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| − | ==Sprint== | + | ==Problem 1== |
| − | # 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-x</math>?
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| − | # 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.
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| − | # 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?
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| − | # 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.
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| − | # 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>.
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| − | # Simplify <math>\frac{2^{2005}-2^{2004}}{2^{2005}+2^{2006}}</math>.
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| − | # Find the number of ways <math>5</math> people can be arranged in a circle of <math>7</math> chairs.
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| − | # Find the tenths digit of <math>\left(\frac{1}{4}\right)^{2005}</math>.
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| − | # Find the number of real solutions to <math>-x^2+2x-1=0</math>.
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| − | # What is the vertex of the parabola <math>y=x^2+2x+5</math>?
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| − | # 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.
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| − | # Find the number of natural numbers <math>(x,y)</math> that satisfy <math>x+y=15</math>.
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| − | # Find the tens digit of <math>17^{10}</math>.
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| − | # Find the number of values of <math>t</math> that give <math>x^2+tx+4=0</math> exactly one solution.
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| − | # How many perfect squares less than <math>1000</math> are divisible by two?
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| − | ==Target==
| + | Evaluate the following expressions: |
| − | # 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> miles apart, how much farther will the faster train travel than the slower once they meet?
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| − | # What is the probability that a point placed on a <math>30-60-90</math> triangle is placed on the hypotenuse? Express your answer as a common fraction in simplest radical form.
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| − | # If the median of a triangle is <math>5</math>, and its two base angles are congruent, find the other two side's lengths in terms of the base.
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| − | # 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.
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| − | ==Team==
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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.
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| − | # 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?
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| − | # 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.
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| − | # 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.
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| − | # 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)
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| − | ==Countdown==
| + | (a) <math>\tan(45^\circ)</math> |
| − | # Evaluate <math>\sqrt{2+\sqrt{4+\sqrt{196}}}</math>.
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| − | # Evaluate <math>2^6*6^2*5^5*10^2</math>.
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| − | # Find the surface area of a cube with volume <math>343000</math> cubic units.
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| − | # If a triangle has a longest side of <math>4</math> units and has two base angles of <math>45</math>, find its area.
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| − | # 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.
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| − | # If <math>y^2-x^2=8</math>, and <math>y+x=2</math>, what is <math>x-y</math>?
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| − | # How many vertices does a dodecahedron have?
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| − | # Find the area of a triangle with sides <math>1</math>, <math>2</math>, and <math>\sqrt 5</math>.
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| − | # What is the maximum number of points needed such that not all of them must be coplanar?
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| − | # What is the maximum number of points needed such that there is only one plane through them?
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| − | # Find the number of perfect squares less than <math>100</math>.
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| − | # What is the sum of the first five triangle numbers?
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| − | # 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>.
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| − | # Let <math>p=q^2</math> and <math>q=3y</math>. What is <math>y+2p</math> in terms of <math>q</math>?
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| − | # Find the number of ways of rearranging the letters in THEE into distinct four-letter words.
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| − | # 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>.
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| − | # In how many different ways can a person wiggle exactly two of his fingers (including thumbs)?
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| − | # 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?
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| − | # Find <math>GCD(346,254)</math>.
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| − | # 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?
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| − | # If <math>-11\le a\le 10</math> and <math>b=a^2</math>, what is the maximum value for <math>b</math>?
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| − | # What is the volume of a cone with height <math>2</math> and radius <math>4</math>? Express in terms of <math>\pi</math>.
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| − | # 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>.
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| − | # If <math>x`=x+|x-1|`</math> for all <math>x\ne 1</math>, find 4`.
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| − | # 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>.
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| − | # 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>?
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| − | # What is the sum of the digits of <math>17^2</math>?
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| − | # Which is greater, <math>\frac{4}{5}</math> or <math>\frac{10}{12}</math>?
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| − | # Which is less, <math>2^{2006}</math>, or <math>4^{2005}</math>?
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| − | # 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>?
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| − | ==Masters== | + | (b) <math>\cos\left(\frac {7\pi}{4}\right)</math> |
| − | # 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.
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| − | ## Find the square root of <math>i</math>.
| + | (c) <math>\sin\left(\frac {5\pi}{3}\right)</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>.
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| | + | (d) <math>\csc(135^\circ)</math> |
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| | + | (e) <math>\cot(945^\circ)</math> |
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| | + | (f) <math>\sin(\pi \sin(\pi/6))</math> |
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| | + | (g) <math>\tan(21\pi)</math> |
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| | + | (h) <math>\sec( - 585^\circ)</math> |
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| | + | ==Problem 2== |
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| | + | Using the unit circle, find <math>\sin \left( x + \frac {\pi}{2} \right)</math> and <math>\cos \left( x + \frac {\pi}{2} \right)</math> in terms of <math>\sin x</math> and <math>\cos x</math>. |
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| | + | ==Problem 3== |
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| | + | In triangle <math>ABC</math>, <math>\angle B = 90^\circ</math>, <math>\sin A = 7/9</math>, and <math>BC = 21</math>. What is <math>AB</math>? |
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| | + | ==Problem 4== |
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| | + | What does the graph of <math>\sin 4x</math> look like compared to the graphs of <math>\sin x</math> and <math>\cos x</math>? What about the graph of <math>2\sin \left( 3x + \frac {\pi}{4} \right) - 1</math>? |
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| | + | ==Problem 5== |
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| | + | Find the value of <math>\tan(\pi/12) \cdot \tan(2\pi/12) \cdot \tan(3\pi/12) \cdots \tan(5\pi/12).</math> |
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| | + | ==Problem 6== |
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| | + | Suppose that parallelogram <math>ABCD</math> has <math>\angle A = \angle C = 30^\circ</math>, <math>\angle B = \angle D = 150^\circ</math>, and the shorter diagonal <math>BD</math> has length 2. If the height of the parallelogram is <math>a</math>, find the perimeter of <math>ABCD</math> in terms of <math>a</math>. |
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| | + | ==Problem 7== |
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| | + | Given a positive number <math>n</math> and a number <math>c</math> satisfying <math>- 1 < c < 1</math>, for how many values of <math>q</math> with <math>0 \leq q < 2\pi</math> is <math>\sin nq = c</math>? What if <math>c = 1</math> or <math>c = - 1</math>? |
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| | + | ==Problem 8== |
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| | + | How many solutions are there to the equation <math>\cos x = \frac {x^2}{1000}</math>, where <math>x</math> is in radians? |
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| | + | ==Problem 9== |
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| | + | Determine all <math>\theta</math> such that <math>0 \le \theta \le \frac {\pi}{2}</math> and <math>\sin^5\theta + \cos^5\theta = 1</math>. |
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| | + | ==Problem 10== |
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| | + | Find the value of <math>\sin(15^\circ)</math>. Hint: Draw an isosceles triangle with vertex angle <math>30^\circ</math>. |
and 
in terms of 
and 
.
, 
, 
, and 
. What is 
?
look like compared to the graphs of 
?
has 
, 
, and the shorter diagonal 
has length 2. If the height of the parallelogram is 
, find the perimeter of 
and a number 
satisfying 
, for how many values of 
with 
is 
? What if 
or 
?
, where 
is in radians?
such that 
and 
.
. Hint: Draw an isosceles triangle with vertex angle 
.
