Difference between revisions of "1983 AHSME Problems"
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label("$P$", P, S); | label("$P$", P, S); | ||
label("$40^\circ$", C+(-0.15,0), NW); | label("$40^\circ$", C+(-0.15,0), NW); | ||
− | label("$10^\circ$", B+(0,0.05), | + | label("$10^\circ$", B+(0,0.05), dir(200); |
− | label("$10^\circ$", A+(0.05,0.02), | + | label("$10^\circ$", A+(0.05,0.02), dir(330));</asy> |
<math>\textbf{(A)}\ 10^{\circ}\qquad | <math>\textbf{(A)}\ 10^{\circ}\qquad | ||
Line 443: | Line 443: | ||
\textbf{(D)}\ 25^{\circ}\qquad | \textbf{(D)}\ 25^{\circ}\qquad | ||
\textbf{(E)}\ 30^{\circ} </math> | \textbf{(E)}\ 30^{\circ} </math> | ||
− | |||
== See also == | == See also == |
Revision as of 09:57, 9 July 2018
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 Problem 16
- 17 Problem 17
- 18 Problem 18
- 19 Problem 19
- 20 Problem 20
- 21 Problem 21
- 22 Problem 22
- 23 Problem 23
- 24 Problem 24
- 25 Problem 25
- 26 Problem 26
- 27 Problem 27
- 28 Problem 28
- 29 Problem 29
- 30 Problem 30
- 31 See also
Problem 1
If and , then equals
Problem 2
Point is outside circle on the plane. At most how many points on are from P?
Problem 3
Three primes , and satisfy and . Then equals
Problem 4
Position such that and are parallel, as are sides and , and sides and . Each side has length of and it is given that . The area of the figure is
Problem 5
Triangle has a right angle at . If , then is
Problem 6
When and are multiplied, the product is a polynomial of degree.
Problem 7
Alice sells an item at $ less than the list price and receives of her selling price as her commission. Bob sells the same item at $ less than the list price and receives of his selling price as his commission. If they both get the same commission, then the list price in dollars is
Problem 8
Let . Then for is
Problem 9
In a certain population the ratio of the number of women to the number of men is to . If the average (arithmetic mean) age of the women is and the average age of the men is , then the average age of the population is
Problem 10
Segment is both a diameter of a circle of radius and a side of an equilateral triangle . The circle also intersects and at points and , respectively. The length of is
Problem 11
Simplify .
Problem 12
If , then equals
Problem 13
If and , and none of these quantities is zero, then equals:
Problem 14
The units digit of is
Problem 15
Three balls marked , and , are placed in an urn. One ball is drawn, its number is recorded, then the ball is returned to the urn. This process is repeated and then repeated once more, and each ball is equally likely to be drawn on each occasion. If the sum of the numbers recorded is , what is the probability that the ball numbered was drawn all three times?
Problem 16
Let , where the digits are obtained by writing the integers through in order. The rd digit to the right of the decimal point is
Problem 17
The diagram to the right shows several numbers in the complex plane. The circle is the unit circle centered at the origin. One of these numbers is the reciprocal of . Which one?
Problem 18
Let be a polynomial function such that, for all real , . For all real is
Problem 19
Point is on side of triangle . If , then the length of is
Problem 20
If and are the roots of , and and are the roots of , then is necessarily
Problem 21
Find the smallest positive number from the numbers below
Problem 22
Consider the two functions and , where the variable and the constants and are real numbers. Each such pair of the constants a and b may be considered as a point in an -plane. Let S be the set of such points for which the graphs of and do NOT intersect (in the - plane.). The area of is
Problem 23
In the adjoining figure the five circles are tangent to one another consecutively and to the lines and ( is the line that is above the circles and is the line that goes under the circles). If the radius of the largest circle is and that of the smallest one is , then the radius of the middle circle is
Problem 24
How many non-congruent right triangles are there such that the perimeter in and the area in are numerically equal?
Problem 25
If and , then
Problem 26
The probability that event occurs is ; the probability that event B occurs is . Let be the probability that both and occur. The smallest interval necessarily containing is the interval
Problem 27
A large sphere is on a horizontal field on a sunny day. At a certain time the shadow of the sphere reaches out a distance of m from the point where the sphere touches the ground. At the same instant a meter stick (held vertically with one end on the ground) casts a shadow of length m. What is the radius of the sphere in meters? (Assume the sun's rays are parallel and the meter stick is a line segment.)
Problem 28
Triangle in the figure has area . Points and , all distinct from and , are on sides and respectively, and . If triangle and quadrilateral have equal areas, then that area is
Problem 29
A point lies in the same plane as a given square of side . Let the vertices of the square, taken counterclockwise, be and . Also, let the distances from to and , respectively, be and . What is the greatest distance that can be from if ?
Problem 30
Distinct points and are on a semicircle with diameter and center . The point is on and . If , then equals
defaultpen(linewidth(0.7)+fontsize(10)); pair C=origin, N=dir(0), B=dir(20), A=dir(135), M=dir(180), P=(3/7)*dir(C--N); draw(M--N^^C--A--P--B--C^^Arc(origin,1,0,180)); markscalefactor=0.03; draw(anglemark(C,A,P)); draw(anglemark(C,B,P)); pair point=C; label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, S); label("$M$", M, dir(point--M)); label("$N$", N, dir(point--N)); label("$P$", P, S); label("$40^\circ$", C+(-0.15,0), NW); label("$10^\circ$", B+(0,0.05), dir(200); label("$10^\circ$", A+(0.05,0.02), dir(330)); (Error compiling LaTeX. label("$10^\circ$", B+(0,0.05), dir(200); ^ 06b83feca9382d82ffdac2d759ae462661f81de3.asy: 17.41: syntax error error: could not load module '06b83feca9382d82ffdac2d759ae462661f81de3.asy')
See also
1983 AHSME (Problems • Answer Key • Resources) | ||
Preceded by 1982 AHSME |
Followed by 1984 AHSME | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.