Difference between revisions of "1996 AIME Problems"

Latest revision as of 16:49, 21 December 2018

1996 AIME (Answer Key) Printable version \| AoPS Contest Collections • PDF
Instructions This is a 15-question, 3-hour examination. All answers are integers ranging from $000$ to $999$ , inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers. No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, calculators and computers are not permitted.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

Problem 1

In a magic square, the sum of the three entries in any row, column, or diagonal is the same value. The figure shows four of the entries of a magic square. Find $x$ .

Solution

Problem 2

For each real number $x$ , let $\lfloor x \rfloor$ denote the greatest integer that does not exceed $x$ . For how many positive integers $n$ is it true that $n<1000$ and that $\lfloor \log_{2} n \rfloor$ is a positive even integer?

Solution

Problem 3

Find the smallest positive integer $n$ for which the expansion of $(xy-3x+7y-21)^n$ , after like terms have been collected, has at least 1996 terms.

Solution

Problem 4

A wooden cube, whose edges are one centimeter long, rests on a horizontal surface. Illuminated by a point source of light that is $x$ centimeters directly above an upper vertex, the cube casts a shadow on the horizontal surface. The area of a shadow, which does not include the area beneath the cube is 48 square centimeters. Find the greatest integer that does not exceed $1000x$ .

Solution

Problem 5

Suppose that the roots of $x^3+3x^2+4x-11=0$ are $a$ , $b$ , and $c$ , and that the roots of $x^3+rx^2+sx+t=0$ are $a+b$ , $b+c$ , and $c+a$ . Find $t$ .

Solution

Problem 6

In a five-team tournament, each team plays one game with every other team. Each team has a $50\%$ chance of winning any game it plays. (There are no ties.) Let $\dfrac{m}{n}$ be the probability that the tournament will produce neither an undefeated team nor a winless team, where $m$ and $n$ are relatively prime integers. Find $m+n$ .

Solution

Problem 7

Two squares of a $7\times 7$ checkerboard are painted yellow, and the rest are painted green. Two color schemes are equivalent if one can be obtained from the other by applying a rotation in the plane board. How many inequivalent color schemes are possible?

Solution

Problem 8

The harmonic mean of two positive integers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers $(x,y)$ with $x<y$ is the harmonic mean of $x$ and $y$ equal to $6^{20}$ ?

Solution

Problem 9

A bored student walks down a hall that contains a row of closed lockers, numbered 1 to 1024. He opens the locker numbered 1, and then alternates between skipping and opening each locker thereafter. When he reaches the end of the hall, the student turns around and starts back. He opens the first closed locker he encounters, and then alternates between skipping and opening each closed locker thereafter. The student continues wandering back and forth in this manner until every locker is open. What is the number of the last locker he opens?

Solution

Problem 10

Find the smallest positive integer solution to $\tan{19x^{\circ}}=\dfrac{\cos{96^{\circ}}+\sin{96^{\circ}}}{\cos{96^{\circ}}-\sin{96^{\circ}}}$ .

Solution

Problem 11

Let $\mathrm {P}$ be the product of the roots of $z^6+z^4+z^3+z^2+1=0$ that have a positive imaginary part, and suppose that $\mathrm {P}=r(\cos{\theta^{\circ}}+i\sin{\theta^{\circ}})$ , where $0<r$ and $0\leq \theta <360$ . Find $\theta$ .

Solution

Problem 12

For each permutation $a_1,a_2,a_3,\cdots,a_{10}$ of the integers $1,2,3,\cdots,10$ , form the sum

$|a_1-a_2|+|a_3-a_4|+|a_5-a_6|+|a_7-a_8|+|a_9-a_{10}|$ .

The average value of all such sums can be written in the form $\dfrac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Find $p+q$ .

Solution

Problem 13

In triangle $ABC$ , $AB=\sqrt{30}$ , $AC=\sqrt{6}$ , and $BC=\sqrt{15}$ . There is a point $D$ for which $\overline{AD}$ bisects $\overline{BC}$ , and $\angle ADB$ is a right angle. The ratio

$\dfrac{\text{Area}(\triangle ADB)}{\text{Area}(\triangle ABC)}$

can be written in the form $\dfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ .

Solution

Problem 14

A $150\times 324\times 375$ rectangular solid is made by gluing together $1\times 1\times 1$ cubes. An internal diagonal of this solid passes through the interiors of how many of the $1\times 1\times 1$ cubes?

Solution

Problem 15

In parallelogram $ABCD,$ let $O$ be the intersection of diagonals $\overline{AC}$ and $\overline{BD}$ . Angles $CAB$ and $DBC$ are each twice as large as angle $DBA,$ and angle $ACB$ is $r$ times as large as angle $AOB$ . Find the greatest integer that does not exceed $1000r$ .

