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

Solution

Problem 2

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

Solution

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 $\displaystyle m+n.$

Solution

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.

Solution

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 $\displaystyle b.$

Solution

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 $\displaystyle n_{}$?

Solution

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 $\displaystyle k+m+n.$

Solution

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?

Solution

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

Solution

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?

Solution

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?

Solution

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 $\displaystyle m+n.$

Solution

Problem 13

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

Solution

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 $\displaystyle m+n+d.$

Solution

Problem 15

Solution

See also

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