# Difference between revisions of "User:Rowechen"

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− | + | Have fun. Don't die. | |

− | == Problem | + | == Problem 13 == |

− | + | A given sequence <math>r_1, r_2, \dots, r_n</math> of distinct real numbers can be put in ascending order by means of one or more "bubble passes". A bubble pass through a given sequence consists of comparing the second term with the first term, and exchanging them if and only if the second term is smaller, then comparing the third term with the second term and exchanging them if and only if the third term is smaller, and so on in order, through comparing the last term, <math>r_n</math>, with its current predecessor and exchanging them if and only if the last term is smaller. | |

+ | |||

+ | The example below shows how the sequence 1, 9, 8, 7 is transformed into the sequence 1, 8, 7, 9 by one bubble pass. The numbers compared at each step are underlined. | ||

+ | <center><math>\underline{1 \quad 9} \quad 8 \quad 7</math></center> | ||

+ | <center><math>1 \quad {}\underline{9 \quad 8} \quad 7</math></center> | ||

+ | <center><math>1 \quad 8 \quad \underline{9 \quad 7}</math></center> | ||

+ | <center><math>1 \quad 8 \quad 7 \quad 9</math></center> | ||

+ | Suppose that <math>n = 40</math>, and that the terms of the initial sequence <math>r_1, r_2, \dots, r_{40}</math> are distinct from one another and are in random order. Let <math>p/q</math>, in lowest terms, be the probability that the number that begins as <math>r_{20}</math> will end up, after one bubble pass, in the <math>30^{\mbox{th}}</math> place. Find <math>p + q</math>. | ||

+ | |||

+ | [[1987 AIME Problems/Problem 13|Solution]] | ||

+ | == Problem 14 == | ||

+ | Given a positive integer <math>n^{}_{}</math>, it can be shown that every complex number of the form <math>r+si^{}_{}</math>, where <math>r^{}_{}</math> and <math>s^{}_{}</math> are integers, can be uniquely expressed in the base <math>-n+i^{}_{}</math> using the integers <math>1,2^{}_{},\ldots,n^2</math> as digits. That is, the equation | ||

+ | <center><math>r+si=a_m(-n+i)^m+a_{m-1}(-n+i)^{m-1}+\cdots +a_1(-n+i)+a_0</math></center> | ||

+ | is true for a unique choice of non-negative integer <math>m^{}_{}</math> and digits <math>a_0,a_1^{},\ldots,a_m</math> chosen from the set <math>\{0^{}_{},1,2,\ldots,n^2\}</math>, with <math>a_m\ne 0^{}){}</math>. We write | ||

+ | <center><math>r+si=(a_ma_{m-1}\ldots a_1a_0)_{-n+i}</math></center> | ||

+ | to denote the base <math>-n+i^{}_{}</math> expansion of <math>r+si^{}_{}</math>. There are only finitely many integers <math>k+0i^{}_{}</math> that have four-digit expansions | ||

+ | <center><math>k=(a_3a_2a_1a_0)_{-3+i^{}_{}}~~~~a_3\ne 0.</math></center> | ||

+ | Find the sum of all such <math>k^{}_{}</math>. | ||

+ | |||

+ | [[1989 AIME Problems/Problem 14|Solution]] | ||

+ | == Problem 13 == | ||

+ | Let <math>f(n)</math> be the integer closest to <math>\sqrt[4]{n}.</math> Find <math>\sum_{k=1}^{1995}\frac 1{f(k)}.</math> | ||

+ | |||

+ | [[1995 AIME Problems/Problem 13|Solution]] | ||

+ | == Problem 14 == | ||

+ | In a circle of radius 42, two chords of length 78 intersect at a point whose distance from the center is 18. The two chords divide the interior of the circle into four regions. Two of these regions are bordered by segments of unequal lengths, and the area of either of them can be expressed uniquely in the form <math>m\pi-n\sqrt{d},</math> where <math>m, n,</math> and <math>d_{}</math> are positive integers and <math>d_{}</math> is not divisible by the square of any prime number. Find <math>m+n+d.</math> | ||

− | [[ | + | [[1995 AIME Problems/Problem 14|Solution]] |

+ | == Problem 13 == | ||

+ | If <math>\{a_1,a_2,a_3,\ldots,a_n\}</math> is a [[set]] of [[real numbers]], indexed so that <math>a_1 < a_2 < a_3 < \cdots < a_n,</math> its complex power sum is defined to be <math>a_1i + a_2i^2+ a_3i^3 + \cdots + a_ni^n,</math> where <math>i^2 = - 1.</math> Let <math>S_n</math> be the sum of the complex power sums of all nonempty [[subset]]s of <math>\{1,2,\ldots,n\}.</math> Given that <math>S_8 = - 176 - 64i</math> and <math> S_9 = p + qi,</math> where <math>p</math> and <math>q</math> are integers, find <math>|p| + |q|.</math> | ||

