to EFEF at point Q.Q. Find PQ.PQ.
8
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Cubic Polynomial

The polynomial P(x)P(x) is cubic. What is the largest value of kk for which the polynomials Q1(x)=x2+(k−29)x−kQ_{1}(x) = x^{2}+(k-29)x-k and Q2(x)=2x2+(2k−43)x+kQ_{2}(x) = 2x^{2}+(2k-43)x+k are both factors of P(x)P(x)?

Basic Rectangles

A rectangular piece of of paper measures 4 units by 5 units. Several lines are drawn parallel to the edges of the paper. A rectangle determined by the intersections of some of these lines is called basic if (i) all four sides of the rectangle are segments of drawn line segments, and (ii) no segments of drawn lines lie inside the rectangle. Given that the total length of all lines drawn is exactly 2007 units, let NN be the maximum possible number of basic rectangles determined. Find the remainder when NN is divided by 1000.
7
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Summation, Ceiling, and Floor Functions

Let N=∑k=11000k(⌈log⁡2k⌉−⌊log⁡2k⌋).N= \sum_{k=1}^{1000}k(\lceil \log_{\sqrt{2}}k\rceil-\lfloor \log_{\sqrt{2}}k \rfloor). Find the remainder when N is divided by 1000. (Here ⌊x⌋\lfloor x \rfloor denotes the greatest integer that is less than or equal to x, and ⌈x⌉\lceil x \rceil denotes the least integer that is greater than or equal to x.)

Floor of cube roots

Given a real number x,x, let ⌊x⌋\lfloor x \rfloor denote the greatest integer less than or equal to x.x. For a certain integer k,k, there are exactly 7070 positive integers n1,n2,…,n70n_{1}, n_{2}, \ldots, n_{70} such that k=⌊n13⌋=⌊n23⌋=⋯=⌊n703⌋k=\lfloor\sqrt[3]{n_{1}}\rfloor = \lfloor\sqrt[3]{n_{2}}\rfloor = \cdots = \lfloor\sqrt[3]{n_{70}}\rfloor and kk divides nin_{i} for all ii such that 1≤i≤70.1 \leq i \leq 70. Find the maximum value of nik\frac{n_{i}}{k} for 1≤i≤70.1\leq i \leq 70.
6
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Frog on a Number Line

A frog is placed at the origin on a number line, and moves according to the following rule: in a given move, the frog advanced to either the closest integer point with a greater integer coordinate that is a multiple of 3, or to the closest integer point with a greater integer coordinate that is a multiple of 13. A move sequence is a sequence of coordinates which correspond to valid moves, beginning with 0, and ending with 39. For example, 0, 3, 6, 13, 15, 26, 39 is a move sequence. How many move sequences are possible for the frog?

Parity-Monotonic Numbers

An integer is called parity-monotonic if its decimal representation a1a2a3⋯aka_{1}a_{2}a_{3}\cdots a_{k} satisfies ai<ai+1a_{i}<a_{i+1} if aia_{i} is odd, and ai>ai+1a_{i}>a_{i+1} is aia_{i} is even. How many four-digit parity-monotonic integers are there?
5
2

Converting Temperatures

The formula for converting a Fahrenheit temperature FF to the corresponding Celsius temperature CC is C=59(F−32)C=\frac{5}{9}(F-32). An integer Fahrenheit temperature is converted to Celsius and rounded to the nearest integer; the resulting integer Celsius temperature is converted back to Fahrenheit and rounded to the nearest integer. For how many integer Fahrenheit temperatures TT with 32≤T≤100032 \leq T \leq 1000 does the original temperature equal the final temperature?

Graph Squares

The graph of the equation 9x+223y=20079x+223y=2007 is drawn on graph paper with each square representing one unit in each direction. How many of the 11 by 11 graph paper squares have interiors lying entirely below the graph and entirely in the first quadrant?
4
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3
2

Complex Number Problem

The complex number zz is equal to 9+bi9+bi, where bb is a positive real number and i2=−1i^{2}=-1. Given that the imaginary parts of z2z^{2} and z3z^{3} are equal, find bb.

Long Diagonal

Square ABCDABCD has side length 1313, and points EE and FF are exterior to the square such that BE=DF=5BE=DF=5 and AE=CF=12AE=CF=12. Find EF2EF^{2}. [asy] size(200); defaultpen(fontsize(10)); real x=22.61986495; pair A=(0,26), B=(26,26), C=(26,0), D=origin, E=A+24*dir(x), F=C+24*dir(180+x); draw(B--C--F--D--C^^D--A--E--B--A, linewidth(0.7)); dot(A^^B^^C^^D^^E^^F); pair point=(13,13); label("AA", A, dir(point--A)); label("BB", B, dir(point--B)); label("CC", C, dir(point--C)); label("DD", D, dir(point--D)); label("EE", E, dir(point--E)); label("FF", F, dir(point--F));[/asy]
2
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Moving Walkway

A 100 foot long moving walkway moves at a constant rate of 6 feet per second. Al steps onto the start of the walkway and stands. Bob steps onto the start of the walkway two seconds later and strolls forward along the walkway at a constant rate of 4 feet per second. Two seconds after that, Cy reaches the start of the walkway and walks briskly forward beside the walkway at a constant rate of 8 feet per second. At a certain time, one of these three persons is exactly halfway between the other two. At that time, find the distance in feet between the start of the walkway and the middle person.

Ordered Triples

Find the number of ordered triple (a,b,c)(a,b,c) where aa, bb, and cc are positive integers, aa is a factor of bb, aa is a factor of cc, and a+b+c=100a+b+c=100.
1
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