Difference between revisions of "1980 AHSME Problems"
Line 174: | Line 174: | ||
[[1980 AHSME Problems/Problem 18|Solution]] | [[1980 AHSME Problems/Problem 18|Solution]] | ||
− | == Problem 19 == | + | ==Problem 19== |
− | <math> | + | Let <math>C_1, C_2</math> and <math>C_3</math> be three parallel chords of a circle on the same side of the center. |
+ | The distance between <math>C_1</math> and <math>C_2</math> is the same as the distance between <math>C_2</math> and <math>C_3</math>. | ||
+ | The lengths of the chords are <math>20, 16</math>, and <math>8</math>. The radius of the circle is | ||
+ | <math>\text{(A)} \ 12 \qquad | ||
+ | \text{(B)} \ 4\sqrt{7} \qquad | ||
+ | \text{(C)} \ \frac{5\sqrt{65}}{3} \qquad | ||
+ | \text{(D)}\ \frac{5\sqrt{22}}{2}\qquad | ||
+ | \text{(E)}\ \text{not uniquely determined} </math> | ||
+ | |||
[[1980 AHSME Problems/Problem 19|Solution]] | [[1980 AHSME Problems/Problem 19|Solution]] | ||
− | + | ||
− | == Problem 20 == | + | ==Problem 20== |
− | + | ||
− | <math> \ | + | A box contains <math>2</math> pennies, <math>4</math> nickels, and <math>6</math> dimes. Six coins are drawn without replacement, |
− | + | with each coin having an equal probability of being chosen. What is the probability that the value of coins drawn is at least <math>50</math> cents? | |
+ | <math> | ||
+ | \text{(A)} \ \frac{37}{924} \qquad | ||
+ | \text{(B)} \ \frac{91}{924} \qquad | ||
+ | \text{(C)} \ \frac{127}{924} \qquad | ||
+ | \text{(D)}\ \frac{132}{924}\qquad | ||
+ | \text{(E)}\ \text{none of these} </math> | ||
+ | |||
[[1980 AHSME Problems/Problem 20|Solution]] | [[1980 AHSME Problems/Problem 20|Solution]] | ||
− | == Problem 21 == | + | ==Problem 21== |
+ | |||
+ | <asy> | ||
+ | defaultpen(linewidth(0.7)+fontsize(10)); | ||
+ | pair B=origin, C=(15,3), D=(5,1), A=7*dir(72)*dir(B--C), E=midpoint(A--C), F=intersectionpoint(A--D, B--E); | ||
+ | draw(E--B--A--C--B^^A--D); | ||
+ | label("$A$", A, dir(D--A)); | ||
+ | label("$B$", B, dir(E--B)); | ||
+ | label("$C$", C, dir(0)); | ||
+ | label("$D$", D, SE); | ||
+ | label("$E$", E, N); | ||
+ | label("$F$", F, dir(80));</asy> | ||
− | <math> \ | + | In triangle <math>ABC</math>, <math>\measuredangle CBA=72^\circ</math>, <math>E</math> is the midpoint of side <math>AC</math>, |
+ | and <math>D</math> is a point on side <math>BC</math> such that <math>2BD=DC</math>; <math>AD</math> and <math>BE</math> intersect at <math>F</math>. | ||
+ | The ratio of the area of triangle <math>BDF</math> to the area of quadrilateral <math>FDCE</math> is | ||
+ | <math>\text{(A)} \ \frac 15 \qquad | ||
+ | \text{(B)} \ \frac 14 \qquad | ||
+ | \text{(C)} \ \frac 13 \qquad | ||
+ | \text{(D)}\ \frac{2}{5}\qquad | ||
+ | \text{(E)}\ \text{none of these}</math> | ||
+ | |||
[[1980 AHSME Problems/Problem 21|Solution]] | [[1980 AHSME Problems/Problem 21|Solution]] | ||
− | == Problem 22 == | + | ==Problem 22== |
− | |||
− | |||
+ | <math>For each real number x, let f(x) be the minimum of the numbers 4x+1, x+2, and -2x+4. Then the maximum value of f(x) is | ||
+ | \text{(A)} \ \frac 13 \qquad | ||
+ | \text{(B)} \ \frac 12 \qquad | ||
+ | \text{(C)} \ \frac 23 \qquad | ||
+ | \text{(D)} \ \frac 52 \qquad | ||
+ | \text{(E)}\ \frac{8}{3} </math> | ||
+ | |||
[[1980 AHSME Problems/Problem 22|Solution]] | [[1980 AHSME Problems/Problem 22|Solution]] | ||
− | == Problem 23 == | + | ==Problem 23== |
− | + | ||
− | + | Line segments drawn from the vertex opposite the hypotenuse of a right triangle to the points trisecting the hypotenuse have lengths \sin x and \cos x, where x is a real number such that 0<x<\frac{\pi}2. The length of the hypotenuse is | |
