Difference between revisions of "User:Temperal/sandbox"
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# Consider a <math>4</math> by <math>5</math> rectangle. Each square is randomly either coloured or not. Express the probability that less than half of the "border" of the rectangle is coloured. | # Consider a <math>4</math> by <math>5</math> rectangle. Each square is randomly either coloured or not. Express the probability that less than half of the "border" of the rectangle is coloured. | ||
− | # The country of Larepmet is undergoing a famine. Each day, <math>\left\ | + | # The country of Larepmet is undergoing a famine. Each day, <math>\left\lfloor\frac{x^3}{x^2+1}\right\rfloor</math> sacks of wheat are lost, where <math>x</math> is the number of sacks previously. If Larepmet starts with <math>100</math> sacks of wheat on day <math>1</math>, on what day will they run out? |
# If <math>a</math> is picked from <math>\{1,2,3\}</math>, <math>b</math> from <math>\{11,12,13\}</math> and <math>c</math> from <math>\{21,22,23\}</math>, find the probability that <math>a+b+c+abc</math> is even as a common fraction. | # If <math>a</math> is picked from <math>\{1,2,3\}</math>, <math>b</math> from <math>\{11,12,13\}</math> and <math>c</math> from <math>\{21,22,23\}</math>, find the probability that <math>a+b+c+abc</math> is even as a common fraction. | ||
# If <math>9</math> people are seated in a circle, and three stand up, find the probability that none of the three who stood are adjacent to each other. | # If <math>9</math> people are seated in a circle, and three stand up, find the probability that none of the three who stood are adjacent to each other. |
Revision as of 15:59, 11 December 2007
Contents
Sprint
- A regular ten-sided polygon has a perimeter of
units and an circumradius of
. If it's cut in half through two of it's vertices and
is the sum of the perimeter of the two new figures, what is
?
- Five rabbits are in a field, two black and three white. If two rabbits are chosen at random, find the probability of the second being black.
- An annoying kid named Aishvar decides to participate in Mathcounts. If he manages to make it onto the team by scoring
% better than the person scoring just below him, and he scored
points on the participation exam, how many points better did he score than the person below him?
- A strange box, which is a prism with isosceles trapezoids as bases, has a height of 3 units. If the volume is
cubic units, and has bases of
and
, find the height of trapezoid bases.
- A die is rolled
times, such that at the
th roll, the die has
faces, each numbered with the natural numbers up to
. Find the probability that one of the rolls is
in terms of
.
- Simplify
.
- Find the number of ways
people can be arranged in a circle of
chairs.
- Find the tenths digit of
.
- Find all the number of real solutions to
.
- What is the vertex of the parabola
?
- If
is randomly chosen from
and
from
, find the probability that
is odd. Express your answer as a common fraction.
- Find the number of natural numbers
that satisfy
.
- Find the tens digit of
.
- Find the number of values of
that give
exactly one solution.
- How many perfect squares less than
are divisible by two?
Target
- A train leaves Omaha at
pm going at
km/h headed for San Fransisco. A train leaves San Fransisco at
pm of the same day at twice that speed, heading for Omaha. If Omaha and San Fransisco are
miles apart, how much farther will the faster train travel than the slower once they meet?
- What is the probability that a point placed on a
triangle is placed on the hypotenuse? Express your answer as a common fraction in simplest radical form.
- If the median of a triangle is
, and its two base angles are congruent, find the other two side's lengths in terms of the base.
- Bob votes randomly in a poll. If the probability that he voted for either licorice or chocolate is
, find the number of options in the poll.
Team
- Consider a
by
rectangle. Each square is randomly either coloured or not. Express the probability that less than half of the "border" of the rectangle is coloured.
- The country of Larepmet is undergoing a famine. Each day,
sacks of wheat are lost, where
is the number of sacks previously. If Larepmet starts with
sacks of wheat on day
, on what day will they run out?
- If
is picked from
,
from
and
from
, find the probability that
is even as a common fraction.
- If
people are seated in a circle, and three stand up, find the probability that none of the three who stood are adjacent to each other.
- Given that at most five people are in a room, and that less than three of them are nine years old, find the probability as a common fraction that more than
of them are nine years old. (If there are no nine-year olds nor any people, consider this to be that 100% of them are nine-year olds)
Countdown
- Evaluate
.
- Evaluate
.
- Find the surface area of a cube with volume
cubic units.
- If a triangle has a base of
units and has two angles of
, find its area.
- If you have a paper with an area of
units squared, find the fraction of it needed to cover a circle with radius
. Express your answer in terms of
as a common fraction.
- If
, and
, what is
?
- How many vertices does a dodecahedron have?
- Find the area of a triangle with sides
,
, and
.
- What is the maximum number of points needed such that not all of them must be coplanar?
- What is the maximum number of points needed such that there is only one plane through them?
- Find the number of perfect squares less than
.
- What is the sum of the first five triangle numbers?
- Let
be the largest integer less than or equal to
. Find the sum of the three smallest solutions to
.
- Let
and
. What is
in terms of
?
- Find the number of ways of rearranging the letters in THEE into distinct four-letter words.
- If
is an operation such that
, find
.
- In how many different ways can a person wiggle exactly two of his fingers (including thumbs)?
- A kid made Blue MOP this year. If he studied
hours a day not including weekends, what is the probability he studied more than
hours if five arbitrary days are picked from his studying schedule?
- Find
.
- Jack randomly throws darts at a
b
square. What's the probability, in terms of
, that he hits within
units of the center?
- If
and
, what is the maximum value for
?
- What is the volume of a cone with height
and radius
? Express in terms of
.
- A triangle has sides of
,
, and
. If the triangle has area
, find
.
- If
for all
, find 4`.
- Find the the number of a selected from
satisfies
.
- The equation for a parabola containing
and
is
. What is
?
- What is the sum of the digits of
?
- Which is greater,
or
?
- Which is less,
, or
?
- A kid wants to make USAMO this year. What is the probability, as an integer, of him doing so, if his score on AIME is
(if above 15,
) randomly chosen from
and the cutoff is
?
Masters
- The set of complex numbers, known as
, consists of the square roots of any negative real number plus a real number. The base complex number, called
, is the square root of negative one. Every complex number can be expressed as
, where
is the square root of
, and
and
are real constants.
i) Find the square root of .
ii) Every complex number
has a square root in the form of
. Find
and
in terms of
and
.