Difference between revisions of "1980 AHSME Problems"
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[[1980 AHSME Problems/Problem 30|Solution]] | [[1980 AHSME Problems/Problem 30|Solution]] | ||
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== See also == | == See also == |
Revision as of 10:27, 7 August 2012
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
The largest whole number such that seven times the number is less than 100 is
Problem 2
The degree of as a polynomial in
is
Problem 3
If the ratio of to
is
, what is the ratio of
to
?
Problem 4
In the adjoining figure, CDE is an equilateral triangle and ABCD and DEFG are squares. The measure of is
[asy]
defaultpen(linewidth(0.7)+fontsize(10));
pair D=origin, C=D+dir(240), E=D+dir(300), F=E+dir(30), G=D+dir(30), A=D+dir(150), B=C+dir(150);
draw(E--D--G--F--E--C--D--A--B--C);
pair point=(0,0.5);
label("", A, dir(point--A));
label("
", B, dir(point--B));
label("
", C, dir(point--C));
label("
", D, dir(-15));
label("
", E, dir(point--E));
label("
", F, dir(point--F));
label("
", G, dir(point--G));[/asy]
Problem 5
If and
are perpendicular diameters of circle
,
in
, and
, then the length of
divided by the length of
is
[asy]
defaultpen(linewidth(0.7)+fontsize(10));
pair A=(-1,0), B=(1,0), C=(0,1), D=(0,-1), Q=origin, P=(-0.5,0);
draw(P--C--D^^A--B^^Circle(Q,1));
label("", A, W);
label("
", B, E);
label("
", C, N);
label("
", D, S);
label("
", P, S);
label("
", Q, SE);
label("
", P+0.0.5*dir(30), dir(30));[/asy]
Problem 6
A positive number satisfies the inequality
if and only if
$\text{(A)} \ x > \frac{1}{4} \qquad \text{(B)} \ x > 2 \qquad \text{(C)} \x > 4 \qquad \text{(D)} \ x < \frac{1}{4}\qquad \text{(E)} \x < 4$ (Error compiling LaTeX. Unknown error_msg)
Problem 7
Sides and
of convex polygon
have lengths 3,4,12, and 13, respectively, and
is a right angle. The area of the quadrilateral is
[asy]
defaultpen(linewidth(0.7)+fontsize(10));
real r=degrees((12,5)), s=degrees((3,4));
pair D=origin, A=(13,0), C=D+12*dir(r), B=A+3*dir(180-(90-r+s));
draw(A--B--C--D--cycle);
markscalefactor=0.05;
draw(rightanglemark(A,B,C));
pair point=incenter(A,C,D);
label("", A, dir(point--A));
label("
", B, dir(point--B));
label("
", C, dir(point--C));
label("
", D, dir(point--D));
label("
", A--B, dir(A--B)*dir(-90));
label("
", B--C, dir(B--C)*dir(-90));
label("
", C--D, dir(C--D)*dir(-90));
label("
", D--A, dir(D--A)*dir(-90));[/asy]
Problem 8
How many pairs of non-zero real numbers satisfy the equation
Problem 9
A man walks miles due west, turns
to his left and walks 3 miles in the new direction. If he finishes a a point
from his starting point, then
is
Problem 10
The number of teeth in three meshed gears ,
, and
are
,
, and
, respectively. (The teeth on all gears are the same size and regularly spaced.) The angular speeds, in revolutions per minutes of
,
, and
are in the proportion
Problem 11
If the sum of the first 10 terms and the sum of the first 100 terms of a given arithmetic progression are 100 and 10, respectively, then the sum of first 110 terms is:
Problem 12
The equations of and
are
and
, respectively. Suppose
makes twice as large of an angle with the horizontal (measured counterclockwise from the positive x-axis ) as does
, and that
has 4 times the slope of
. If
is not horizontal, then
is
Problem 13
A bug (of negligible size) starts at the origin on the coordinate plane. First, it moves one unit right to . Then it makes a
counterclockwise and travels
a unit to
. If it continues in this fashion, each time making a
degree turn counterclockwise and traveling half as far as the previous move, to which of the following points will it come closest?
Problem 14
If the function is defined by
satisfies
for all rea numbers
except
, then
is
Problem 15
A store prices an item in dollars and cents so that when 4% sales tax is added, no rounding is necessary because the result is exactly dollars where
is a positive integer. The smallest value of
is
Problem 16
Four of the eight vertices of a cube are the vertices of a regular tetrahedron. Find the ratio of the surface area of the cube to the surface area of the tetrahedron.
Problem 17
Given that , for how many integers
is
an integer?
Problem 18
If ,
,
, and
, then
equals
Problem 19
Problem 20
Problem 21
Problem 22
Problem 23
Problem 24
Problem 25
Problem 26
Problem 27
Problem 28
Problem 29
Problem 30