Difference between revisions of "1992 AHSME Problems"

Revision as of 02:34, 28 September 2014

Problem 1

If $3(4x+\pi)=P$ then $6(8x+10\pi)=$

$\text{(A) } 2P\quad \text{(B) } 3P\quad \text{(C) } 6P\quad \text{(D) } 8P\quad \text{(E) } 18P$

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?

$\text{(A) } 40\%\quad \text{(B) } 48\%\quad \text{(C) } 52\%\quad \text{(D) } 60\%\quad \text{(E) } 80\%$

Solution

Problem 3

If $m>0$ and the points $(m,3)$ and $(1,m)$ lie on a line with slope $m$ , then $m=$

$\text{(A) } 1\quad \text{(B) } \sqrt{2}\quad \text{(C) } \sqrt{3}\quad \text{(D) } 2\quad \text{(E) } \sqrt{5}$

Solution

Problem 4

If $a,b$ and $c$ are positive integers and $a$ and $b$ are odd, then $3^a+(b-1)^2c$ is

$\text{(A) odd for all choices of c} \quad \text{(B) even for all choices of c} \quad\\ \text{(C) odd if c is even; even if c is odd} \quad\\ \text{(D) odd if c is odd; even if c is even} \quad\\ \text{(E) odd if c is not a multiple of 3;evn if c is a multiple of 3}$

Solution

Problem 5

$6^6+6^6+6^6+6^6+6^6+6^6=$

$\text{(A) } 6^6 \quad \text{(B) } 6^7\quad \text{(C) } 36^6\quad \text{(D) } 6^{36}\quad \text{(E) } 36^{36}$

Solution

Problem 6

If $x>y>0$ , then $\frac{x^y y^x}{y^y x^x}=$

$\text{(A) } (x-y)^{y/x}\quad \text{(B) } \left(\frac{x}{y}\right)^{x-y}\quad \text{(C) } 1\quad \text{(D) } \left(\frac{x}{y}\right)^{y-x}\quad \text{(E) } (x-y)^{x/y}$

Solution

Problem 7

The ratio of $w$ to $x$ is $4:3$ , of $y$ to $z$ is $3:2$ and of $z$ to $x$ is $1:6$ . What is the ratio of $w$ to $y$ ?

$\text{(A) } 1:3\quad \text{(B) } 16:3\quad \text{(C) } 20:3\quad \text{(D) } 27:4\quad \text{(E) } 12:1$

Solution

Problem 8

$[asy] draw((-10,-10)--(-10,10)--(10,10)--(10,-10)--cycle,dashed+linewidth(.75)); draw((-7,-7)--(-7,7)--(7,7)--(7,-7)--cycle,dashed+linewidth(.75)); draw((-10,-10)--(10,10),dashed+linewidth(.75)); draw((-10,10)--(10,-10),dashed+linewidth(.75)); fill((10,10)--(10,9)--(9,9)--(9,10)--cycle,black); fill((9,9)--(9,8)--(8,8)--(8,9)--cycle,black); fill((8,8)--(8,7)--(7,7)--(7,8)--cycle,black); fill((-10,-10)--(-10,-9)--(-9,-9)--(-9,-10)--cycle,black); fill((-9,-9)--(-9,-8)--(-8,-8)--(-8,-9)--cycle,black); fill((-8,-8)--(-8,-7)--(-7,-7)--(-7,-8)--cycle,black); fill((10,-10)--(10,-9)--(9,-9)--(9,-10)--cycle,black); fill((9,-9)--(9,-8)--(8,-8)--(8,-9)--cycle,black); fill((8,-8)--(8,-7)--(7,-7)--(7,-8)--cycle,black); fill((-10,10)--(-10,9)--(-9,9)--(-9,10)--cycle,black); fill((-9,9)--(-9,8)--(-8,8)--(-8,9)--cycle,black); fill((-8,8)--(-8,7)--(-7,7)--(-7,8)--cycle,black); [/asy]$

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

$\text{(A) } 121\quad \text{(B) } 625\quad \text{(C) } 676\quad \text{(D) } 2500\quad \text{(E) } 2601$

Solution

Problem 9

$[asy] draw((-7,0)--(7,0),black+linewidth(.75)); draw((-3*sqrt(3),0)--(-2*sqrt(3),3)--(-sqrt(3),0)--(0,3)--(sqrt(3),0)--(2*sqrt(3),3)--(3*sqrt(3),0),black+linewidth(.75)); draw((-2*sqrt(3),0)--(-1*sqrt(3),3)--(0,0)--(sqrt(3),3)--(2*sqrt(3),0),black+linewidth(.75)); [/asy]$

Five equilateral triangles, each with side $2\sqrt{3}$ , 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