− | [[ | + | [[1998 AIME Problems/Problem 13|Solution]] |

− | == Problem | + | == Problem 14 == |

− | In | + | In triangle <math>ABC,</math> it is given that angles <math>B</math> and <math>C</math> are congruent. Points <math>P</math> and <math>Q</math> lie on <math>\overline{AC}</math> and <math>\overline{AB},</math> respectively, so that <math>AP = PQ = QB = BC.</math> Angle <math>ACB</math> is <math>r</math> times as large as angle <math>APQ,</math> where <math>r</math> is a positive real number. Find the greatest integer that does not exceed <math>1000r</math>. |

+ | |||

+ | [[2000 AIME I Problems/Problem 14|Solution]] | ||

+ | == Problem 14 == | ||

+ | Every positive integer <math>k</math> has a unique factorial base expansion <math>(f_1,f_2,f_3,\ldots,f_m)</math>, meaning that <math>k=1!\cdot f_1+2!\cdot f_2+3!\cdot f_3+\cdots+m!\cdot f_m</math>, where each <math>f_i</math> is an integer, <math>0\le f_i\le i</math>, and <math>0<f_m</math>. Given that <math>(f_1,f_2,f_3,\ldots,f_j)</math> is the factorial base expansion of <math>16!-32!+48!-64!+\cdots+1968!-1984!+2000!</math>, find the value of <math>f_1-f_2+f_3-f_4+\cdots+(-1)^{j+1}f_j</math>. | ||

− | [[ | + | [[2000 AIME II Problems/Problem 14|Solution]] |

+ | == Problem 13 == | ||

+ | In a certain circle, the chord of a <math>d</math>-degree arc is 22 centimeters long, and the chord of a <math>2d</math>-degree arc is 20 centimeters longer than the chord of a <math>3d</math>-degree arc, where <math>d < 120.</math> The length of the chord of a <math>3d</math>-degree arc is <math>- m + \sqrt {n}</math> centimeters, where <math>m</math> and <math>n</math> are positive integers. Find <math>m + n.</math> | ||

+ | [[2001 AIME I Problems/Problem 13|Solution]] | ||

== Problem 13 == | == Problem 13 == | ||

− | + | Let <math> N </math> be the number of positive integers that are less than or equal to 2003 and whose base-2 representation has more 1's than 0's. Find the remainder when <math> N </math> is divided by 1000. | |

+ | |||

+ | [[2003 AIME I Problems/Problem 13|Solution]] | ||

+ | == Problem 15 == | ||

+ | A long thin strip of paper is 1024 units in length, 1 unit in width, and is divided into 1024 unit squares. The paper is folded in half repeatedly. For the first fold, the right end of the paper is folded over to coincide with and lie on top of the left end. The result is a 512 by 1 strip of double thickness. Next, the right end of this strip is folded over to coincide with and lie on top of the left end, resulting in a 256 by 1 strip of quadruple thickness. This process is repeated 8 more times. After the last fold, the strip has become a stack of 1024 unit squares. How many of these squares lie below the square that was originally the 942nd square counting from the left? | ||

− | [[ | + | [[2004 AIME II Problems/Problem 15|Solution]] |

+ | == Problem 15 == | ||

+ | Let <math> w_1 </math> and <math> w_2 </math> denote the circles <math> x^2+y^2+10x-24y-87=0 </math> and <math> x^2 +y^2-10x-24y+153=0, </math> respectively. Let <math> m </math> be the smallest positive value of <math> a </math> for which the line <math> y=ax </math> contains the center of a circle that is externally tangent to <math> w_2 </math> and internally tangent to <math> w_1. </math> Given that <math> m^2=\frac pq, </math> where <math> p </math> and <math> q </math> are relatively prime integers, find <math> p+q. </math> | ||

− | == Problem | + | [[2005 AIME II Problems/Problem 15|Solution]] |

− | + | == Problem 15 == | |

− | < | + | In <math>\triangle{ABC}</math> with <math>AB = 12</math>, <math>BC = 13</math>, and <math>AC = 15</math>, let <math>M</math> be a point on <math>\overline{AC}</math> such that the incircles of <math>\triangle{ABM}</math> and <math>\triangle{BCM}</math> have equal radii. Let <math>p</math> and <math>q</math> be positive relatively prime integers such that <math>\frac {AM}{CM} = \frac {p}{q}</math>. Find <math>p + q</math>. |

