Difference between revisions of "1992 AHSME Problems"
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\text{(D) odd if c is odd; even if c is even} \quad\\ | \text{(D) odd if c is odd; even if c is even} \quad\\ | ||
\text{(E) odd if c is not a multiple of 3;evn if c is a multiple of 3} </math> | \text{(E) odd if c is not a multiple of 3;evn if c is a multiple of 3} </math> | ||
+ | |||
[[1992 AHSME Problems/Problem 4|Solution]] | [[1992 AHSME Problems/Problem 4|Solution]] | ||
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\text{(D) } 6^{36}\quad | \text{(D) } 6^{36}\quad | ||
\text{(E) } 36^{36}</math> | \text{(E) } 36^{36}</math> | ||
+ | |||
[[1992 AHSME Problems/Problem 5|Solution]] | [[1992 AHSME Problems/Problem 5|Solution]] | ||
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\text{(D) } \left(\frac{x}{y}\right)^{y-x}\quad | \text{(D) } \left(\frac{x}{y}\right)^{y-x}\quad | ||
\text{(E) } (x-y)^{x/y}</math> | \text{(E) } (x-y)^{x/y}</math> | ||
+ | |||
[[1992 AHSME Problems/Problem 6|Solution]] | [[1992 AHSME Problems/Problem 6|Solution]] | ||
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== Problem 11 == | == Problem 11 == | ||
+ | <asy> | ||
+ | draw(circle((0,0),18),black+linewidth(.75)); | ||
+ | draw(circle((0,0),6),black+linewidth(.75)); | ||
+ | draw((-18,0)--(18,0)--(-14,8*sqrt(2))--cycle,black+linewidth(.75)); | ||
+ | dot((-18,0));dot((18,0));dot((-14,8*sqrt(2))); | ||
+ | MP("A",(-18,0),W);MP("C",(18,0),E);MP("B",(-14,8*sqrt(2)),W); | ||
+ | </asy> | ||
+ | The ratio of the radii of two concentric circles is <math>1:3</math>. If <math>\overline{AC}</math> is a diameter of the larger circle, <math>\overline{BC}</math> is a chord of the larger circle that is tangent to the smaller circle, and <math>AB=12</math>, then the radius of the larger circle is | ||
+ | |||
+ | <math>\text{(A) } 13\quad | ||
+ | \text{(B) } 18\quad | ||
+ | \text{(C) } 21\quad | ||
+ | \text{(D) } 24\quad | ||
+ | \text{(E) } 26</math> | ||
[[1992 AHSME Problems/Problem 11|Solution]] | [[1992 AHSME Problems/Problem 11|Solution]] | ||
== Problem 12 == | == Problem 12 == | ||
+ | Let <math>y=mx+b</math> be the image when the line <math>x-3y+11=0</math> is reflected across the <math>x</math>-axis. The value of <math>m+b</math> is | ||
+ | <math>\text{(A) -6} \quad | ||
+ | \text{(B) } -5\quad | ||
+ | \text{(C) } -4\quad | ||
+ | \text{(D) } -3\quad | ||
+ | \text{(E) } -2</math> | ||
[[1992 AHSME Problems/Problem 12|Solution]] | [[1992 AHSME Problems/Problem 12|Solution]] | ||
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== Problem 13 == | == Problem 13 == | ||
+ | How many pairs of positive integers (a,b) with <math>a+b\le 100</math> satisfy the equation | ||
+ | |||
+ | <cmath>\frac{a+b^{-1}}{a^{-1}+b}=13?</cmath> | ||
+ | |||
+ | <math>\text{(A) } 1\quad | ||
+ | \text{(B) } 5\quad | ||
+ | \text{(C) } 7\quad | ||
+ | \text{(D) } 9\quad | ||
+ | \text{(E) } 13</math> | ||
[[1992 AHSME Problems/Problem 13|Solution]] | [[1992 AHSME Problems/Problem 13|Solution]] | ||
== Problem 14 == | == Problem 14 == | ||
+ | Which of the following equations have the same graph? | ||
+ | <math>I.\quad y=x-2 \qquad II.\quad y=\frac{x^2-4}{x+2}\qquad III.\quad (x+2)y=x^2-4</math> | ||
+ | |||
+ | <math>\text{(A) I and II only} \quad | ||
+ | \text{(B) I and III only} \quad | ||
+ | \text{(C) II and III only} \quad | ||
+ | \text{(D) I,II,and III} \quad \\ | ||
+ | \text{(E) None. All of the equations have different graphs} </math> | ||
[[1992 AHSME Problems/Problem 14|Solution]] | [[1992 AHSME Problems/Problem 14|Solution]] | ||
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== Problem 15 == | == Problem 15 == | ||
+ | Let <math>I=\sqrt{-1}</math>. Define a sequence of complex numbers by | ||
+ | |||
+ | <cmath>z_1=0,\quad z_{n+1}=z_{n}^2+i \text{ for } n\ge1.</cmath> | ||
+ | In the complex plane, how far from the origin is <math>z_{111}</math>? | ||
+ | |||
+ | <math>\text{(A) } 1\quad | ||