$\text{(A) 10} \quad \text{(B) } 12\quad \text{(C) } 15\quad \text{(D) } 10\sqrt{3}\quad \text{(E) } 12\sqrt{3}$

Solution

Problem 10

The number of positive integers $k$ for which the equation $kx-12=3k$ has an integer solution for $x$ is

$\text{(A) } 3\quad \text{(B) } 4\quad \text{(C) } 5\quad \text{(D) } 6\quad \text{(E) } 7$

Solution

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 $1:3$ . If $\overline{AC}$ is a diameter of the larger circle, $\overline{BC}$ is a chord of the larger circle that is tangent to the smaller circle, and $AB=12$ , then the radius of the larger circle is

$\text{(A) } 13\quad \text{(B) } 18\quad \text{(C) } 21\quad \text{(D) } 24\quad \text{(E) } 26$

Solution

Problem 12

Let $y=mx+b$ be the image when the line $x-3y+11=0$ is reflected across the $x$ -axis. The value of $m+b$ is

$\text{(A) -6} \quad \text{(B) } -5\quad \text{(C) } -4\quad \text{(D) } -3\quad \text{(E) } -2$

Solution

Problem 13

How many pairs of positive integers (a,b) with $a+b\le 100$ satisfy the equation

$\frac{a+b^{-1}}{a^{-1}+b}=13?$

$\text{(A) } 1\quad \text{(B) } 5\quad \text{(C) } 7\quad \text{(D) } 9\quad \text{(E) } 13$

Solution

Problem 14

Which of the following equations have the same graph?

$I.\quad y=x-2 \qquad II.\quad y=\frac{x^2-4}{x+2}\qquad III.\quad (x+2)y=x^2-4$

$\text{(A) I and II only} \quad \text{(B) I and III only} \quad \text{(C) II and III only} \quad \text{(D) I,II,and III} \quad \\ \text{(E) None. All of the equations have different graphs}$

Solution

Problem 15

Let $I=\sqrt{-1}$ . Define a sequence of complex numbers by

$z_1=0,\quad z_{n+1}=z_{n}^2+i \text{ for } n\ge1.$ In the complex plane, how far from the origin is $z_{111}$ ?

$\text{(A) } 1\quad \text{(B) } \sqrt{2}\quad \text{(C) } \sqrt{3}\quad \text{(D) } \sqrt{110}\quad \text{(E) } \sqrt{2^{55}}$

Solution

Problem 16

If $\frac{y}{x-z}=\frac{x+y}{z}=\frac{x}{y}$ for three positive numbers $x,y$ and $z$ , all different, then $\frac{x}{y}=$

$\text{(A) } \frac{1}{2}\quad \text{(B) } \frac{3}{5}\quad \text{(C) } \frac{2}{3}\quad \text{(D) } \frac{5}{3}\quad \text{(E) } 2$

Solution

Problem 17

The 2-digit integers from 19 to 92 are written consecutively to form the integer $N=192021\cdots9192$ . Suppose that $3^k$ is the highest power of 3 that is a factor of $N$ . What is $k$ ?

$\text{(A) } 0\quad \text{(B) } 1\quad \text{(C) } 2\quad \text{(D) } 3\quad \text{(E) more than } 3$

Solution

Problem 18

The increasing sequence of positive integers $a_1,a_2,a_3,\cdots$ has the property that

$a_{n+2}=a_n+a_{n+1} \text{ for all } n\ge 1.$

If $a_7=120$ , then $a_8$ is

$\text{(A) } 128\quad \text{(B) } 168\quad \text{(C) } 193\quad \text{(D) } 194\quad \text{(E) } 210$

Solution

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?

$\text{(A) } 75\%\quad \text{(B) } 78\%\quad \text{(C) } 81\%\quad \text{(D) } 84\%\quad \text{(E) } 87\%$

Solution

Problem 20

$[asy] draw((1,0)--(2*cos(pi/8),2*sin(pi/8))--(cos(pi/4),sin(pi/4))--(2*cos(3*pi/8),2*sin(3*pi/8))--(cos(pi/2),sin(pi/2))--(2*cos(5*pi/8),2*sin(5*pi/8))--(cos(3*pi/4),sin(3*pi/4))--(2*cos(7*pi/8),2*sin(7*pi/8))--(-1,0),black+linewidth(.75)); MP("A_1",(2*cos(5*pi/8),2*sin(5*pi/8)),N);MP("A_2",(2*cos(3*pi/8),2*sin(3*pi/8)),N);MP("A_3",(2*cos(1*pi/8),2*sin(1*pi/8)),N); MP("A_n",(2*cos(7*pi/8),2*sin(7*pi/8)),N); MP("B_1",(cos(4*pi/8),sin(4*pi/8)),S);MP("B_2",(cos(2*pi/8),sin(2*pi/8)),S);MP("B_n",(cos(6*pi/8),sin(6*pi/8)),S); [/asy]$ Part of an "n-pointed regular star" is shown. It is a simple closed polygon in which all $2n$ edges are congruent, angles $A_1,A_2,\cdots,A_n$ are congruent, and angles $B_1,B_2,\cdots,B_n$ are congruent. If the acute angle at $A_1$ is $10^\circ$ less than the acute angle at $B_1$ , then $n=$