− | |||

− | == Problem | + | [[2010 AIME I Problems/Problem 15|Solution]] |

− | + | == Problem 15 == | |

− | < | + | In triangle <math>ABC</math>, <math>AC=13</math>, <math>BC=14</math>, and <math>AB=15</math>. Points <math>M</math> and <math>D</math> lie on <math>AC</math> with <math>AM=MC</math> and <math>\angle ABD = \angle DBC</math>. Points <math>N</math> and <math>E</math> lie on <math>AB</math> with <math>AN=NB</math> and <math>\angle ACE = \angle ECB</math>. Let <math>P</math> be the point, other than <math>A</math>, of intersection of the circumcircles of <math>\triangle AMN</math> and <math>\triangle ADE</math>. Ray <math>AP</math> meets <math>BC</math> at <math>Q</math>. The ratio <math>\frac{BQ}{CQ}</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 | + | [[2010 AIME II Problems/Problem 15|Solution]] |

− | Let | + | == Problem 14 == |

− | < | + | Let <math>A_1 A_2 A_3 A_4 A_5 A_6 A_7 A_8</math> be a regular octagon. Let <math>M_1</math>, <math>M_3</math>, <math>M_5</math>, and <math>M_7</math> be the midpoints of sides <math>\overline{A_1 A_2}</math>, <math>\overline{A_3 A_4}</math>, <math>\overline{A_5 A_6}</math>, and <math>\overline{A_7 A_8}</math>, respectively. For <math>i = 1, 3, 5, 7</math>, ray <math>R_i</math> is constructed from <math>M_i</math> towards the interior of the octagon such that <math>R_1 \perp R_3</math>, <math>R_3 \perp R_5</math>, <math>R_5 \perp R_7</math>, and <math>R_7 \perp R_1</math>. Pairs of rays <math>R_1</math> and <math>R_3</math>, <math>R_3</math> and <math>R_5</math>, <math>R_5</math> and <math>R_7</math>, and <math>R_7</math> and <math>R_1</math> meet at <math>B_1</math>, <math>B_3</math>, <math>B_5</math>, <math>B_7</math> respectively. If <math>B_1 B_3 = A_1 A_2</math>, then <math>\cos 2 \angle A_3 M_3 B_1</math> can be written in the form <math>m - \sqrt{n}</math>, where <math>m</math> and <math>n</math> are positive integers. Find <math>m + n</math>. |

− | |||

− | == Problem | + | [[2011 AIME I Problems/Problem 14|Solution]] |

− | + | == Problem 15 == | |

+ | For some integer <math>m</math>, the polynomial <math>x^3 - 2011x + m</math> has the three integer roots <math>a</math>, <math>b</math>, and <math>c</math>. Find <math>|a| + |b| + |c|</math>. | ||

− | [[ | + | [[2011 AIME I Problems/Problem 15|Solution]] |

+ | ==Problem 15== | ||

+ | Let <math>N</math> be the number of ordered triples <math>(A,B,C)</math> of integers satisfying the conditions (a) <math>0\le A<B<C\le99</math>, (b) there exist integers <math>a</math>, <math>b</math>, and <math>c</math>, and prime <math>p</math> where <math>0\le b<a<c<p</math>, (c) <math>p</math> divides <math>A-a</math>, <math>B-b</math>, and <math>C-c</math>, and (d) each ordered triple <math>(A,B,C)</math> and each ordered triple <math>(b,a,c)</math> form arithmetic sequences. Find <math>N</math>. | ||

− | ==Problem | + | [[2013 AIME I Problems/Problem 15|Solution]] |

− | <math> | + | ==Problem 14== |

− | < | + | For positive integers <math>n</math> and <math>k</math>, let <math>f(n, k)</math> be the remainder when <math>n</math> is divided by <math>k</math>, and for <math>n > 1</math> let <math>F(n) = \max_{\substack{1\le k\le \frac{n}{2}}} f(n, k)</math>. Find the remainder when <math>\sum\limits_{n=20}^{100} F(n)</math> is divided by <math>1000</math>. |

− | < | ||

− | < | ||

− | |||

− | ==Problem | + | [[2013 AIME II Problems/Problem 14|Solution]] |

− | <math> | + | == Problem 14 == |

− | + | In a group of nine people each person shakes hands with exactly two of the other people from the group. Let <math>N</math> be the number of ways this handshaking can occur. Consider two handshaking arrangements different if and only if at least two people who shake hands under one arrangement do not shake hands under the other arrangement. Find the remainder when <math>N</math> is divided by <math>1000</math>. | |

− | |||

− | |||

− | + | [[2012 AIME II Problems/Problem 14|Solution]] | |

+ | ==Problem 14== | ||

− | + | For each integer <math>n \ge 2</math>, let <math>A(n)</math> be the area of the region in the coordinate plane defined by the inequalities <math>1\le x \le n</math> and <math>0\le y \le x \left\lfloor \sqrt x \right\rfloor</math>, where <math>\left\lfloor \sqrt x \right\rfloor</math> is the greatest integer not exceeding <math>\sqrt x</math>. Find the number of values of <math>n</math> with <math>2\le n \le 1000</math> for which <math>A(n)</math> is an integer. | |

