1995 AIME Problems

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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 $\displaystyle m-n.$

AIME 1995 Problem 1.png


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 $\displaystyle a+b.$

AIME 1995 Problem 9.png


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


Problem 15


See also

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