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− | ==Problem 5== | + | ==Problem 5*== |
Suppose that <math>x</math>, <math>y</math>, and <math>z</math> are complex numbers such that <math>xy = -80 - 320i</math>, <math>yz = 60</math>, and <math>zx = -96 + 24i</math>, where <math>i</math> <math>=</math> <math>\sqrt{-1}</math>. Then there are real numbers <math>a</math> and <math>b</math> such that <math>x + y + z = a + bi</math>. Find <math>a^2 + b^2</math>. | Suppose that <math>x</math>, <math>y</math>, and <math>z</math> are complex numbers such that <math>xy = -80 - 320i</math>, <math>yz = 60</math>, and <math>zx = -96 + 24i</math>, where <math>i</math> <math>=</math> <math>\sqrt{-1}</math>. Then there are real numbers <math>a</math> and <math>b</math> such that <math>x + y + z = a + bi</math>. Find <math>a^2 + b^2</math>. | ||
− | ==Problem 6== | + | ==Problem 6 (check)== |
A real number <math>a</math> is chosen randomly and uniformly from the interval <math>[-20, 18]</math>. The probability that the roots of the polynomial | A real number <math>a</math> is chosen randomly and uniformly from the interval <math>[-20, 18]</math>. The probability that the roots of the polynomial | ||
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Note: Homothety | Note: Homothety | ||
− | ==Problem 13== | + | ==Problem 13*== |
Misha rolls a standard, fair six-sided die until she rolls 1-2-3 in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is <math>\dfrac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | Misha rolls a standard, fair six-sided die until she rolls 1-2-3 in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is <math>\dfrac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | ||
− | ==Problem 6== | + | Note: Find relation between odd & even. |
+ | |||
+ | ==Problem 6 (check)== | ||
Let <math>N</math> be the number of complex numbers <math>z</math> with the properties that <math>|z|=1</math> and <math>z^{6!}-z^{5!}</math> is a real number. Find the remainder when <math>N</math> is divided by <math>1000</math>. | Let <math>N</math> be the number of complex numbers <math>z</math> with the properties that <math>|z|=1</math> and <math>z^{6!}-z^{5!}</math> is a real number. Find the remainder when <math>N</math> is divided by <math>1000</math>. | ||
− | ==Problem 9== | + | ==Problem 9*== |
Find the number of four-element subsets of <math>\{1,2,3,4,\dots, 20\}</math> with the property that two distinct elements of a subset have a sum of <math>16</math>, and two distinct elements of a subset have a sum of <math>24</math>. For example, <math>\{3,5,13,19\}</math> and <math>\{6,10,20,18\}</math> are two such subsets. | Find the number of four-element subsets of <math>\{1,2,3,4,\dots, 20\}</math> with the property that two distinct elements of a subset have a sum of <math>16</math>, and two distinct elements of a subset have a sum of <math>24</math>. For example, <math>\{3,5,13,19\}</math> and <math>\{6,10,20,18\}</math> are two such subsets. | ||
− | ==Problem 10== | + | ==Problem 10*== |
The wheel shown below consists of two circles and five spokes, with a label at each point where a spoke meets a circle. A bug walks along the wheel, starting at point <math>A</math>. At every step of the process, the bug walks from one labeled point to an adjacent labeled point. Along the inner circle the bug only walks in a counterclockwise direction, and along the outer circle the bug only walks in a clockwise direction. For example, the bug could travel along the path <math>AJABCHCHIJA</math>, which has <math>10</math> steps. Let <math>n</math> be the number of paths with <math>15</math> steps that begin and end at point <math>A.</math> Find the remainder when <math>n</math> is divided by <math>1000</math>. | The wheel shown below consists of two circles and five spokes, with a label at each point where a spoke meets a circle. A bug walks along the wheel, starting at point <math>A</math>. At every step of the process, the bug walks from one labeled point to an adjacent labeled point. Along the inner circle the bug only walks in a counterclockwise direction, and along the outer circle the bug only walks in a clockwise direction. For example, the bug could travel along the path <math>AJABCHCHIJA</math>, which has <math>10</math> steps. Let <math>n</math> be the number of paths with <math>15</math> steps that begin and end at point <math>A.</math> Find the remainder when <math>n</math> is divided by <math>1000</math>. | ||