+ | <math>\text{(A)} \ \frac 43 \qquad | ||
+ | \text{(B)} \ \frac 32 \qquad | ||
+ | \text{(C)} \ \frac{3\sqrt{5}}{5} \qquad | ||
+ | \text{(D)}\ \frac{2\sqrt{5}}{3}\qquad | ||
+ | \text{(E)}\ \text{not uniquely determined}</math> | ||
+ | |||
[[1980 AHSME Problems/Problem 23|Solution]] | [[1980 AHSME Problems/Problem 23|Solution]] | ||
− | == Problem 24 == | + | ==Problem 24== |
− | + | For some real number r, the polynomial 8x^3-4x^2-42x+45 is divisible by (x-r)^2. Which of the following numbers is closest to r? | |
− | [[1980 AHSME Problems/Problem 24|Solution | + | <math>\text{(A)} \ 1.22 \qquad |
+ | \text{(B)} \ 1.32 \qquad | ||
+ | \text{(C)} \ 1.42 \qquad | ||
+ | \text{(D)} \ 1.52 \qquad | ||
+ | \text{(E)} \ 1.62 </math> | ||
+ | |||
+ | [[1980 AHSME Problems/Problem 24|Solution | ||
+ | |||
+ | ==Problem 25== | ||
− | + | In the non-decreasing sequence of odd integers <math>\{a_1,a_2,a_3,\ldots \}=\{1,3,3,3,5,5,5,5,5,\ldots \}</math> each odd positive integer <math>k</math> | |
− | + | appears <math>k</math> times. It is a fact that there are integers <math>b, c</math>, and <math>d</math> such that for all positive integers <math>n</math>, | |
− | <math> \ | + | <math>a_n=b\lfloor \sqrt{n+c} \rfloor +d</math>, |
+ | where <math>\lfloor x \rfloor</math> denotes the largest integer not exceeding <math>x</math>. The sum <math>b+c+d</math> equals | ||
+ | <math>\text{(A)} \ 0 \qquad | ||
+ | \text{(B)} \ 1 \qquad | ||
+ | \text{(C)} \ 2 \qquad | ||
+ | \text{(D)} \ 3 \qquad | ||
+ | \text{(E)} \ 4 </math> | ||
+ | |||
[[1980 AHSME Problems/Problem 25|Solution]] | [[1980 AHSME Problems/Problem 25|Solution]] | ||
− | == Problem 26 == | + | ==Problem 26== |
− | <math> | + | Four balls of radius <math>1</math> are mutually tangent, three resting on the floor and the fourth resting on the others. |
+ | A tetrahedron, each of whose edges have length <math>s</math>, is circumscribed around the balls. Then <math>s</math> equals | ||
+ | <math>\text{(A)} \ 4\sqrt 2 \qquad | ||
+ | \text{(B)} \ 4\sqrt 3 \qquad | ||
+ | \text{(C)} \ 2\sqrt 6 \qquad | ||
+ | \text{(D)}\ 1+2\sqrt 6\qquad | ||
+ | \text{(E)}\ 2+2\sqrt 6</math> | ||
+ | |||
[[1980 AHSME Problems/Problem 26|Solution]] | [[1980 AHSME Problems/Problem 26|Solution]] | ||
− | == Problem 27 == | + | ==Problem 27== |
− | <math> \ | + | The sum <math>\sqrt[3] {5+2\sqrt{13}}+\sqrt[3]{5-2\sqrt{13}}</math> equals |
+ | <math>\text{(A)} \ \frac 32 \qquad | ||
+ | \text{(B)} \ \frac{\sqrt[3]{65}}{4} \qquad | ||
+ | \text{(C)} \ \frac{1+\sqrt[6]{13}}{2} \qquad | ||
+ | \text{(D)}\ \sqrt[3]{2}\qquad | ||
+ | \text{(E)}\ \text{none of these} </math> | ||
+ | |||
[[1980 AHSME Problems/Problem 27|Solution]] | [[1980 AHSME Problems/Problem 27|Solution]] | ||
− | == Problem 28 == | + | ==Problem 28== |
+ | The polynomial <math>x^{2n}+1+(x+1)^{2n}</math> is not divisible by <math>x^2+x+1</math> if <math>n</math> equals | ||
− | <math> \ | + | <math>\text{(A)} \ 17 \qquad |
+ | \text{(B)} \ 20 \qquad | ||
+ | \text{(C)} \ 21 \qquad | ||
+ | \text{(D)} \ 64 \qquad | ||
+ | \text{(E)} \ 65 </math> | ||
+ | |||
+ | [[1980 AHSME Problems/Problem 28|Solution]] | ||
− | + | ==Problem 29== | |
− | + | How many ordered triples (x,y,z) of integers satisfy the system of equations below? | |
− | < | + | <cmath>\begin{array}{l} x^2-3xy+2yz-z^2=31 \\ -x^2+6yz+2z^2=44 \\ x^2+xy+8z^2=100\\ \end{array} </cmath> |
+ | <math>\text{(A)} \ 0 \qquad | ||
+ | \text{(B)} \ 1 \qquad | ||
+ | \text{(C)} \ 2 \qquad | ||
+ | \text{(D)}\ \text{a finite number greater than 2}\qquad | ||