− | <math> | ||

+ | [[2015 AIME I Problems/Problem 14|Solution]] | ||

==Problem 13== | ==Problem 13== | ||

− | Point <math> | + | Let <math>\triangle ABC</math> have side lengths <math>AB=30</math>, <math>BC=32</math>, and <math>AC=34</math>. Point <math>X</math> lies in the interior of <math>\overline{BC}</math>, and points <math>I_1</math> and <math>I_2</math> are the incenters of <math>\triangle ABX</math> and <math>\triangle ACX</math>, respectively. Find the minimum possible area of <math>\triangle AI_1I_2</math> as <math>X</math> varies along <math>\overline{BC}</math>. |

+ | |||

+ | [[2018 AIME I Problems/Problem 13 | Solution]] | ||

+ | == Problem 14 == | ||

+ | The sequence <math>(a_n)</math> satisfies <math>a_0=0</math> and <math>a_{n + 1} = \frac{8}{5}a_n + \frac{6}{5}\sqrt{4^n - a_n^2}</math> for <math>n \geq 0</math>. Find the greatest integer less than or equal to <math>a_{10}</math>. | ||

+ | |||

+ | [[2009 AIME II Problems/Problem 14|Solution]] | ||

+ | ==Problem 14== | ||

+ | |||

+ | The incircle <math>\omega</math> of triangle <math>ABC</math> is tangent to <math>\overline{BC}</math> at <math>X</math>. Let <math>Y \neq X</math> be the other intersection of <math>\overline{AX}</math> with <math>\omega</math>. Points <math>P</math> and <math>Q</math> lie on <math>\overline{AB}</math> and <math>\overline{AC}</math>, respectively, so that <math>\overline{PQ}</math> is tangent to <math>\omega</math> at <math>Y</math>. Assume that <math>AP = 3</math>, <math>PB = 4</math>, <math>AC = 8</math>, and <math>AQ = \dfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | ||

+ | |||

+ | [[2018 AIME II Problems/Problem 14 | Solution]] | ||

+ | ==Problem 15== | ||

+ | David found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals, <math>A,\text{ }B,\text{ }C</math>, which can each be inscribed in a circle with radius <math>1</math>. Let <math>\varphi_A</math> denote the measure of the acute angle made by the diagonals of quadrilateral <math>A</math>, and define <math>\varphi_B</math> and <math>\varphi_C</math> similarly. Suppose that <math>\sin\varphi_A=\frac{2}{3}</math>, <math>\sin\varphi_B=\frac{3}{5}</math>, and <math>\sin\varphi_C=\frac{6}{7}</math>. All three quadrilaterals have the same area <math>K</math>, which can be written in the form <math>\dfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math>. | ||

+ | |||

+ | [[2018 AIME I Problems/Problem 15 | Solution]] | ||

+ | ==Problem 15== | ||

+ | |||

+ | Circles <math>\omega_1</math> and <math>\omega_2</math> intersect at points <math>X</math> and <math>Y</math>. Line <math>\ell</math> is tangent to <math>\omega_1</math> and <math>\omega_2</math> at <math>A</math> and <math>B</math>, respectively, with line <math>AB</math> closer to point <math>X</math> than to <math>Y</math>. Circle <math>\omega</math> passes through <math>A</math> and <math>B</math> intersecting <math>\omega_1</math> again at <math>D \neq A</math> and intersecting <math>\omega_2</math> again at <math>C \neq B</math>. The three points <math>C</math>, <math>Y</math>, <math>D</math> are collinear, <math>XC = 67</math>, <math>XY = 47</math>, and <math>XD = 37</math>. Find <math>AB^2</math>. | ||

− | [[ | + | [[2016 AIME I Problems/Problem 15 | Solution]] |

==Problem 15== | ==Problem 15== | ||

Let <math>\triangle ABC</math> be an acute triangle with circumcircle <math>\omega,</math> and let <math>H</math> be the intersection of the altitudes of <math>\triangle ABC.</math> Suppose the tangent to the circumcircle of <math>\triangle HBC</math> at <math>H</math> intersects <math>\omega</math> at points <math>X</math> and <math>Y</math> with <math>HA=3,HX=2,</math> and <math>HY=6.</math> The area of <math>\triangle ABC</math> can be written as <math>m\sqrt{n},</math> where <math>m</math> and <math>n</math> are positive integers, and <math>n</math> is not divisible by the square of any prime. Find <math>m+n.</math> | Let <math>\triangle ABC</math> be an acute triangle with circumcircle <math>\omega,</math> and let <math>H</math> be the intersection of the altitudes of <math>\triangle ABC.</math> Suppose the tangent to the circumcircle of <math>\triangle HBC</math> at <math>H</math> intersects <math>\omega</math> at points <math>X</math> and <math>Y</math> with <math>HA=3,HX=2,</math> and <math>HY=6.</math> The area of <math>\triangle ABC</math> can be written as <math>m\sqrt{n},</math> where <math>m</math> and <math>n</math> are positive integers, and <math>n</math> is not divisible by the square of any prime. Find <math>m+n.</math> | ||