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<asy> size(6cm); draw(unitcircle); draw(scale(2) * unitcircle); for(int d = 90; d < 360 + 90; d += 72){ draw(2 * dir(d) -- dir(d)); } dot(1 * dir( 90), linewidth(5)); dot(1 * dir(162), linewidth(5)); dot(1 * dir(234), linewidth(5)); dot(1 * dir(306), linewidth(5)); dot(1 * dir(378), linewidth(5)); dot(2 * dir(378), linewidth(5)); dot(2 * dir(306), linewidth(5)); dot(2 * dir(234), linewidth(5)); dot(2 * dir(162), linewidth(5)); dot(2 * dir( 90), linewidth(5)); label("$A$", 1 * dir( 90), -dir( 90)); label("$B$", 1 * dir(162), -dir(162)); label("$C$", 1 * dir(234), -dir(234)); label("$D$", 1 * dir(306), -dir(306)); label("$E$", 1 * dir(378), -dir(378)); label("$F$", 2 * dir(378), dir(378)); label("$G$", 2 * dir(306), dir(306)); label("$H$", 2 * dir(234), dir(234)); label("$I$", 2 * dir(162), dir(162)); label("$J$", 2 * dir( 90), dir( 90)); </asy> | <asy> size(6cm); draw(unitcircle); draw(scale(2) * unitcircle); for(int d = 90; d < 360 + 90; d += 72){ draw(2 * dir(d) -- dir(d)); } dot(1 * dir( 90), linewidth(5)); dot(1 * dir(162), linewidth(5)); dot(1 * dir(234), linewidth(5)); dot(1 * dir(306), linewidth(5)); dot(1 * dir(378), linewidth(5)); dot(2 * dir(378), linewidth(5)); dot(2 * dir(306), linewidth(5)); dot(2 * dir(234), linewidth(5)); dot(2 * dir(162), linewidth(5)); dot(2 * dir( 90), linewidth(5)); label("$A$", 1 * dir( 90), -dir( 90)); label("$B$", 1 * dir(162), -dir(162)); label("$C$", 1 * dir(234), -dir(234)); label("$D$", 1 * dir(306), -dir(306)); label("$E$", 1 * dir(378), -dir(378)); label("$F$", 2 * dir(378), dir(378)); label("$G$", 2 * dir(306), dir(306)); label("$H$", 2 * dir(234), dir(234)); label("$I$", 2 * dir(162), dir(162)); label("$J$", 2 * dir( 90), dir( 90)); </asy> | ||
− | ==Problem 5== | + | ==Problem 5 (check)== |
A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are <math>189</math>, <math>320</math>, <math>287</math>, <math>234</math>, <math>x</math>, and <math>y</math>. Find the greatest possible value of <math>x+y</math>. | A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are <math>189</math>, <math>320</math>, <math>287</math>, <math>234</math>, <math>x</math>, and <math>y</math>. Find the greatest possible value of <math>x+y</math>. | ||
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A special deck of cards contains <math>49</math> cards, each labeled with a number from <math>1</math> to <math>7</math> and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of eight cards from the deck at random. Given that she gets at least one card of each color and at least one card with each number, the probability that Sharon can discard one of her cards and <math>\textit{still}</math> have at least one card of each color and at least one card with each number is <math>\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>. | A special deck of cards contains <math>49</math> cards, each labeled with a number from <math>1</math> to <math>7</math> and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of eight cards from the deck at random. Given that she gets at least one card of each color and at least one card with each number, the probability that Sharon can discard one of her cards and <math>\textit{still}</math> have at least one card of each color and at least one card with each number is <math>\frac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>. | ||
− | ==Problem 11== | + | ==Problem 11 (check)== |
Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing through other towns on the way). | Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing through other towns on the way). | ||