+ | \text{(B) } \sqrt{2}\quad | ||
+ | \text{(C) } \sqrt{3}\quad | ||
+ | \text{(D) } \sqrt{110}\quad | ||
+ | \text{(E) } \sqrt{2^{55}}</math> | ||
[[1992 AHSME Problems/Problem 15|Solution]] | [[1992 AHSME Problems/Problem 15|Solution]] | ||
== Problem 16 == | == Problem 16 == | ||
+ | If | ||
+ | <cmath>\frac{y}{x-z}=\frac{x+y}{z}=\frac{x}{y}</cmath> | ||
+ | for three positive numbers <math>x,y</math> and <math>z</math>, all different, then <math>\frac{x}{y}=</math> | ||
+ | <math>\text{(A) } \frac{1}{2}\quad | ||
+ | \text{(B) } \frac{3}{5}\quad | ||
+ | \text{(C) } \frac{2}{3}\quad | ||
+ | \text{(D) } \frac{5}{3}\quad | ||
+ | \text{(E) } 2</math> | ||
[[1992 AHSME Problems/Problem 16|Solution]] | [[1992 AHSME Problems/Problem 16|Solution]] | ||
== Problem 17 == | == Problem 17 == | ||
+ | The 2-digit integers from 19 to 92 are written consecutively to form the integer <math>N=192021\cdots9192</math>. Suppose that <math>3^k</math> is the highest power of 3 that is a factor of <math>N</math>. What is <math>k</math>? | ||
+ | <math>\text{(A) } 0\quad | ||
+ | \text{(B) } 1\quad | ||
+ | \text{(C) } 2\quad | ||
+ | \text{(D) } 3\quad | ||
+ | \text{(E) more than } 3</math> | ||
[[1992 AHSME Problems/Problem 17|Solution]] | [[1992 AHSME Problems/Problem 17|Solution]] | ||
== Problem 18 == | == Problem 18 == | ||
+ | The increasing sequence of positive integers <math>a_1,a_2,a_3,\cdots </math> has the property that | ||
+ | <cmath>a_{n+2}=a_n+a_{n+1} \text{ for all } n\ge 1.</cmath> | ||
+ | |||
+ | If <math>a_7=120</math>, then <math>a_8</math> is | ||
+ | |||
+ | <math>\text{(A) } 128\quad | ||
+ | \text{(B) } 168\quad | ||
+ | \text{(C) } 193\quad | ||
+ | \text{(D) } 194\quad | ||
+ | \text{(E) } 210</math> | ||
[[1992 AHSME Problems/Problem 18|Solution]] | [[1992 AHSME Problems/Problem 18|Solution]] | ||
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\text{(D) } 84\%\quad | \text{(D) } 84\%\quad | ||
\text{(E) } 87\%</math> | \text{(E) } 87\%</math> | ||
+ | |||
[[1992 AHSME Problems/Problem 19|Solution]] | [[1992 AHSME Problems/Problem 19|Solution]] | ||
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\text{(D) } 36\quad | \text{(D) } 36\quad | ||
\text{(E) } 60</math> | \text{(E) } 60</math> | ||
+ | |||
[[1992 AHSME Problems/Problem 20|Solution]] | [[1992 AHSME Problems/Problem 20|Solution]] | ||
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\text{(D) } 1001\quad | \text{(D) } 1001\quad | ||
\text{(E) } 1009</math> | \text{(E) } 1009</math> | ||
+ | |||
[[1992 AHSME Problems/Problem 21|Solution]] | [[1992 AHSME Problems/Problem 21|Solution]] | ||
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\text{(D) } 1250\quad | \text{(D) } 1250\quad | ||
\text{(E) } 2500</math> | \text{(E) } 2500</math> | ||
+ | |||
[[1992 AHSME Problems/Problem 22|Solution]] | [[1992 AHSME Problems/Problem 22|Solution]] | ||
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\text{(D) } 22\quad | \text{(D) } 22\quad | ||
\text{(E) } 23</math> | \text{(E) } 23</math> | ||
+ | |||
[[1992 AHSME Problems/Problem 23|Solution]] | [[1992 AHSME Problems/Problem 23|Solution]] | ||
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\text{(D) } 5.5\quad | \text{(D) } 5.5\quad | ||
\text{(E) } 6</math> | \text{(E) } 6</math> | ||
+ | |||
[[1992 AHSME Problems/Problem 24|Solution]] | [[1992 AHSME Problems/Problem 24|Solution]] | ||
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\text{(D) } 2109\quad | \text{(D) } 2109\quad | ||
\text{(E) } 8825</math> | \text{(E) } 8825</math> | ||
+ | |||
[[1992 AHSME Problems/Problem 30|Solution]] | [[1992 AHSME Problems/Problem 30|Solution]] | ||
Revision as of 02:34, 28 September 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
If then
Problem 2
An urn is filled with coins and beads, all of which are either silver or gold. Twenty percent of the objects in the urn are beads. Forty percent of the coins in the urn are silver. What percent of objects in the urn are gold coins?
