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1995 AIME Problems

From AoPSWiki

Contents

1 Problem 1
2 Problem 2
3 Problem 3
4 Problem 4
5 Problem 5
6 Problem 6
7 Problem 7
8 Problem 8
9 Problem 9
10 Problem 10
11 Problem 11
12 Problem 12
13 Problem 13
14 Problem 14
15 Problem 15
16 See also

Problem 1

Square $\displaystyle S_{1}$ is $1\times 1.$ For $i\ge 1,$ the lengths of the sides of square $\displaystyle S_{i+1}$ are half the lengths of the sides of square $\displaystyle S_{i},$ two adjacent sides of square $\displaystyle S_{i}$ are perpendicular bisectors of two adjacent sides of square $\displaystyle S_{i+1},$ and the other two sides of square $\displaystyle S_{i+1},$ are the perpendicular bisectors of two adjacent sides of square $\displaystyle S_{i+2}.$ The total area enclosed by at least one of $\displaystyle S_{1}, S_{2}, S_{3}, S_{4}, S_{5}$ can be written in the form $\displaystyle m/n,$ where $\displaystyle m$ and $\displaystyle n$ are relatively prime positive integers. Find $\displaystyle m-n.$

Problem 2

Find the last three digits of the product of the positive roots of $\sqrt{1995}x^{\log_{1995}x}=x^2.$

Problem 3

Starting at $\displaystyle (0,0),$ an object moves in the coordinate plane via a sequence of steps, each of length one. Each step is left, right, up, or down, all four equally likely. Let $\displaystyle p$ be the probability that the object reaches $\displaystyle (2,2)$ in six or fewer steps. Given that $\displaystyle p$ can be written in the form $\displaystyle m/n,$ where $\displaystyle m$ and $\displaystyle n$ are relatively prime positive integers, find $\displaystyle m+n.$

Problem 4

Circles of radius $\displaystyle 3$ and $\displaystyle 6$ are externally tangent to each other and are internally tangent to a circle of radius $\displaystyle 9$ . The circle of radius $\displaystyle 9$ has a chord that is a common external tangent of the other two circles. Find the square of the length of this chord.

Problem 5

For certain real values of $\displaystyle a, b, c,$ and $\displaystyle d_{},$ the equation $\displaystyle x^4+ax^3+bx^2+cx+d=0$ has four non-real roots. The product of two of these roots is $\displaystyle 13+i$ and the sum of the other two roots is $\displaystyle 3+4i,$ where $i=\sqrt{-1}.$ Find $\displaystyle b.$

Problem 6

Let $\displaystyle n=2^{31}3^{19}.$ How many positive integer divisors of $\displaystyle n^2$ are less than $\displaystyle n_{}$ but do not divide $\displaystyle n_{}$ ?

Problem 7

Given that $\displaystyle (1+\sin t)(1+\cos t)=5/4$ and

$(1-\sin t)(1-\cos t)=\frac mn-\sqrt{k},$

where $\displaystyle k, m,$ and $\displaystyle n_{}$ are positive integers with $\displaystyle m_{}$ and $\displaystyle n_{}$ relatively prime, find $\displaystyle k+m+n.$

Problem 8

For how many ordered pairs of positive integers $\displaystyle (x,y),$ with $\displaystyle y<x\le 100,$ are both $\displaystyle \frac xy$ and $\displaystyle \frac{x+1}{y+1}$ integers?

Problem 9

Triangle $\displaystyle ABC$ is isosceles, with $\displaystyle AB=AC$ and altitude $\displaystyle AM=11.$ Suppose that there is a point $\displaystyle D$ on $\displaystyle \overline{AM}$ with $\displaystyle AD=10$ and $\displaystyle \angle BDC=3\angle BAC.$ Then the perimeter of $\displaystyle \triangle ABC$ may be written in the form $\displaystyle a+\sqrt{b},$ where $\displaystyle a$ and $\displaystyle b$ are integers. Find $\displaystyle a+b.$

Problem 10

What is the largest positive integer that is not the sum of a positive integral multiple of 42 and a positive composite integer?

Problem 11

A right rectangular prism $\displaystyle P_{}$ (i.e., a rectangular parallelpiped) has sides of integral length $\displaystyle a, b, c,$ with $\displaystyle a\le b\le c.$ A plane parallel to one of the faces of $\displaystyle P_{}$ cuts $\displaystyle P_{}$ into two prisms, one of which is similar to $\displaystyle P_{},$ and both of which have nonzero volume. Given that $\displaystyle b=1995,$ for how many ordered triples $\displaystyle (a, b, c)$ does such a plane exist?

Problem 12

Pyramid $\displaystyle OABCD$ has square base $\displaystyle ABCD,$ congruent edges $\displaystyle \overline{OA}, \overline{OB}, \overline{OC},$ and $\displaystyle \overline{OD},$ and $\displaystyle \angle AOB=45^\circ.$ Let $\displaystyle \theta$ be the measure of the dihedral angle formed by faces $\displaystyle OAB$ and $\displaystyle OBC.$ Given that $\displaystyle \cos \theta=m+\sqrt{n},$ where $\displaystyle m_{}$ and $\displaystyle n_{}$ are integers, find $\displaystyle m+n.$

Problem 13

Let $\displaystyle f(n)$ be the integer closest to $\displaystyle \sqrt[4]{n}.$ Find $\displaystyle \sum_{k=1}^{1995}\frac 1{f(k)}.$

Problem 14

In a circle of radius 42, two chords of length 78 intersect at a point whose distance from the center is 18. The two chords divide the interior of the circle into four regions. Two of these regions are bordered by segments of unequal lenghts, and the area of either of them can be expressed uniquley in the form $\displaystyle m\pi-n\sqrt{d},$ where $\displaystyle m, n,$ and $\displaystyle d_{}$ are positive integers and $\displaystyle d_{}$ is not divisible by the square of any prime number. Find $\displaystyle m+n+d.$

Problem 15

Let $\displaystyle p_{}$ be the probability that, in the process of repeatedly flipping a fair coin, one will encounter a run of 5 heads before one encounters a run of 2 tails. Given that $\displaystyle p_{}$ can be written in the form $\displaystyle m/n$ where $\displaystyle m_{}$ and $\displaystyle n_{}$ are relatively prime positive integers, find $\displaystyle m+n$ .

See also

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Looking for a challenging geometry text? Preparing for MATHCOUNTS or the AMC exams? Check out Art of Problem Solving's Introduction to Geometry by Richard Rusczyk.

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