Solution

@@ Line 1: / Line 1: @@
+{{AIME Problems|year=1996}}
 == Problem 1 ==
+In a magic square, the sum of the three entries in any row, column, or diagonal is the same value. The figure shows four of the entries of a magic square. Find <math>x</math>.
+<center>[[Image:AIME_1996_Problem_01.png]]</center>
 [[1996 AIME Problems/Problem 1|Solution]]
 == Problem 2 ==
-For each real number <math>x</math>, let <math>\lfloor x \rfloor</math> denote the greatest integer that does not exceed x. For how man positive integers <math>n</math> is it true that <math>n<1000</math> and that <math>\lfloor \log_{2} n \rfloor</math> is a positive even integer?
+For each real number <math>x</math>, let <math>\lfloor x \rfloor</math> denote the greatest integer that does not exceed <math>x</math>. For how many positive integers <math>n</math> is it true that <math>n<1000</math> and that <math>\lfloor \log_{2} n \rfloor</math> is a positive even integer?
 [[1996 AIME Problems/Problem 2|Solution]]
@@ Line 24: / Line 29: @@
 == Problem 6 ==
-In a five-team tournament, each team plays one game with every other team. Each team has a <math>50%</math> chance of winning any game it plays. (There are no ties.) Let <math>\dfrac{m}{n}</math> be the probability that the tournament will product neither an undefeated team nor a winless team, where <math>m</math> and <math>n</math> are relatively prime integers. Find <math>m+n</math>.
+In a five-team tournament, each team plays one game with every other team. Each team has a <math>50\%</math> chance of winning any game it plays. (There are no ties.) Let <math>\dfrac{m}{n}</math> be the probability that the tournament will produce neither an undefeated team nor a winless team, where <math>m</math> and <math>n</math> are relatively prime integers. Find <math>m+n</math>.
 [[1996 AIME Problems/Problem 6|Solution]]
 == Problem 7 ==
+Two squares of a <math>7\times 7</math> checkerboard are painted yellow, and the rest are painted green. Two color schemes are equivalent if one can be obtained from the other by applying a rotation in the plane board. How many inequivalent color schemes are possible?
 [[1996 AIME Problems/Problem 7|Solution]]
 == Problem 8 ==
+The harmonic mean of two positive integers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers <math>(x,y)</math> with <math>x<y</math> is the harmonic mean of <math>x</math> and <math>y</math> equal to <math>6^{20}</math>?
 [[1996 AIME Problems/Problem 8|Solution]]
 == Problem 9 ==
+A bored student walks down a hall that contains a row of closed lockers, numbered 1 to 1024. He opens the locker numbered 1, and then alternates between skipping and opening each locker thereafter. When he reaches the end of the hall, the student turns around and starts back. He opens the first closed locker he encounters, and then alternates between skipping and opening each closed locker thereafter. The student continues wandering back and forth in this manner until every locker is open. What is the number of the last locker he opens?
 [[1996 AIME Problems/Problem 9|Solution]]
 == Problem 10 ==
+Find the smallest positive integer solution to <math>\tan{19x^{\circ}}=\dfrac{\cos{96^{\circ}}+\sin{96^{\circ}}}{\cos{96^{\circ}}-\sin{96^{\circ}}}</math>.
 [[1996 AIME Problems/Problem 10|Solution]]
 == Problem 11 ==
+Let <math>\mathrm {P}</math> be the product of the roots of <math>z^6+z^4+z^3+z^2+1=0</math> that have a positive imaginary part, and suppose that <math>\mathrm {P}=r(\cos{\theta^{\circ}}+i\sin{\theta^{\circ}})</math>, where <math>0<r</math> and <math>0\leq \theta <360</math>. Find <math>\theta</math>.
 [[1996 AIME Problems/Problem 11|Solution]]
 == Problem 12 ==
+For each permutation <math>a_1,a_2,a_3,\cdots,a_{10}</math> of the integers <math>1,2,3,\cdots,10</math>, form the sum
+<math>|a_1-a_2|+|a_3-a_4|+|a_5-a_6|+|a_7-a_8|+|a_9-a_{10}|</math>.
+The average value of all such sums can be written in the form <math>\dfrac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>.
 [[1996 AIME Problems/Problem 12|Solution]]
 == Problem 13 ==
+In triangle <math>ABC</math>, <math>AB=\sqrt{30}</math>, <math>AC=\sqrt{6}</math>, and <math>BC=\sqrt{15}</math>. There is a point <math>D</math> for which <math>\overline{AD}</math> bisects <math>\overline{BC}</math>, and <math>\angle ADB</math> is a right angle. The ratio
+<cmath>\dfrac{\text{Area}(\triangle ADB)}{\text{Area}(\triangle ABC)}</cmath>
+can be written in the form <math>\dfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>.
 [[1996 AIME Problems/Problem 13|Solution]]
@@ Line 63: / Line 82: @@
 == Problem 15 ==
+In parallelogram <math>ABCD,</math> let <math>O</math> be the intersection of diagonals <math>\overline{AC}</math> and <math>\overline{BD}</math>. Angles <math>CAB</math> and <math>DBC</math> are each twice as large as angle <math>DBA,</math> and angle <math>ACB</math> is <math>r</math> times as large as angle <math>AOB</math>. Find the greatest integer that does not exceed <math>1000r</math>.
 [[1996 AIME Problems/Problem 15|Solution]]
 == See also ==
+{{AIME box|year=1996|before=[[1995 AIME Problems]]|after=[[1997 AIME Problems]]}}
 * [[American Invitational Mathematics Examination]]
 * [[AIME Problems and Solutions]]
 * [[Mathematics competition resources]]
+{{MAA Notice}}

1996 AIME (Problems • Answer Key • Resources)
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Difference between revisions of "1996 AIME Problems"

Latest revision as of 16:49, 21 December 2018

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

See also