Problem 3
If and the points
and
lie on a line with slope
, then
Problem 4
If and
are positive integers and
and
are odd, then
is
Problem 5
Problem 6
If , then
Problem 7
The ratio of to
is
, of
to
is
and of
to
is
. What is the ratio of
to
?
Problem 8
A square floor is tiled with congruent square tiles. The tiles on the two diagonals of the floor are black. The rest of the tiles are white. If there are 101 black tiles, then the total number of tiles is
Problem 9
Five equilateral triangles, each with side , are arranged so they are all on the same side of a line containing one side of each vertex. Along this line, the midpoint of the base of one triangle is a vertex of the next. The area of the region of the plane that is covered by the union of the five triangular regions is
Problem 10
The number of positive integers for which the equation
has an integer solution for
is
Problem 11
The ratio of the radii of two concentric circles is . If
is a diameter of the larger circle,
is a chord of the larger circle that is tangent to the smaller circle, and
, then the radius of the larger circle is
Problem 12
Let be the image when the line
is reflected across the
-axis. The value of
is
Problem 13
How many pairs of positive integers (a,b) with satisfy the equation
Problem 14
Which of the following equations have the same graph?
Problem 15
Let . Define a sequence of complex numbers by
In the complex plane, how far from the origin is
?
Problem 16
If
for three positive numbers
and
, all different, then
Problem 17
The 2-digit integers from 19 to 92 are written consecutively to form the integer . Suppose that
is the highest power of 3 that is a factor of
. What is
?
Problem 18
The increasing sequence of positive integers has the property that
If , then
is
Problem 19
For each vertex of a solid cube, consider the tetrahedron determined by the vertex and the midpoints of the three edges that meet at that vertex. The portion of the cube that remains when these eight tetrahedra are cut away is called a cubeoctahedron. The ratio of the volume of the cubeoctahedron to the volume of the original cube is closest to which of these?
Problem 20
Part of an "n-pointed regular star" is shown. It is a simple closed polygon in which all
edges are congruent, angles
are congruent, and angles
are congruent. If the acute angle at
is
less than the acute angle at
, then
Problem 21
For a finite sequence of numbers, the Cesáro sum of A is defined to be
, where
and
. If the Cesáro sum of
the 99-term sequence
is 1000, what is the Cesáro sum of the 100-term sequence
?
Problem 22
Ten points are selected on the positive -axis,
, and five points are selected on the positive
-axis,
. The fifty segments connecting the ten points on
to the five points on
are drawn. What is the maximum possible number of points of intersection of these fifty segments that could lie in the interior of the first quadrant?
Problem 23
Let be a subset of
such that no pair of distinct elements in
has a sum divisible by
. What is the maximum number of elements in
?
Problem 24
Let be a parallelogram of area
with
and
. Locate
and
on segments
and
, respectively, with
. Let the line through
parallel to
intersect
at
. The area of quadrilateral
is
Problem 25
In , $\angle{ABC=120^\circ,AB=3$ (Error compiling LaTeX. Unknown error_msg) and
. If perpendiculars constructed to
at
and to
at
meet at
, then
Problem 26
Semicircle has center
and radius
. Point
is on
and
. Extend
and
to
and
, respectively, so that circular arcs
and
have
and
as their respective centers. Circular arc
has center
. The area of the shaded "smile"
, is
Problem 27
A circle of radius has chords
of length
and
of length 7. When
and
are extended through
and
, respectively, they intersect at
, which is outside of the circle. If
and
, then
Problem 28
Let . The product of the real parts of the roots of
is
Problem 29
An "unfair" coin has a probability of turning up heads. If this coin is tossed
times, what is the probability that the total number of heads is even?
Problem 30
Let be an isosceles trapezoid with bases
and
. Suppose
and a circle with center on
is tangent to segments
and
. If
is the smallest possible value of
, then
=
See also
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
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