[[2020 AIME I Problems/Problem 15 | Solution]] | [[2020 AIME I Problems/Problem 15 | Solution]] | ||

− | |||

==Problem 14== | ==Problem 14== | ||

Let <math>P(x)</math> be a quadratic polynomial with complex coefficients whose <math>x^2</math> coefficient is <math>1.</math> Suppose the equation <math>P(P(x))=0</math> has four distinct solutions, <math>x=3,4,a,b.</math> Find the sum of all possible values of <math>(a+b)^2.</math> | Let <math>P(x)</math> be a quadratic polynomial with complex coefficients whose <math>x^2</math> coefficient is <math>1.</math> Suppose the equation <math>P(P(x))=0</math> has four distinct solutions, <math>x=3,4,a,b.</math> Find the sum of all possible values of <math>(a+b)^2.</math> | ||

[[2020 AIME I Problems/Problem 14 | Solution]] | [[2020 AIME I Problems/Problem 14 | Solution]] | ||

+ | == Problem 13 == | ||

+ | How many integers <math> N </math> less than 1000 can be written as the sum of <math> j </math> consecutive positive odd integers from exactly 5 values of <math> j\ge 1 </math>? | ||

+ | |||

+ | [[2006 AIME II Problems/Problem 13|Solution]] | ||

+ | == Problem 11 == | ||

+ | Define a <i>T-grid</i> to be a <math>3\times3</math> matrix which satisfies the following two properties: | ||

+ | |||

+ | <OL> | ||

+ | <LI>Exactly five of the entries are <math>1</math>'s, and the remaining four entries are <math>0</math>'s.</LI> | ||

+ | <LI>Among the eight rows, columns, and long diagonals (the long diagonals are <math>\{a_{13},a_{22},a_{31}\}</math> and <math>\{a_{11},a_{22},a_{33}\})</math>, no more than one of the eight has all three entries equal.</LI></OL> | ||

+ | |||

+ | Find the number of distinct <i>T-grids</i>. | ||

+ | |||

+ | |||

+ | [[2010 AIME II Problems/Problem 11|Solution]] | ||

+ | == Problem 12 == | ||

+ | Six men and some number of women stand in a line in random order. Let <math>p</math> be the probability that a group of at least four men stand together in the line, given that every man stands next to at least one other man. Find the least number of women in the line such that <math>p</math> does not exceed 1 percent. | ||

+ | |||

+ | [[2011 AIME I Problems/Problem 12|Solution]] | ||

+ | ==Problem 11== | ||

+ | Consider arrangements of the <math>9</math> numbers <math>1, 2, 3, \dots, 9</math> in a <math>3 \times 3</math> array. For each such arrangement, let <math>a_1</math>, <math>a_2</math>, and <math>a_3</math> be the medians of the numbers in rows <math>1</math>, <math>2</math>, and <math>3</math> respectively, and let <math>m</math> be the median of <math>\{a_1, a_2, a_3\}</math>. Let <math>Q</math> be the number of arrangements for which <math>m = 5</math>. Find the remainder when <math>Q</math> is divided by <math>1000</math>. | ||

+ | |||

+ | [[2017 AIME I Problems/Problem 11 | Solution]] | ||

+ | ==Problem 10== | ||

+ | |||

+ | Find the number of functions <math>f(x)</math> from <math>\{1, 2, 3, 4, 5\}</math> to <math>\{1, 2, 3, 4, 5\}</math> that satisfy <math>f(f(x)) = f(f(f(x)))</math> for all <math>x</math> in <math>\{1, 2, 3, 4, 5\}</math>. | ||

+ | |||

+ | [[2018 AIME II Problems/Problem 10 | Solution]] | ||

+ | ==Problem 11== | ||

+ | For integers <math>a,b,c</math> and <math>d,</math> let <math>f(x)=x^2+ax+b</math> and <math>g(x)=x^2+cx+d.</math> Find the number of ordered triples <math>(a,b,c)</math> of integers with absolute values not exceeding <math>10</math> for which there is an integer <math>d</math> such that <math>g(f(2))=g(f(4))=0.</math> | ||

+ | |||

+ | [[2020 AIME I Problems/Problem 11 | Solution]] | ||

+ | == Problem 11 == | ||

+ | A sequence is defined as follows <math> a_1=a_2=a_3=1, </math> and, for all positive integers <math> n, a_{n+3}=a_{n+2}+a_{n+1}+a_n. </math> Given that <math> a_{28}=6090307, a_{29}=11201821, </math> and <math> a_{30}=20603361, </math> find the remainder when <math>\sum^{28}_{k=1} a_k </math> is divided by 1000. | ||