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For each integer <math>n\geq3</math>, let <math>f(n)</math> be the number of <math>3</math>-element subsets of the vertices of a regular <math>n</math>-gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of <math>n</math> such that <math>f(n+1)=f(n)+78</math>. | For each integer <math>n\geq3</math>, let <math>f(n)</math> be the number of <math>3</math>-element subsets of the vertices of a regular <math>n</math>-gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of <math>n</math> such that <math>f(n+1)=f(n)+78</math>. | ||
− | ==Problem 5== | + | ==Problem 5 (check)== |
A moving particle starts at the point <math>(4,4)</math> and moves until it hits one of the coordinate axes for the first time. When the particle is at the point <math>(a,b)</math>, it moves at random to one of the points <math>(a-1,b)</math>, <math>(a,b-1)</math>, or <math>(a-1,b-1)</math>, each with probability <math>\tfrac{1}{3}</math>, independently of its previous moves. The probability that it will hit the coordinate axes at <math>(0,0)</math> is <math>\tfrac{m}{3^n}</math>, where <math>m</math> and <math>n</math> are positive integers, and <math>m</math> is not divisible by <math>3</math>. Find <math>m + n</math>. | A moving particle starts at the point <math>(4,4)</math> and moves until it hits one of the coordinate axes for the first time. When the particle is at the point <math>(a,b)</math>, it moves at random to one of the points <math>(a-1,b)</math>, <math>(a,b-1)</math>, or <math>(a-1,b-1)</math>, each with probability <math>\tfrac{1}{3}</math>, independently of its previous moves. The probability that it will hit the coordinate axes at <math>(0,0)</math> is <math>\tfrac{m}{3^n}</math>, where <math>m</math> and <math>n</math> are positive integers, and <math>m</math> is not divisible by <math>3</math>. Find <math>m + n</math>. | ||
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In convex quadrilateral <math>KLMN</math>, side <math>\overline{MN}</math> is perpendicular to diagonal <math>\overline{KM}</math>, side <math>\overline{KL}</math> is perpendicular to diagonal <math>\overline{LN}</math>, <math>MN = 65</math>, and <math>KL = 28</math>. The line through <math>L</math> perpendicular to side <math>\overline{KN}</math> intersects diagonal <math>\overline{KM}</math> at <math>O</math> with <math>KO = 8</math>. Find <math>MO</math>. | In convex quadrilateral <math>KLMN</math>, side <math>\overline{MN}</math> is perpendicular to diagonal <math>\overline{KM}</math>, side <math>\overline{KL}</math> is perpendicular to diagonal <math>\overline{LN}</math>, <math>MN = 65</math>, and <math>KL = 28</math>. The line through <math>L</math> perpendicular to side <math>\overline{KN}</math> intersects diagonal <math>\overline{KM}</math> at <math>O</math> with <math>KO = 8</math>. Find <math>MO</math>. | ||
− | ==Problem 14== | + | ==Problem 14* (check)== |
Find the least odd prime factor of <math>2019^8 + 1</math>. | Find the least odd prime factor of <math>2019^8 + 1</math>. | ||
Note: Use FLT | Note: Use FLT | ||
− | ==Problem 8== | + | ==Problem 8 (almost)== |
Find the number of sets <math>\{a,b,c\}</math> of three distinct positive integers with the property that the product of <math>a,b,</math> and <math>c</math> is equal to the product of <math>11,21,31,41,51,</math> and <math>61</math>. | Find the number of sets <math>\{a,b,c\}</math> of three distinct positive integers with the property that the product of <math>a,b,</math> and <math>c</math> is equal to the product of <math>11,21,31,41,51,</math> and <math>61</math>. | ||
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Note: SFFT | Note: SFFT | ||
+ | |||
+ | ==Problem 11== | ||
+ | The circumcircle of acute <math>\triangle ABC</math> has center <math>O</math>. The line passing through point <math>O</math> perpendicular to <math>\overline{OB}</math> intersects lines <math>AB</math> and <math>BC</math> at <math>P</math> and <math>Q</math>, respectively. Also <math>AB=5</math>, <math>BC=4</math>, <math>BQ=4.5</math>, and <math>BP=\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | ||