$\text{(A) } 12\quad \text{(B) } 18\quad \text{(C) } 24\quad \text{(D) } 36\quad \text{(E) } 60$

Solution

Problem 21

For a finite sequence $A=(a_1,a_2,...,a_n)$ of numbers, the Cesáro sum of A is defined to be $\frac{S_1+\cdots+S_n}{n}$ , where $S_k=a_1+\cdots+a_k$ and $1\leq k\leq n$ . If the Cesáro sum of the 99-term sequence $(a_1,...,a_{99})$ is 1000, what is the Cesáro sum of the 100-term sequence $(1,a_1,...,a_{99})$ ?

$\text{(A) } 991\quad \text{(B) } 999\quad \text{(C) } 1000\quad \text{(D) } 1001\quad \text{(E) } 1009$

Solution

Problem 22

Ten points are selected on the positive $x$ -axis, $X^+$ , and five points are selected on the positive $y$ -axis, $Y^+$ . The fifty segments connecting the ten points on $X^+$ to the five points on $Y^+$ 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?

$\text{(A) } 250\quad \text{(B) } 450\quad \text{(C) } 500\quad \text{(D) } 1250\quad \text{(E) } 2500$

Solution

Problem 23

Let $S$ be a subset of $\{1,2,3,...,50\}$ such that no pair of distinct elements in $S$ has a sum divisible by $7$ . What is the maximum number of elements in $S$ ?

$\text{(A) } 6\quad \text{(B) } 7\quad \text{(C) } 14\quad \text{(D) } 22\quad \text{(E) } 23$

Solution

Problem 24

Let $ABCD$ be a parallelogram of area $10$ with $AB=3$ and $BC=5$ . Locate $E,F$ and $G$ on segments $\overline{AB},\overline{BC}$ and $\overline{AD}$ , respectively, with $AE=BF=AG=2$ . Let the line through $G$ parallel to $\overline{EF}$ intersect $\overline{CD}$ at $H$ . The area of quadrilateral $EFGH$ is

$\text{(A) } 4\quad \text{(B) } 4.5\quad \text{(C) } 5\quad \text{(D) } 5.5\quad \text{(E) } 6$

Solution

Problem 25

In $\triangle{ABC}$ , $\angle{ABC=120^\circ,AB=3$ (Error compiling LaTeX. Unknown error_msg) and $BC=4$ . If perpendiculars constructed to $\overline{AB}$ at $A$ and to $\overline{BC}$ at $C$ meet at $D$ , then $CD=$

$\text{(A) } 3\quad \text{(B) } \frac{8}{\sqrt{3}}\quad \text{(C) } 5\quad \text{(D) } \frac{11}{2}\quad \text{(E) } \frac{10}{\sqrt{3}}$

Solution

Problem 26

$[asy] fill((1,0)--arc((1,0),2,180,225)--cycle,grey); fill((-1,0)--arc((-1,0),2,315,360)--cycle,grey); fill((0,-1)--arc((0,-1),2-sqrt(2),225,315)--cycle,grey); fill((0,0)--arc((0,0),1,180,360)--cycle,white); draw((1,0)--arc((1,0),2,180,225)--(1,0),black+linewidth(1)); draw((-1,0)--arc((-1,0),2,315,360)--(-1,0),black+linewidth(1)); draw((0,0)--arc((0,0),1,180,360)--(0,0),black+linewidth(1)); draw(arc((0,-1),2-sqrt(2),225,315),black+linewidth(1)); draw((0,0)--(0,-1),black+linewidth(1)); MP("C",(0,0),N);MP("A",(-1,0),N);MP("B",(1,0),N); MP("D",(0,-.8),NW);MP("E",(1-sqrt(2),-sqrt(2)),SW);MP("F",(-1+sqrt(2),-sqrt(2)),SE); [/asy]$

Semicircle $\widehat{AB}$ has center $C$ and radius $1$ . Point $D$ is on $\widehat{AB}$ and $\overline{CD}\perp\overline{AB}$ . Extend $\overline{BD}$ and $\overline{AD}$ to $E$ and $F$ , respectively, so that circular arcs $\widehat{AE}$ and $\widehat{BF}$ have $B$ and $A$ as their respective centers. Circular arc $\widehat{EF}$ has center $D$ . The area of the shaded "smile" $AEFBDA$ , is