+ | |||

+ | [[2006 AIME II Problems/Problem 11|Solution]] | ||

+ | |||

==Problem 13== | ==Problem 13== | ||

− | + | Point <math>D</math> lies on side <math>\overline{BC}</math> of <math>\triangle ABC</math> so that <math>\overline{AD}</math> bisects <math>\angle BAC.</math> The perpendicular bisector of <math>\overline{AD}</math> intersects the bisectors of <math>\angle ABC</math> and <math>\angle ACB</math> in points <math>E</math> and <math>F,</math> respectively. Given that <math>AB=4,BC=5,</math> and <math>CA=6,</math> the area of <math>\triangle AEF</math> can be written as <math>\tfrac{m\sqrt{n}}p,</math> where <math>m</math> and <math>p</math> are relatively prime positive integers, and <math>n</math> is a positive integer not divisible by the square of any prime. Find <math>m+n+p.</math> | |

+ | |||

+ | [[2020 AIME I Problems/Problem 13 | Solution]] | ||

− | |||

==Problem 15== | ==Problem 15== | ||

In triangle <math>ABC</math>, we have <math>BC = 13</math>, <math>CA = 37</math>, and <math>AB = 40</math>. Points <math>D</math>, <math>E</math>, and <math>F</math> are selected | In triangle <math>ABC</math>, we have <math>BC = 13</math>, <math>CA = 37</math>, and <math>AB = 40</math>. Points <math>D</math>, <math>E</math>, and <math>F</math> are selected | ||

on <math>BC</math>, <math>CA</math>, and <math>AB</math> respectively such that <math>AD</math>, <math>BE</math>, and <math>CF</math> concur at the circumcenter of <math>ABC</math>. The value of <math>\frac{1}{AD}+\frac{1}{BE}+\frac{1}{CF}</math> can be expressed as <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Determine <math>m+n</math>. | on <math>BC</math>, <math>CA</math>, and <math>AB</math> respectively such that <math>AD</math>, <math>BE</math>, and <math>CF</math> concur at the circumcenter of <math>ABC</math>. The value of <math>\frac{1}{AD}+\frac{1}{BE}+\frac{1}{CF}</math> can be expressed as <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Determine <math>m+n</math>. | ||

+ | |||

+ | ==Problem 12== | ||

+ | <math>ABC</math> is a scalene triangle. The circle with diameter <math>AB</math> intersects <math>BC</math> at <math>D</math>, and <math>E</math> is the foot of the altitude from <math>C</math>. <math>P</math> is the intersection of <math>AD</math> and <math>CE</math>. Given that <math>AP = 136</math>, <math>BP = 80</math>, and <math>CP = 26</math>, determine the circumradius of <math>ABC</math>. | ||

+ | |||

+ | ==Problem 15== | ||

+ | <math>ABCD</math> is a convex quadrilateral in which <math>AB \parallel CD</math>. Let <math>U</math> denote the intersection of the extensions of <math>AD</math> and <math>BC</math>. <math>\Omega_1</math> is the circle tangent to line segment <math>BC</math> which also passes through <math>A</math> and <math>D</math>, and <math>\Omega_2</math> is the circle tangent to <math>AD</math> which passes through <math>B</math> and <math>C</math>. Call the points of tangency <math>M</math> and <math>S</math>. Let <math>O</math> and <math>P</math> be the points of intersection between <math>\Omega_1</math> and <math>\Omega_2</math>. | ||

+ | Finally, <math>MS</math> intersects <math>OP</math> at <math>V</math>. If <math>AB = 2</math>, <math>BC = 2005</math>, <math>CD = 4</math>, and <math>DA = 2004</math>, then the value of <math>UV^2</math> is some integer <math>n</math>. Determine the remainder obtained when <math>n</math> is divided by <math>1000</math>. | ||

+ | |||

+ | ==Problem 13== | ||

+ | <math>P(x)</math> is the polynomial of minimal degree that satisfies | ||

+ | <cmath>P(k) = \frac{1}{k(k+1)}</cmath> | ||

+ | |||

+ | for <math>k = 1, 2, 3, . . . , 10</math>. The value of <math>P(11)</math> can be written as <math>-\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively | ||

+ | prime positive integers. Determine <math>m + n</math>. | ||

+ | |||

+ | ==Problem 12== | ||

+ | |||

+ | <math>ABCD</math> is a cyclic quadrilateral with <math>AB = 8</math>, <math>BC = 4</math>, <math>CD = 1</math>, and <math>DA = 7</math>. Let <math>O</math> and <math>P</math> denote the circumcenter and intersection of <math>AC</math> and <math>BD</math> respectively. The value of <math>OP^2</math> can be expressed as <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime, positive integers. Determine the remainder obtained when <math>m + n</math> is divided by <math>1000</math>. | ||

+ | |||

+ | ==Problem 11== | ||

+ | <math>10</math> lines and <math>10</math> circles divide the plane into at most <math>n</math> disjoint regions. Compute <math>n</math>. |

## Latest revision as of 19:29, 2 June 2020

Have fun. Don't die.