+ | |||
+ | Note: easy but pay attention to the wording | ||
+ | |||
+ | ==Problem 6== | ||
+ | Point <math>A,B,C,D,</math> and <math>E</math> are equally spaced on a minor arc of a circle. Points <math>E,F,G,H,I</math> and <math>A</math> are equally spaced on a minor arc of a second circle with center <math>C</math> as shown in the figure below. The angle <math>\angle ABD</math> exceeds <math>\angle AHG</math> by <math>12^\circ</math>. Find the degree measure of <math>\angle BAG</math>. | ||
+ | |||
+ | <asy> pair A,B,C,D,E,F,G,H,I,O; O=(0,0); C=dir(90); B=dir(70); A=dir(50); D=dir(110); E=dir(130); draw(arc(O,1,50,130)); real x=2*sin(20*pi/180); F=x*dir(228)+C; G=x*dir(256)+C; H=x*dir(284)+C; I=x*dir(312)+C; draw(arc(C,x,200,340)); label("$A$",A,dir(0)); label("$B$",B,dir(75)); label("$C$",C,dir(90)); label("$D$",D,dir(105)); label("$E$",E,dir(180)); label("$F$",F,dir(225)); label("$G$",G,dir(260)); label("$H$",H,dir(280)); label("$I$",I,dir(315));</asy> | ||
+ | |||
+ | Note: Arc angles | ||
+ | |||
+ | ==Problem 5== | ||
+ | Real numbers <math>r</math> and <math>s</math> are roots of <math>p(x)=x^3+ax+b</math>, and <math>r+4</math> and <math>s-3</math> are roots of <math>q(x)=x^3+ax+b+240</math>. Find the sum of all possible values of <math>|b|</math>. | ||
+ | |||
+ | ==Problem 13== | ||
+ | With all angles measured in degrees, the product <math>\prod_{k=1}^{45} \csc^2(2k-1)^\circ=m^n</math>, where <math>m</math> and <math>n</math> are integers greater than 1. Find <math>m+n</math>. | ||
+ | |||
+ | ==Problem 7== | ||
+ | Let <math>w</math> and <math>z</math> be complex numbers such that <math>|w| = 1</math> and <math>|z| = 10</math>. Let <math>\theta = \arg \left(\tfrac{w-z}{z}\right)</math>. The maximum possible value of <math>\tan^2 \theta</math> can be written as <math>\tfrac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>. (Note that <math>\arg(w)</math>, for <math>w \neq 0</math>, denotes the measure of the angle that the ray from <math>0</math> to <math>w</math> makes with the positive real axis in the complex plane.) | ||
+ | |||
+ | ==Problem 9== | ||
+ | Let <math>x_1< x_2 < x_3</math> be the three real roots of the equation <math>\sqrt{2014} x^3 - 4029x^2 + 2 = 0</math>. Find <math>x_2(x_1+x_3)</math>. | ||
+ | |||
+ | ==Problem 8== | ||
+ | The domain of the function <math>f(x) = \arcsin(\log_{m}(nx))</math> is a closed interval of length <math>\frac{1}{2013}</math> , where <math>m</math> and <math>n</math> are positive integers and <math>m>1</math>. Find the remainder when the smallest possible sum <math>m+n</math> is divided by <math>1000</math>. | ||
+ | |||
+ | Note: stupid problem - need to test <math>m</math> | ||
+ | |||
+ | ==Problem 8== | ||
+ | |||
+ | The complex numbers <math>z</math> and <math>w</math> satisfy the system<cmath>z + \frac{20i}w = 5+i</cmath><cmath>w+\frac{12i}z = -4+10i</cmath>Find the smallest possible value of <math>\vert zw\vert^2</math>. | ||
+ | |||
+ | ==Problem 11== | ||
+ | A frog begins at <math>P_0 = (0,0)</math> and makes a sequence of jumps according to the following rule: from <math>P_n = (x_n, y_n),</math> the frog jumps to <math>P_{n+1},</math> which may be any of the points <math>(x_n + 7, y_n + 2),</math> <math>(x_n + 2, y_n + 7),</math> <math>(x_n - 5, y_n - 10),</math> or <math>(x_n - 10, y_n - 5).</math> There are <math>M</math> points <math>(x, y)</math> with <math>|x| + |y| \le 100</math> that can be reached by a sequence of such jumps. Find the remainder when <math>M</math> is divided by <math>1000.</math> | ||
+ | |||
+ | Note: Don't silly answer extraction | ||
+ | |||
+ | ==Problem 6== | ||
+ | Find the smallest positive integer <math>n</math> with the property that the polynomial <math>x^4 - nx + 63</math> can be written as a product of two nonconstant polynomials with integer coefficients. | ||