+ | \text{(E)}\ \text{infinitely many} </math> | ||
+ | |||
[[1980 AHSME Problems/Problem 29|Solution]] | [[1980 AHSME Problems/Problem 29|Solution]] | ||
− | == Problem 30 == | + | ==Problem 30== |
+ | |||
+ | A six digit number (base 10) is squarish if it satisfies the following conditions: | ||
+ | |||
+ | (i) none of its digits are zero; | ||
+ | (ii) it is a perfect square; and | ||
+ | (iii) the first of two digits, the middle two digits and the last two digits of the number are all perfect squares when considered as two digit numbers. | ||
+ | |||
+ | How many squarish numbers are there? | ||
− | <math> \ | + | <math>\text{(A)} \ 0 \qquad |
+ | \text{(B)} \ 2 \qquad | ||
+ | \text{(C)} \ 3 \qquad | ||
+ | \text{(D)} \ 8 \qquad | ||
+ | \text{(E)} \ 9 </math> | ||
[[1980 AHSME Problems/Problem 30|Solution]] | [[1980 AHSME Problems/Problem 30|Solution]] | ||
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* [[AHSME]] | * [[AHSME]] | ||
* [[AHSME Problems and Solutions]] | * [[AHSME Problems and Solutions]] | ||
+ | * [[AMC 12 Problems and Solutions]] | ||
* [[Mathematics competition resources]] | * [[Mathematics competition resources]] | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 23:29, 2 October 2014
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
Problem 5
If and
are perpendicular diameters of circle
,
in
, and
, then the length of
divided by the length of
is
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
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
Let and
be three parallel chords of a circle on the same side of the center.
The distance between
and
is the same as the distance between
and
.
The lengths of the chords are
, and
. The radius of the circle is
Problem 20
A box contains pennies,
nickels, and
dimes. Six coins are drawn without replacement,
with each coin having an equal probability of being chosen. What is the probability that the value of coins drawn is at least
cents?
Problem 21
In triangle ,
,
is the midpoint of side
,
and
is a point on side
such that
;
and
intersect at
.
The ratio of the area of triangle
to the area of quadrilateral
is
Problem 22
Problem 23
Line segments drawn from the vertex opposite the hypotenuse of a right triangle to the points trisecting the hypotenuse have lengths \sin x and \cos x, where x is a real number such that 0<x<\frac{\pi}2. The length of the hypotenuse is
Problem 24
For some real number r, the polynomial 8x^3-4x^2-42x+45 is divisible by (x-r)^2. Which of the following numbers is closest to r?
[[1980 AHSME Problems/Problem 24|Solution
Problem 25
In the non-decreasing sequence of odd integers each odd positive integer
appears
times. It is a fact that there are integers
, and
such that for all positive integers
,
,
where
denotes the largest integer not exceeding
. The sum
equals
Problem 26
Four balls of radius are mutually tangent, three resting on the floor and the fourth resting on the others.
A tetrahedron, each of whose edges have length
, is circumscribed around the balls. Then
equals
Problem 27
The sum equals
Problem 28
The polynomial is not divisible by
if
equals
Problem 29
How many ordered triples (x,y,z) of integers satisfy the system of equations below?
Problem 30
A six digit number (base 10) is squarish if it satisfies the following conditions:
(i) none of its digits are zero; (ii) it is a perfect square; and (iii) the first of two digits, the middle two digits and the last two digits of the number are all perfect squares when considered as two digit numbers.
How many squarish numbers are there?
See also
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.