## Contents

- 1 Problem 13
- 2 Problem 14
- 3 Problem 13
- 4 Problem 14
- 5 Problem 13
- 6 Problem 14
- 7 Problem 14
- 8 Problem 13
- 9 Problem 13
- 10 Problem 15
- 11 Problem 15
- 12 Problem 15
- 13 Problem 15
- 14 Problem 14
- 15 Problem 15
- 16 Problem 15
- 17 Problem 14
- 18 Problem 14
- 19 Problem 14
- 20 Problem 13
- 21 Problem 14
- 22 Problem 14
- 23 Problem 15
- 24 Problem 15
- 25 Problem 15
- 26 Problem 14
- 27 Problem 13
- 28 Problem 11
- 29 Problem 12
- 30 Problem 11
- 31 Problem 10
- 32 Problem 11
- 33 Problem 11
- 34 Problem 13
- 35 Problem 15
- 36 Problem 12
- 37 Problem 15
- 38 Problem 13
- 39 Problem 12
- 40 Problem 11

## Problem 13

A given sequence of distinct real numbers can be put in ascending order by means of one or more "bubble passes". A bubble pass through a given sequence consists of comparing the second term with the first term, and exchanging them if and only if the second term is smaller, then comparing the third term with the second term and exchanging them if and only if the third term is smaller, and so on in order, through comparing the last term, , with its current predecessor and exchanging them if and only if the last term is smaller.

The example below shows how the sequence 1, 9, 8, 7 is transformed into the sequence 1, 8, 7, 9 by one bubble pass. The numbers compared at each step are underlined.

Suppose that , and that the terms of the initial sequence are distinct from one another and are in random order. Let , in lowest terms, be the probability that the number that begins as will end up, after one bubble pass, in the place. Find .

## Problem 14

Given a positive integer , it can be shown that every complex number of the form , where and are integers, can be uniquely expressed in the base using the integers as digits. That is, the equation

is true for a unique choice of non-negative integer and digits chosen from the set , with . We write

to denote the base expansion of . There are only finitely many integers that have four-digit expansions

Find the sum of all such .

## Problem 13

Let be the integer closest to Find

## Problem 14

In a circle of radius 42, two chords of length 78 intersect at a point whose distance from the center is 18. The two chords divide the interior of the circle into four regions. Two of these regions are bordered by segments of unequal lengths, and the area of either of them can be expressed uniquely in the form where and are positive integers and is not divisible by the square of any prime number. Find

## Problem 13

If is a set of real numbers, indexed so that its complex power sum is defined to be where Let be the sum of the complex power sums of all nonempty subsets of Given that and where and are integers, find

## Problem 14

In triangle it is given that angles and are congruent. Points and lie on and respectively, so that Angle is times as large as angle where is a positive real number. Find the greatest integer that does not exceed .

## Problem 14

Every positive integer has a unique factorial base expansion , meaning that , where each is an integer, , and . Given that is the factorial base expansion of , find the value of .

## Problem 13

In a certain circle, the chord of a -degree arc is 22 centimeters long, and the chord of a -degree arc is 20 centimeters longer than the chord of a -degree arc, where The length of the chord of a -degree arc is centimeters, where and are positive integers. Find

## Problem 13

Let be the number of positive integers that are less than or equal to 2003 and whose base-2 representation has more 1's than 0's. Find the remainder when is divided by 1000.

## Problem 15

A long thin strip of paper is 1024 units in length, 1 unit in width, and is divided into 1024 unit squares. The paper is folded in half repeatedly. For the first fold, the right end of the paper is folded over to coincide with and lie on top of the left end. The result is a 512 by 1 strip of double thickness. Next, the right end of this strip is folded over to coincide with and lie on top of the left end, resulting in a 256 by 1 strip of quadruple thickness. This process is repeated 8 more times. After the last fold, the strip has become a stack of 1024 unit squares. How many of these squares lie below the square that was originally the 942nd square counting from the left?

## Problem 15

Let and denote the circles and respectively. Let be the smallest positive value of for which the line contains the center of a circle that is externally tangent to and internally tangent to Given that where and are relatively prime integers, find

## Problem 15

In with , , and , let be a point on such that the incircles of and have equal radii. Let and be positive relatively prime integers such that . Find .

## Problem 15

In triangle , , , and . Points and lie on with and . Points and lie on with and . Let be the point, other than , of intersection of the circumcircles of and . Ray meets at . The ratio can be written in the form , where and are relatively prime positive integers. Find .