+ | |||
+ | Note: <math>(x^2 + ax + b)(x^2+cx+d)</math> | ||
+ | |||
+ | ==Problem 7== | ||
+ | Define an ordered triple <math>(A, B, C)</math> of sets to be <math>\textit{minimally intersecting}</math> if <math>|A \cap B| = |B \cap C| = |C \cap A| = 1</math> and <math>A \cap B \cap C = \emptyset</math>. For example, <math>(\{1,2\},\{2,3\},\{1,3,4\})</math> is a minimally intersecting triple. Let <math>N</math> be the number of minimally intersecting ordered triples of sets for which each set is a subset of <math>\{1,2,3,4,5,6,7\}</math>. Find the remainder when <math>N</math> is divided by <math>1000</math>. | ||
+ | |||
+ | ==Problem 9== | ||
+ | Let <math>(a,b,c)</math> be a real solution of the system of equations <math>x^3 - xyz = 2</math>, <math>y^3 - xyz = 6</math>, <math>z^3 - xyz = 20</math>. The greatest possible value of <math>a^3 + b^3 + c^3</math> can be written in the form <math>\frac {m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. | ||
+ | |||
+ | ==Problem 9== | ||
+ | Ten identical crates each of dimensions <math>3</math> ft <math>\times</math> <math>4</math> ft <math>\times</math> <math>6</math> ft. The first crate is placed flat on the floor. Each of the remaining nine crates is placed, in turn, flat on top of the previous crate, and the orientation of each crate is chosen at random. Let <math>\frac {m}{n}</math> be the probability that the stack of crates is exactly <math>41</math> ft tall, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m</math>. | ||
+ | |||
+ | ==Problem 9== | ||
+ | Rectangle <math>ABCD</math> is given with <math>AB=63</math> and <math>BC=448.</math> Points <math>E</math> and <math>F</math> lie on <math>AD</math> and <math>BC</math> respectively, such that <math>AE=CF=84.</math> The inscribed circle of triangle <math>BEF</math> is tangent to <math>EF</math> at point <math>P,</math> and the inscribed circle of triangle <math>DEF</math> is tangent to <math>EF</math> at point <math>Q.</math> Find <math>PQ.</math> | ||
+ | |||
+ | Note: Don't trust diagrams. | ||
+ | |||
+ | ==Problem 6== | ||
+ | A frog is placed at the origin on the number line, and moves according to the following rule: in a given move, the frog advances to either the closest point with a greater integer coordinate that is a multiple of <math>3</math>, or to the closest point with a greater integer coordinate that is a multiple of <math>13</math>. A move sequence is a sequence of coordinates which correspond to valid moves, beginning with <math>0</math>, and ending with <math>39</math>. For example, <math>0,\ 3,\ 6,\ 13,\ 15,\ 26,\ 39</math> is a move sequence. How many move sequences are possible for the frog? | ||
+ | |||
+ | Note: 0 => 13 => 26 => 39 | ||
+ | |||
+ | ==Problem 15== | ||
+ | |||
+ | The area of the smallest equilateral triangle with one vertex on each of the sides of the right triangle with side lengths <math>2\sqrt3</math>, <math>5</math>, and <math>\sqrt{37}</math>, as shown, is <math>\tfrac{m\sqrt{p}}{n}</math>, where <math>m</math>, <math>n</math>, and <math>p</math> are positive integers, <math>m</math> and <math>n</math> are relatively prime, and <math>p</math> is not divisible by the square of any prime. Find <math>m+n+p</math>. | ||
+ | |||
+ | <asy> | ||
+ | size(5cm); | ||
+ | pair C=(0,0),B=(0,2*sqrt(3)),A=(5,0); | ||
+ | real t = .385, s = 3.5*t-1; | ||
+ | pair R = A*t+B*(1-t), P=B*s; | ||
+ | pair Q = dir(-60) * (R-P) + P; | ||
+ | fill(P--Q--R--cycle,gray); | ||
+ | draw(A--B--C--A^^P--Q--R--P); | ||
+ | dot(A--B--C--P--Q--R); | ||
+ | </asy> |
Revision as of 10:53, 27 August 2024
Contents
[hide]- 1 Problem 5*
- 2 Problem 6 (check)
- 3 Problem 9
- 4 Problem 13*
- 5 Problem 6 (check)
- 6 Problem 9*
- 7 Problem 10*
- 8 Problem 5 (check)
- 9 Problem 9
- 10 Problem 11 (check)
- 11 Problem 13
- 12 Problem 5 (check)
- 13 Problem 6
- 14 Problem 14* (check)
- 15 Problem 8 (almost)
- 16 Problem 10
- 17 Problem 6
- 18 Problem 7
- 19 Problem 9
- 20 Problem 12
- 21 Problem 8
- 22 Problem 11
- 23 Problem 6