$\text{(A) } (2-\sqrt{2})\pi\quad \text{(B) } 2\pi-\pi \sqrt{2}-1\quad \text{(C) } (1-\frac{\sqrt{2}}{2})\pi\quad\\ \text{(D) } \frac{5\pi}{2}-\pi\sqrt{2}-1\quad \text{(E) } (3-2\sqrt{2})\pi$

Solution

Problem 27

A circle of radius $r$ has chords $\overline{AB}$ of length $10$ and $\overline{CD}$ of length 7. When $\overline{AB}$ and $\overline{CD}$ are extended through $B$ and $C$ , respectively, they intersect at $P$ , which is outside of the circle. If $\angle{APD}=60^\circ$ and $BP=8$ , then $r^2=$

$\text{(A) } 70\quad \text{(B) } 71\quad \text{(C) } 72\quad \text{(D) } 73\quad \text{(E) } 74$ Solution

Problem 28

Let $i=\sqrt{-1}$ . The product of the real parts of the roots of $z^2-z=5-5i$ is

$\text{(A) } -25\quad \text{(B) } -6\quad \text{(C) } -5\quad \text{(D) } \frac{1}{4}\quad \text{(E) } 25$

Solution

Problem 29

An "unfair" coin has a $2/3$ probability of turning up heads. If this coin is tossed $50$ times, what is the probability that the total number of heads is even?

$\text{(A) } 25(\frac{2}{3})^{50}\quad \text{(B) } \frac{1}{2}(1-\frac{1}{3^{50}})\quad \text{(C) } \frac{1}{2}\quad \text{(D) } \frac{1}{2}(1+\frac{1}{3^{50}})\quad \text{(E) } \frac{2}{3}$

Solution

Problem 30

Let $ABCD$ be an isosceles trapezoid with bases $AB=92$ and $CD=19$ . Suppose $AD=BC=x$ and a circle with center on $\overline{AB}$ is tangent to segments $\overline{AD}$ and $\overline{BC}$ . If $m$ is the smallest possible value of $x$ , then $m^2$ =

$\text{(A) } 1369\quad \text{(B) } 1679\quad \text{(C) } 1748\quad \text{(D) } 2109\quad \text{(E) } 8825$

Solution

@@ Line 43: / Line 43: @@
 \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>
 [[1992 AHSME Problems/Problem 4|Solution]]
@@ Line 54: / Line 55: @@
 \text{(D) } 6^{36}\quad
 \text{(E) } 36^{36}</math>
 [[1992 AHSME Problems/Problem 5|Solution]]
@@ Line 66: / Line 68: @@
 \text{(D) } \left(\frac{x}{y}\right)^{y-x}\quad
 \text{(E) } (x-y)^{x/y}</math>
 [[1992 AHSME Problems/Problem 6|Solution]]
@@ Line 143: / Line 146: @@
 == 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]]
 == 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]]
@@ Line 154: / Line 177: @@
 == 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]]
 == 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]]
@@ Line 164: / Line 204: @@
 == 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]]
 == 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]]
 == 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]]
 == 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]]
@@ Line 192: / Line 266: @@
 \text{(D) } 84\%\quad
 \text{(E) } 87\%</math>
 [[1992 AHSME Problems/Problem 19|Solution]]
@@ Line 209: / Line 284: @@
 \text{(D) } 36\quad
 \text{(E) } 60</math>
 [[1992 AHSME Problems/Problem 20|Solution]]
@@ Line 223: / Line 299: @@
 \text{(D) } 1001\quad
 \text{(E) } 1009</math>
 [[1992 AHSME Problems/Problem 21|Solution]]
@@ Line 234: / Line 311: @@
 \text{(D) } 1250\quad
 \text{(E) } 2500</math>
 [[1992 AHSME Problems/Problem 22|Solution]]
@@ Line 246: / Line 324: @@
 \text{(D) } 22\quad
 \text{(E) } 23</math>
 [[1992 AHSME Problems/Problem 23|Solution]]
@@ Line 257: / Line 336: @@
 \text{(D) } 5.5\quad
 \text{(E) } 6</math>
 [[1992 AHSME Problems/Problem 24|Solution]]
@@ Line 341: / Line 421: @@
 \text{(D) } 2109\quad
 \text{(E) } 8825</math>
 [[1992 AHSME Problems/Problem 30|Solution]]

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Difference between revisions of "1992 AHSME Problems"

Revision as of 02:34, 28 September 2014

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30

See also