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Difference between revisions of "User:Temperal/sandbox"

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== Problem ==
+
==Sprint==
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A bunny is nice. Why?
+
# A regular ten-sided polygon has a perimeter of <math>50</math> units and an circumradius of <math>x</math>. If it's cut in half through two of it's vertices and <math>p</math> is the sum of the perimeter of the two new figures, what is <math>p-4x</math>?
 +
# 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 <math>5</math>% better than the person scoring just below him, and he scored <math>84</math> 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 <math>3</math> units.  If the volume is <math>135</math> cubic units, and has trapeziodbases of <math>5</math> and <math>3</math>, find the height of the trapezoid.
 +
# A die is rolled <math>n</math> times, such that at the <math>n</math>th roll, the die has <math>n</math> faces, each numbered with the natural numbers up to <math>n</math>. Find the probability that one of the rolls is <math>12</math> in terms of <math>n</math>.
 +
# Simplify <math>\frac{2^{2005}-2^{2004}}{2^{2005}+2^{2006}}</math>.
 +
# Find the number of ways <math>5</math> people can be arranged in a circle of <math>7</math> chairs.
 +
# Find the tenths digit of <math>\left(\frac{1}{4}\right)^{2005}</math>.
 +
#  Find the number of real solutions to <math>-x^2+2x-1=0</math>.
 +
# What is the vertex of the parabola <math>y=x^2+2x+5</math>?
 +
# If <math>a</math> is randomly chosen from <math>\{1,2,3\}</math> and <math>b</math> from <math>\{4,5,6\}</math>, find the probability that <math>b-a</math> is odd.  Express your answer as a common fraction.
 +
# Find the number of natural numbers <math>(x,y)</math> that satisfy <math>x+y=15</math>.
 +
# Find the tens digit of <math>17^{10}</math>.
 +
# Find the number of values of <math>t</math> that give <math>x^2+tx+4=0</math> exactly one solution.
 +
# How many perfect squares less than <math>1000</math> are divisible by two?
  
−
<math>\mathrm{(A)}\ {{{answera}}} \qquad \mathrm{(B)}\ {{{answerb}}} \qquad \mathrm{(C)}\ {{{answerc}}} \qquad \mathrm{(D)}\ {{{answerd}}} \qquad \mathrm{(E)}\ {{{answere}}}</math>
+
==Target==
 +
# A train leaves Omaha at <math>5</math>pm going at <math>100</math> km/h headed for San Fransisco. A train leaves San Fransisco at <math>6</math> pm of the same day at twice that speed, heading for Omaha. If Omaha and San Fransisco  are <math>1500</math> km 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 <math>30-60-90</math> triangle is  not placed on the hypotenuse? Express your answer as a common fraction in simplest radical form.
 +
# Given that <math>(x+y)^2=z</math>, <math>xy=4</math>, and <math>x^2+y^2=5</math>, find <math>z</math>.
 +
#  Bob votes randomly in a poll. If the probability that he voted for either licorice or chocolate is <math>\frac{1}{16}</math>, find the number of options in the poll.
  
−
== Solution ==
+
==Team==
−
Because he is nice. yay, bunnies!
+
# Consider a <math>4</math> by <math>5</math> rectangular grid of unit squares. Each square is randomly either coloured or not.  The squares containing a side of the rectangle are the "border". 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\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>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.
 +
# 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 <math>\frac{1}{3}</math> 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)
  