- 24 Problem 5
- 25 Problem 13
- 26 Problem 7
- 27 Problem 9
- 28 Problem 8
- 29 Problem 8
- 30 Problem 11
- 31 Problem 6
- 32 Problem 7
- 33 Problem 9
- 34 Problem 9
- 35 Problem 9
- 36 Problem 6
- 37 Problem 15
Problem 5*
Suppose that , , and are complex numbers such that , , and , where . Then there are real numbers and such that . Find .
Problem 6 (check)
A real number is chosen randomly and uniformly from the interval . The probability that the roots of the polynomial
are all real can be written in the form , where and are relatively prime positive integers. Find .
Problem 9
Octagon with side lengths and is formed by removing 6-8-10 triangles from the corners of a rectangle with side on a short side of the rectangle, as shown. Let be the midpoint of , and partition the octagon into 7 triangles by drawing segments , , , , , and . Find the area of the convex polygon whose vertices are the centroids of these 7 triangles.
Note: Homothety
Problem 13*
Misha rolls a standard, fair six-sided die until she rolls 1-2-3 in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is where and are relatively prime positive integers. Find .
Note: Find relation between odd & even.
Problem 6 (check)
Let be the number of complex numbers with the properties that and is a real number. Find the remainder when is divided by .
Problem 9*
Find the number of four-element subsets of with the property that two distinct elements of a subset have a sum of , and two distinct elements of a subset have a sum of . For example, and are two such subsets.
Problem 10*
The wheel shown below consists of two circles and five spokes, with a label at each point where a spoke meets a circle. A bug walks along the wheel, starting at point . At every step of the process, the bug walks from one labeled point to an adjacent labeled point. Along the inner circle the bug only walks in a counterclockwise direction, and along the outer circle the bug only walks in a clockwise direction. For example, the bug could travel along the path , which has steps. Let be the number of paths with steps that begin and end at point Find the remainder when is divided by .
Problem 5 (check)
A set contains four numbers. The six pairwise sums of distinct elements of the set, in no particular order, are , , , , , and . Find the greatest possible value of .
Problem 9
A special deck of cards contains cards, each labeled with a number from to and colored with one of seven colors. Each number-color combination appears on exactly one card. Sharon will select a set of eight cards from the deck at random. Given that she gets at least one card of each color and at least one card with each number, the probability that Sharon can discard one of her cards and have at least one card of each color and at least one card with each number is , where and are relatively prime positive integers. Find .
Problem 11 (check)
Five towns are connected by a system of roads. There is exactly one road connecting each pair of towns. Find the number of ways there are to make all the roads one-way in such a way that it is still possible to get from any town to any other town using the roads (possibly passing through other towns on the way).
Note: Complimentary counting + PiE
Problem 13
For each integer , let be the number of -element subsets of the vertices of a regular -gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of such that .
Problem 5 (check)
A moving particle starts at the point and moves until it hits one of the coordinate axes for the first time. When the particle is at the point , it moves at random to one of the points , , or , each with probability , independently of its previous moves. The probability that it will hit the coordinate axes at is , where and are positive integers, and is not divisible by . Find .
Note: recursion with probability
Problem 6
In convex quadrilateral , side is perpendicular to diagonal , side is perpendicular to diagonal , , and . The line through perpendicular to side intersects diagonal at with . Find .
