1983 AIME Problems

Problem 1

Let $x$ , $y$ , and $z$ all exceed $1$ , and let $w$ be a positive number such that $\log_xw=24$ , $\log_y w = 40$ , and $\log_{xyz}w=12$ . Find $\log_zw$ .

Solution

Problem 2

Let $f(x)=|x-p|+|x-15|+|x-p-15|$ , where $p \leq x \leq 15$ . Determine the minimum value taken by $f(x)$ by $x$ in the interval $0 < p<15$ .

Solution

Problem 3

What is the product of the real roots of the equation $x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}$ ?

Solution

Problem 4

A machine shop cutting tool is in the shape of a notched circle, as shown. The radius of the circle is 50 cm, the length of $AB$ is 6 cm, and that of $BC$ is 2 cm. The angle $ABC$ is a right angle. Find the square of the distance (in centimeters) from $B$ to the center of the circle.

$[asy] size(150); defaultpen(linewidth(0.6)+fontsize(11)); real r=10; pair O=(0,0),A=r*dir(45),B=(A.x,A.y-r),C; path P=circle(O,r); C=intersectionpoint(B--(B.x+r,B.y),P); draw(P); draw(C--B--O--A--B); dot(O); dot(A); dot(B); dot(C); label("$O$",O,SW); label("$A$",A,NE); label("$B$",B,S); label("$C$",C,SE); [/asy]$

Solution

Problem 5

Suppose that the sum of the squares of two complex numbers $x$ and $y$ is $7$ and the sum of the cubes is $10$ . What is the largest real value of $x + y$ can have?

Solution

Problem 6

Let $a_n$ equal $6^{n}+8^{n}$ . Determine the remainder upon dividing $a_ {83}$ by $49$ .

Solution

Problem 7

Twenty five of King Arthur's knights are seated at their customary round table. Three of them are chosen - all choices being equally likely - and are sent off to slay a troublesome dragon. Let $P$ be the probability that at least two of the three had been sitting next to each other. If $P$ is written as a fraction in lowest terms, what is the sum of the numerator and the denominator?

Solution

Problem 8

What is the largest 2-digit prime factor of the integer ${200\choose 100}$ ?

Solution

Problem 9

Find the minimum value of $\frac{9x^2\sin^2 x + 4}{x\sin x}$ for $0 < x < \pi$ .

Solution

Problem 10

The numbers $1447$ , $1005$ , and $1231$ have something in common. Each is a four-digit number beginning with $1$ that has exactly two identical digits. How many such numbers are there?

Solution

Problem 11

The solid shown has a square base of side length $s$ . The upper edge is parallel to the base and has length $2s$ . All edges have length $s$ . Given that $s=6\sqrt{2}$ , what is the volume of the solid?

size(170);import three; pathpen = black+linewidth(0.65); pointpen = black;
currentprojection = perspective(30,-20,10);real s = 6 * 2^.5;
triple A=(0,0,0),B=(s,0,0),C=(s,s,0),D=(0,s,0),E=(-s/2,s/2,6),F=(3*s/2,s/2,6);
D(A--B--C--D--A--E--D); D(B--F--C); D(E--F);
MP("A",A);MP("B",B);MP("C",C);MP("D",D);MP("E",E,N);MP("F",F,N);
 (Error making remote request. Unknown error_msg)

Solution

Problem 12

The length of diameter $AB$ is a two digit integer. Reversing the digits gives the length of a perpendicular chord $CD$ . The distance from their intersection point $H$ to the center $O$ is a positive rational number. Determine the length of $AB$ .

$[asy]pointpen=black; pathpen=black+linewidth(0.65); pair O=(0,0),A=(-65/2,0),B=(65/2,0); pair H=(-((65/2)^2-28^2)^.5,0),C=(H.x,28),D=(H.x,-28); D(CP(O,A));D(MP("A",A,W)--MP("B",B,E));D(MP("C",C,N)--MP("D",D)); dot(MP("H",H,SE));dot(MP("O",O,SE));[/asy]$

Solution

Problem 13

For $\{1, 2, 3, \ldots, n\}$ and each of its non-empty subsets, an alternating sum is defined as follows. Arrange the number in the subset in decreasing order and then, beginning with the largest, alternately add and subtract succesive numbers. For example, the alternating sum for $\{1, 2, 3, 6,9\}$ is $9-6+3-2+1=6$ and for $\{5\}$ it is simply $5$ . Find the sum of all such alternating sums for $n=7$ .

Solution

Problem 14

In the adjoining figure, two circles with radii $6$ and $8$ are drawn with their centers $12$ units apart. At $P$ , one of the points of intersection, a line is drawn in sich a way that the chords $QP$ and $PR$ have equal length. ( $P$ is the midpoint of $QR$ ) Find the square of the length of $QP$ .

Solution

Problem 15

The adjoining figure shows two intersecting chords in a circle, with $B$ on minor arc $AD$ . Suppose that the radius of the circle is $5$ , that $BC=6$ , and that $AD$ is bisected by $BC$ . Suppose further that $AD$ is the only chord starting at $A$ which is bisected by $BC$ . It follows that the sine of the minor arc $AB$ is a rational number. If this fraction is expressed as a fraction $\frac{m}{n}$ in lowest terms, what is the product $mn$ ?