1996 AIME Problems

Part of AIME Problems

Subcontests

(13)

1

Magic Square

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.

[asy] size(100);defaultpen(linewidth(0.7)); int i; for(i=0; i<4; i=i+1) { draw((0,2*i)--(6,2*i)^^(2*i,0)--(2*i,6)); } label("

x

", (1,5)); label("

1

", (1,3)); label("

19

", (3,5)); label("

96

", (5,5));[/asy]

1

Parallelogram

In parallelogram

ABCD,

O

be the intersection of diagonals

\overline{AC}

\overline{BD}.

CAB

DBC

are each twice as large as angle

DBA,

ACB

r

times as large as angle

AOB.

Find the greatest integer that does not exceed

1000r.

1

Area Ratio

ABC, AB=\sqrt{30}, AC=\sqrt{6},

BC=\sqrt{15}.

There is a point

D

\overline{AD}

\overline{BC}

\angle ADB

is a right angle. The ratio

\frac{\text{Area}(\triangle ADB)}{\text{Area}(\triangle ABC)}

can be written in the form

m/n,

m

n

are relatively prime positive integers. Find

m+n.

1

Sum of Absolute Values

For each permutation

a_1, a_2, a_3, \ldots,a_{10}

of the integers

1,2,3,\ldots,10,

form the sum |a_1 \minus{} a_2| \plus{} |a_3 \minus{} a_4| \plus{} |a_5 \minus{} a_6| \plus{} |a_7 \minus{} a_8| \plus{} |a_9 \minus{} a_{10}|. The average value of all such sums can be written in the form

p/q,

p

q

are relatively prime positive integers. Find p \plus{} q.

1

Complex #'s Fun

P

be the product of the roots of

z^6+z^4+z^3+z^2+1=0

that have positive imaginary part, and suppose that

P=r(\cos \theta^\circ+i\sin \theta^\circ),

0<r

0\le \theta <360.

\theta.

1

Trig Equation

Find the smallest positive integer solution to

\tan 19x^\circ=\frac{\cos 96^\circ+\sin 96^\circ}{\cos 96^\circ-\sin 96^\circ}.

1

Locker Mayhem

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 closed 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?

1

Harmonic Mean

The harmonic mean of two positive numbers is the reciprocal of the arithmetic mean of their reciprocals. For how many ordered pairs of positive integers

(x,y)

x<y

is the harmonic mean of

x

y

6^{20}.

1

Cubic Roots

Suppose that the roots of

x^3+3x^2+4x-11=0

a, b,

c,

and that the roots of

x^3+rx^2+sx+t=0

a+b, b+c,

c+a.

t.

1

Cube Shadow

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 the shadow, which does not inclued the area beneath the cube is 48 square centimeters. Find the greatest integer that does not exceed

1000x.

1

Log Floor

For each real number

x,

\lfloor x\rfloor

denote the greatest integer that does not exceed

x.

For how many positive integers

n

is it true that

n<1000

\lfloor \log_2 n\rfloor

is a positive even integer.

1

Aime 1996 #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.

1

AIME 1996

Two of the 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 of the board. How many inequivalent color schemes are possible?