−
== See also ==
+
==Countdown==
−
{{AIME box|year=2007|ab=|num-b=2|num-a=4|n=I}}
+
# Evaluate <math>\sqrt{2+\sqrt{4+\sqrt{196}}}</math>.
 +
# Evaluate <math>2^6*6^2*5^5*10^2</math>.
 +
# Find the surface area of a cube with volume <math>343000</math> cubic units.
 +
# If a triangle has a longest side of <math>4</math> units and has two base angles of <math>45</math>, find its area.
 +
# If you have a paper with an area of <math>500</math> units squared, find the fraction of it needed to cover a circle with radius <math>10</math>. Express your answer in terms of <math>\pi</math> as a common fraction.
 +
# If <math>y^2-x^2=8</math>, and <math>y+x=2</math>, what is <math>x-y</math>?
 +
# How many vertices does a dodecahedron have?
 +
# Find the area of a triangle with sides <math>1</math>, <math>2</math>, and <math>\sqrt 5</math>.
 +
# 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 <math>100</math>.
 +
# What is the sum of the first five triangle numbers?
 +
# Let <math>\lfloor x\rfloor</math> be the largest integer less than or equal to <math>x</math>. Find the sum of the three smallest solutions to <math>x - \lfloor x\rfloor = \frac {1}{\lfloor x\rfloor}</math>.
 +
# Let <math>p=q^2</math> and <math>q=3y</math>. What is <math>y+2p</math> in terms of <math>q</math>?
 +
# Find the number of ways of rearranging the letters in THEE into distinct four-letter words.
 +
# If <math>\sqrt{+}</math> is an operation such that <math>\sqrt{a+b}=\sqrt{a} \sqrt{+} \sqrt{b}</math>, find <math>4 \sqrt{+} 3</math>.
 +
# 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 <math>5</math> hours a day not including weekends, what is the probability he studied more than <math>30</math> hours if five arbitrary days are picked from his studying schedule?
 +
# Find <math>GCD(346,254)</math>.
 +
# Jack randomly throws darts at a <math>10</math> b <math>10</math> square. What's the probability, in terms of <math>\pi</math>, that he hits within <math>2</math> units of the center?
 +
# If <math>-11\le a\le 10</math> and <math>b=a^2</math>, what is the maximum value for <math>b</math>?
 +
# What is the volume of a cone with height <math>2</math> and radius <math>4</math>? Express in terms of <math>\pi</math>.
 +
# A triangle has sides of <math>10</math>, <math>8</math>, and <math>x</math>. If the triangle has area <math>24</math>, find <math>x</math>. 
 +
# If <math>x`=x+|x-1|`</math> for all <math>x\ne 1</math>, find 4`.
 +
# Find the the number of a selected from <math>\{1,2,3,4,5,6,7,8,9,10\}</math> satisfies <math>a \pmod{3} \equiv 2</math>.
 +
# The equation for a parabola containing <math>(0,0)</math> and <math>(2,0)</math> is <math>x^2-kx=y</math>. What is <math>k</math>?
 +
# What is the sum of the digits of <math>17^2</math>?
 +
# Which is greater, <math>\frac{4}{5}</math> or <math>\frac{10}{12}</math>?
 +
# Which is less, <math>2^{2006}</math>, or <math>4^{2005}</math>?
 +
# 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 <math>s=|k^2-3k|+1</math> (if above 15, <math>s=15</math>) with <math>k</math> randomly chosen from <math>\{1,2,3,4,5,6,7,8,9,10\}</math> and the cutoff is <math>7</math>?
  
−
[[Category:Intermediate Combinatorics Problems]]
+
==Masters==
 +
# The set of complex numbers, known as <math>\mathbb{C}</math>, consists of the square roots of any negative real number plus a real number. The base complex number, called <math>i</math>, is the square root of negative one. Every complex number can be expressed as <math>a+bi</math>, where <math>i</math> is the square root of <math>-1</math>, and <math>a</math> and <math>b</math> are real constants.
 +
## Find the square root of <math>i</math>.
 +
## Every complex number <math>a+bi</math> has a square root in the form of <math>c+di</math>. Find <math>c</math> and <math>d</math> in terms of <math>a</math> and <math>b</math>.

Revision as of 12:46, 19 January 2008

Sprint

  1. A regular ten-sided polygon has a perimeter of $50$ units and an circumradius of $x$. If it's cut in half through two of it's vertices and $p$ is the sum of the perimeter of the two new figures, what is $p-4x$?
  2. 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.
  3. An annoying kid named Aishvar decides to participate in Mathcounts. If he manages to make it onto the team by scoring $5$% better than the person scoring just below him, and he scored $84$ points on the participation exam, how many points better did he score than the person below him?
  4. A strange box, which is a prism with isosceles trapezoids as bases, has a height of $3$ units. If the volume is $135$ cubic units, and has trapeziodbases of $5$ and $3$, find the height of the trapezoid.
  5. A die is rolled $n$ times, such that at the $n$th roll, the die has $n$ faces, each numbered with the natural numbers up to $n$. Find the probability that one of the rolls is $12$ in terms of $n$.
  6. Simplify $\frac{2^{2005}-2^{2004}}{2^{2005}+2^{2006}}$.
  7. Find the number of ways $5$ people can be arranged in a circle of $7$ chairs.
  8. Find the tenths digit of $\left(\frac{1}{4}\right)^{2005}$.
  9. Find the number of real solutions to $-x^2+2x-1=0$.
  10. What is the vertex of the parabola $y=x^2+2x+5$?
  11. If $a$ is randomly chosen from $\{1,2,3\}$ and $b$ from $\{4,5,6\}$, find the probability that $b-a$ is odd. Express your answer as a common fraction.
  12. Find the number of natural numbers $(x,y)$ that satisfy $x+y=15$.
  13. Find the tens digit of $17^{10}$.
  14. Find the number of values of $t$ that give $x^2+tx+4=0$ exactly one solution.
  15. How many perfect squares less than $1000$ are divisible by two?