Problem 14* (check)
Find the least odd prime factor of .
Note: Use FLT
Problem 8 (almost)
Find the number of sets of three distinct positive integers with the property that the product of and is equal to the product of and .
Problem 10
Triangle is inscribed in circle . Points and are on side with . Rays and meet again at and (other than ), respectively. If and , then , where and are relatively prime positive integers. Find .
Problem 6
In let be the center of the inscribed circle, and let the bisector of intersect at . The line through and intersects the circumscribed circle of at the two points and . If and , then , where and are relatively prime positive integers. Find .
Note: angle chase, then angle bisectors.
Problem 7
For integers and consider the complex numberFind the number of ordered pairs of integers such that this complex number is a real number.
Note: and beware of absolute value sign
Problem 9
Triangle has and . This triangle is inscribed in rectangle with on and on . Find the maximum possible area of .
Use:
Problem 12
Find the least positive integer such that is a product of at least four not necessarily distinct primes.
Note: should be multiple of .
Problem 8
Let and be positive integers satisfying . The maximum possible value of is , where and are relatively prime positive integers. Find .
Note: SFFT
Problem 11
The circumcircle of acute has center . The line passing through point perpendicular to intersects lines and at and , respectively. Also , , , and , where and are relatively prime positive integers. Find .
Note: easy but pay attention to the wording
Problem 6
Point and are equally spaced on a minor arc of a circle. Points and are equally spaced on a minor arc of a second circle with center as shown in the figure below. The angle exceeds by . Find the degree measure of .
Note: Arc angles
Problem 5
Real numbers and are roots of , and and are roots of . Find the sum of all possible values of .
Problem 13
With all angles measured in degrees, the product , where and are integers greater than 1. Find .
Problem 7
Let and be complex numbers such that and . Let . The maximum possible value of can be written as , where and are relatively prime positive integers. Find . (Note that , for , denotes the measure of the angle that the ray from to makes with the positive real axis in the complex plane.)
Problem 9
Let be the three real roots of the equation . Find .
Problem 8
The domain of the function is a closed interval of length , where and are positive integers and . Find the remainder when the smallest possible sum is divided by .
Note: stupid problem - need to test
Problem 8
The complex numbers and satisfy the systemFind the smallest possible value of .
Problem 11
A frog begins at and makes a sequence of jumps according to the following rule: from the frog jumps to which may be any of the points or There are points with that can be reached by a sequence of such jumps. Find the remainder when is divided by
Note: Don't silly answer extraction
Problem 6
Find the smallest positive integer with the property that the polynomial can be written as a product of two nonconstant polynomials with integer coefficients.
Note:
Problem 7
Define an ordered triple of sets to be if and . For example, is a minimally intersecting triple. Let be the number of minimally intersecting ordered triples of sets for which each set is a subset of . Find the remainder when is divided by .
Problem 9
Let be a real solution of the system of equations , , . The greatest possible value of can be written in the form , where and are relatively prime positive integers. Find .
Problem 9
Ten identical crates each of dimensions ft ft ft. The first crate is placed flat on the floor. Each of the remaining nine crates is placed, in turn, flat on top of the previous crate, and the orientation of each crate is chosen at random. Let be the probability that the stack of crates is exactly ft tall, where and are relatively prime positive integers. Find .
Problem 9
Rectangle is given with and Points and lie on and respectively, such that The inscribed circle of triangle is tangent to at point and the inscribed circle of triangle is tangent to at point Find
Note: Don't trust diagrams.
Problem 6
A frog is placed at the origin on the number line, and moves according to the following rule: in a given move, the frog advances to either the closest point with a greater integer coordinate that is a multiple of , or to the closest point with a greater integer coordinate that is a multiple of . A move sequence is a sequence of coordinates which correspond to valid moves, beginning with , and ending with . For example, is a move sequence. How many move sequences are possible for the frog?
Note: 0 => 13 => 26 => 39
Problem 15
The area of the smallest equilateral triangle with one vertex on each of the sides of the right triangle with side lengths , , and , as shown, is , where , , and are positive integers, and are relatively prime, and is not divisible by the square of any prime. Find .