Difference between revisions of "1983 AIME Problems"
(Asy now compiles and gives sensible output) |
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path P=circle(O,r); | path P=circle(O,r); | ||
C=intersectionpoint(B--(B.x+r,B.y),P); | C=intersectionpoint(B--(B.x+r,B.y),P); | ||
− | draw( | + | draw(Arc(origin, r, dir(A), dir(C))); |
− | draw(C | + | draw(C--A--B);dot(A); dot(B); dot(C); |
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label("$O$",O,SW); | label("$O$",O,SW); | ||
label("$A$",A,NE); | label("$A$",A,NE); |
Revision as of 17:25, 21 August 2011
1983 AIME (Answer Key) | AoPS Contest Collections | ||
Instructions
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1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
Problem 1
Let ,, and all exceed , and let be a positive number such that , , and . Find .
Problem 2
Let , where . Determine the minimum value taken by by in the interval .
Problem 3
What is the product of the real roots of the equation ?
Problem 4
A machine shop cutting tool is in the shape of a notched circle, as shown. The radius of the circle is cm, the length of is 6 cm, and that of is 2 cm. The angle is a right angle. Find the square of the distance (in centimeters) from to the center of the circle.
size(150); defaultpen(linewidth(0.65)+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(Arc(origin, r, dir(A), dir(C))); draw(C--A--B);dot(A); dot(B); dot(C); label("$O$",O,SW); label("$A$",A,NE); label("$B$",B,S); label("$C$",C,SE); (Error compiling LaTeX. draw(Arc(origin, r, dir(A), dir(C))); ^ 4fbda1241be8e01aaf9d2f949b1b56bf27d7f53b.asy: 8.9: no matching function 'Arc(pair, real, pair, pair)')
Problem 5
Suppose that the sum of the squares of two complex numbers and is and the sum of the cubes is . What is the largest real value that can have?
Problem 6
Let equal . Determine the remainder upon dividing by .
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 be the probability that at least two of the three had been sitting next to each other. If is written as a fraction in lowest terms, what is the sum of the numerator and the denominator?
Problem 8
What is the largest 2-digit prime factor of the integer ?
Problem 9
Find the minimum value of for .
Problem 10
The numbers , , and have something in common. Each is a four-digit number beginning with that has exactly two identical digits. How many such numbers are there?
Problem 11
The solid shown has a square base of side length . The upper edge is parallel to the base and has length . All edges have length . Given that , what is the volume of the solid?
Problem 12
The length of diameter is a two digit integer. Reversing the digits gives the length of a perpendicular chord . The distance from their intersection point to the center is a positive rational number. Determine the length of .
Problem 13
For 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 is and for it is simply . Find the sum of all such alternating sums for .
Problem 14
In the adjoining figure, two circles with radii and are drawn with their centers units apart. At , one of the points of intersection, a line is drawn in such a way that the chords and have equal length. ( is the midpoint of ) Find the square of the length of .
Problem 15
The adjoining figure shows two intersecting chords in a circle, with on minor arc . Suppose that the radius of the circle is , that , and that is bisected by . Suppose further that is the only chord starting at which is bisected by . It follows that the sine of the minor arc is a rational number. If this fraction is expressed as a fraction in lowest terms, what is the product ?