Target

  1. A train leaves Omaha at $5$pm going at $100$ km/h headed for San Fransisco. A train leaves San Fransisco at $6$ pm of the same day at twice that speed, heading for Omaha. If Omaha and San Fransisco are $1500$ km apart, how much farther will the faster train travel than the slower once they meet?
  2. What is the probability that a point placed on a $30-60-90$ triangle is not placed on the hypotenuse? Express your answer as a common fraction in simplest radical form.
  3. Given that $(x+y)^2=z$, $xy=4$, and $x^2+y^2=5$, find $z$.
  4. Bob votes randomly in a poll. If the probability that he voted for either licorice or chocolate is $\frac{1}{16}$, find the number of options in the poll.

Team

  1. Consider a $4$ by $5$ rectangular grid of unit squares. Each square is randomly either coloured or not. The squares containing a side of the rectangle are the "border". Express the probability that less than half of the "border" of the rectangle is coloured.
  2. The country of Larepmet is undergoing a famine. Each day, $\left\lfloor\frac{x^3}{x^2+1}\right\rfloor$ sacks of wheat are lost, where $x$ is the number of sacks previously. If Larepmet starts with $100$ sacks of wheat on day $1$, on what day will they run out?
  3. If $a$ is picked from $\{1,2,3\}$, $b$ from $\{11,12,13\}$ and $c$ from $\{21,22,23\}$, find the probability that $a+b+c+abc$ is even as a common fraction.
  4. If $9$ 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.
  5. 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 $\frac{1}{3}$ 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

  1. Evaluate $\sqrt{2+\sqrt{4+\sqrt{196}}}$.
  2. Evaluate $2^6*6^2*5^5*10^2$.
  3. Find the surface area of a cube with volume $343000$ cubic units.
  4. If a triangle has a longest side of $4$ units and has two base angles of $45$, find its area.
  5. If you have a paper with an area of $500$ units squared, find the fraction of it needed to cover a circle with radius $10$. Express your answer in terms of $\pi$ as a common fraction.
  6. If $y^2-x^2=8$, and $y+x=2$, what is $x-y$?
  7. How many vertices does a dodecahedron have?
  8. Find the area of a triangle with sides $1$, $2$, and $\sqrt 5$.
  9. What is the maximum number of points needed such that not all of them must be coplanar?
  10. What is the maximum number of points needed such that there is only one plane through them?
  11. Find the number of perfect squares less than $100$.
  12. What is the sum of the first five triangle numbers?
  13. Let $\lfloor x\rfloor$ be the largest integer less than or equal to $x$. Find the sum of the three smallest solutions to $x - \lfloor x\rfloor = \frac {1}{\lfloor x\rfloor}$.
  14. Let $p=q^2$ and $q=3y$. What is $y+2p$ in terms of $q$?
  15. Find the number of ways of rearranging the letters in THEE into distinct four-letter words.
  16. If $\sqrt{+}$ is an operation such that $\sqrt{a+b}=\sqrt{a} \sqrt{+} \sqrt{b}$, find $4 \sqrt{+} 3$.
  17. In how many different ways can a person wiggle exactly two of his fingers (including thumbs)?
  18. A kid made Blue MOP this year. If he studied $5$ hours a day not including weekends, what is the probability he studied more than $30$ hours if five arbitrary days are picked from his studying schedule?
  19. Find $GCD(346,254)$.
  20. Jack randomly throws darts at a $10$ b $10$ square. What's the probability, in terms of $\pi$, that he hits within $2$ units of the center?
  21. If $-11\le a\le 10$ and $b=a^2$, what is the maximum value for $b$?
  22. What is the volume of a cone with height $2$ and radius $4$? Express in terms of $\pi$.
  23. A triangle has sides of $10$, $8$, and $x$. If the triangle has area $24$, find $x$.
  24. If $x`=x+|x-1|`$ for all $x\ne 1$, find 4`.
  25. Find the the number of a selected from $\{1,2,3,4,5,6,7,8,9,10\}$ satisfies $a \pmod{3} \equiv 2$.
  26. The equation for a parabola containing $(0,0)$ and $(2,0)$ is $x^2-kx=y$. What is $k$?
  27. What is the sum of the digits of $17^2$?
  28. Which is greater, $\frac{4}{5}$ or $\frac{10}{12}$?
  29. Which is less, $2^{2006}$, or $4^{2005}$?
  30. 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 $s=|k^2-3k|+1$ (if above 15, $s=15$) with $k$ randomly chosen from $\{1,2,3,4,5,6,7,8,9,10\}$ and the cutoff is $7$?

Masters

  1. The set of complex numbers, known as $\mathbb{C}$, consists of the square roots of any negative real number plus a real number. The base complex number, called $i$, is the square root of negative one. Every complex number can be expressed as $a+bi$, where $i$ is the square root of $-1$, and $a$ and $b$ are real constants.
    1. Find the square root of $i$.
    2. Every complex number $a+bi$ has a square root in the form of $c+di$. Find $c$ and $d$ in terms of $a$ and $b$.
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