## Problem 14

Let be a regular octagon. Let , , , and be the midpoints of sides , , , and , respectively. For , ray is constructed from towards the interior of the octagon such that , , , and . Pairs of rays and , and , and , and and meet at , , , respectively. If , then can be written in the form , where and are positive integers. Find .

## Problem 15

For some integer , the polynomial has the three integer roots , , and . Find .

## Problem 15

Let be the number of ordered triples of integers satisfying the conditions (a) , (b) there exist integers , , and , and prime where , (c) divides , , and , and (d) each ordered triple and each ordered triple form arithmetic sequences. Find .

## Problem 14

For positive integers and , let be the remainder when is divided by , and for let . Find the remainder when is divided by .

## Problem 14

In a group of nine people each person shakes hands with exactly two of the other people from the group. Let be the number of ways this handshaking can occur. Consider two handshaking arrangements different if and only if at least two people who shake hands under one arrangement do not shake hands under the other arrangement. Find the remainder when is divided by .

## Problem 14

For each integer , let be the area of the region in the coordinate plane defined by the inequalities and , where is the greatest integer not exceeding . Find the number of values of with for which is an integer.

## Problem 13

Let have side lengths , , and . Point lies in the interior of , and points and are the incenters of and , respectively. Find the minimum possible area of as varies along .

## Problem 14

The sequence satisfies and for . Find the greatest integer less than or equal to .

## Problem 14

The incircle of triangle is tangent to at . Let be the other intersection of with . Points and lie on and , respectively, so that is tangent to at . Assume that , , , and , where and are relatively prime positive integers. Find .

## Problem 15

David found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals, , which can each be inscribed in a circle with radius . Let denote the measure of the acute angle made by the diagonals of quadrilateral , and define and similarly. Suppose that , , and . All three quadrilaterals have the same area , which can be written in the form , where and are relatively prime positive integers. Find .

## Problem 15

Circles and intersect at points and . Line is tangent to and at and , respectively, with line closer to point than to . Circle passes through and intersecting again at and intersecting again at . The three points , , are collinear, , , and . Find .

## Problem 15

Let be an acute triangle with circumcircle and let be the intersection of the altitudes of Suppose the tangent to the circumcircle of at intersects at points and with and The area of can be written as where and are positive integers, and is not divisible by the square of any prime. Find

## Problem 14

Let be a quadratic polynomial with complex coefficients whose coefficient is Suppose the equation has four distinct solutions, Find the sum of all possible values of

## Problem 13

How many integers less than 1000 can be written as the sum of consecutive positive odd integers from exactly 5 values of ?

## Problem 11

Define a *T-grid* to be a matrix which satisfies the following two properties:

- Exactly five of the entries are 's, and the remaining four entries are 's.
- Among the eight rows, columns, and long diagonals (the long diagonals are and , no more than one of the eight has all three entries equal.

Find the number of distinct *T-grids*.

## Problem 12

Six men and some number of women stand in a line in random order. Let be the probability that a group of at least four men stand together in the line, given that every man stands next to at least one other man. Find the least number of women in the line such that does not exceed 1 percent.

## Problem 11

Consider arrangements of the numbers in a array. For each such arrangement, let , , and be the medians of the numbers in rows , , and respectively, and let be the median of . Let be the number of arrangements for which . Find the remainder when is divided by .

## Problem 10

Find the number of functions from to that satisfy for all in .

## Problem 11

For integers and let and Find the number of ordered triples of integers with absolute values not exceeding for which there is an integer such that

## Problem 11

A sequence is defined as follows and, for all positive integers Given that and find the remainder when is divided by 1000.

## Problem 13

Point lies on side of so that bisects The perpendicular bisector of intersects the bisectors of and in points and respectively. Given that and the area of can be written as where and are relatively prime positive integers, and is a positive integer not divisible by the square of any prime. Find

## Problem 15

In triangle , we have , , and . Points , , and are selected on , , and respectively such that , , and concur at the circumcenter of . The value of can be expressed as where and are relatively prime positive integers. Determine .

## Problem 12

is a scalene triangle. The circle with diameter intersects at , and is the foot of the altitude from . is the intersection of and . Given that , , and , determine the circumradius of .

## Problem 15

is a convex quadrilateral in which . Let denote the intersection of the extensions of and . is the circle tangent to line segment which also passes through and , and is the circle tangent to which passes through and . Call the points of tangency and . Let and be the points of intersection between and . Finally, intersects at . If , , , and , then the value of is some integer . Determine the remainder obtained when is divided by .

## Problem 13

is the polynomial of minimal degree that satisfies

for . The value of can be written as , where and are relatively prime positive integers. Determine .

## Problem 12

is a cyclic quadrilateral with , , , and . Let and denote the circumcenter and intersection of and respectively. The value of can be expressed as , where and are relatively prime, positive integers. Determine the remainder obtained when is divided by .

## Problem 11

lines and circles divide the plane into at most disjoint regions. Compute .