Difference between revisions of "1996 AIME Problems"
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== Problem 10 == | == 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]] | [[1996 AIME Problems/Problem 10|Solution]] | ||
Revision as of 15:05, 24 September 2007
Contents
Problem 1
Problem 2
For each real number 
, let 
denote the greatest integer that does not exceed x. For how man positive integers 
is it true that 
and that 
is a positive even integer?
Problem 3
Find the smallest positive integer for which the expansion of 
, after like terms have been collected, has at least 1996 terms.
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 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 
.
Problem 5
Suppose that the roots of 
are 
, 
, and 
, and that the roots of 
are 
, 
, and 
. Find 
.
Problem 6
In a five-team tournament, each team plays one game with every other team. Each team has a 
chance of winning any game it plays. (There are no ties.) Let 
be the probability that the tournament will product neither an undefeated team nor a winless team, where 
and are relatively prime integers. Find 
.
Problem 7
Two squares of a 
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?
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 
with 
is the harmonic mean of and 
equal to 
?
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?
Problem 10
Find the smallest positive integer solution to 
.
Problem 11
Problem 12
Problem 13
Problem 14
A 
rectangular solid is made by gluing together 
cubes. An internal diagonal of this solid passes through the interiors of how many of the cubes?
