Difference between revisions of "User:Rowechen"
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Here's the AIME compilation I will be doing: | Here's the AIME compilation I will be doing: | ||
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− | [[ | + | == Problem 3 == |
+ | Nine people sit down for dinner where there are three choices of meals. Three people order the beef meal, three order the chicken meal, and three order the fish meal. The waiter serves the nine meals in random order. Find the number of ways in which the waiter could serve the meal types to the nine people so that exactly one person receives the type of meal ordered by that person. | ||
+ | |||
+ | [[2012 AIME I Problems/Problem 3|Solution]] | ||
+ | ==Problem 4== | ||
+ | |||
+ | In equiangular octagon <math>CAROLINE</math>, <math>CA = RO = LI = NE =</math> <math>\sqrt{2}</math> and <math>AR = OL = IN = EC = 1</math>. The self-intersecting octagon <math>CORNELIA</math> enclosed six non-overlapping triangular regions. Let <math>K</math> be the area enclosed by <math>CORNELIA</math>, that is, the total area of the six triangular regions. Then <math>K = \frac{a}{b}</math>, where <math>a</math> and <math>b</math> are relatively prime positive integers. Find <math>a + b</math>. | ||
+ | |||
+ | [[2018 AIME II Problems/Problem 4 | Solution]] | ||
==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>. | |
+ | [[2018 AIME II Problems/Problem 5 | Solution]] | ||
+ | == Problem 7 == | ||
+ | Let <math>S_i</math> be the set of all integers <math>n</math> such that <math>100i\leq n < 100(i + 1)</math>. For example, <math>S_4</math> is the set <math>{400,401,402,\ldots,499}</math>. How many of the sets <math>S_0, S_1, S_2, \ldots, S_{999}</math> do not contain a perfect square? | ||
− | [[ | + | [[2008 AIME I Problems/Problem 7|Solution]] |
− | ==Problem | + | == 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>. | |
− | [[ | + | '''Note''': <math>|S|</math> represents the number of elements in the set <math>S</math>. |
+ | |||
+ | [[2010 AIME I Problems/Problem 7|Solution]] | ||
== Problem 7 == | == Problem 7 == | ||
− | + | At each of the sixteen circles in the network below stands a student. A total of <math>3360</math> coins are distributed among the sixteen students. All at once, all students give away all their coins by passing an equal number of coins to each of their neighbors in the network. After the trade, all students have the same number of coins as they started with. Find the number of coins the student standing at the center circle had originally. | |
+ | |||
+ | <center><asy> | ||
+ | import cse5; | ||
+ | unitsize(6mm); | ||
+ | defaultpen(linewidth(.8pt)); | ||
+ | dotfactor = 8; | ||
+ | pathpen=black; | ||
+ | |||
+ | pair A = (0,0); | ||
+ | pair B = 2*dir(54), C = 2*dir(126), D = 2*dir(198), E = 2*dir(270), F = 2*dir(342); | ||
+ | pair G = 3.6*dir(18), H = 3.6*dir(90), I = 3.6*dir(162), J = 3.6*dir(234), K = 3.6*dir(306); | ||
+ | pair M = 6.4*dir(54), N = 6.4*dir(126), O = 6.4*dir(198), P = 6.4*dir(270), L = 6.4*dir(342); | ||
+ | pair[] dotted = {A,B,C,D,E,F,G,H,I,J,K,L,M,N,O,P}; | ||
+ | |||
+ | D(A--B--H--M); | ||
+ | D(A--C--H--N); | ||
+ | D(A--F--G--L); | ||
+ | D(A--E--K--P); | ||
+ | D(A--D--J--O); | ||
+ | D(B--G--M); | ||
+ | D(F--K--L); | ||
+ | D(E--J--P); | ||
+ | D(O--I--D); | ||
+ | D(C--I--N); | ||
+ | D(L--M--N--O--P--L); | ||
+ | |||
+ | dot(dotted); | ||
− | + | </asy></center> | |
− | |||
− | |||
− | [[ | + | [[2012 AIME I Problems/Problem 7|Solution]] |
− | == Problem | + | == Problem 11 == |
− | + | Ms. Math's kindergarten class has 16 registered students. The classroom has a very large number, ''N'', of play blocks which satisfies the conditions: | |
− | [[ | + | (a) If 16, 15, or 14 students are present in the class, then in each case all the blocks can be distributed in equal numbers to each student, and |
+ | |||
+ | (b) There are three integers <math>0 < x < y < z < 14</math> such that when <math>x</math>, <math>y</math>, or <math>z</math> students are present and the blocks are distributed in equal numbers to each student, there are exactly three blocks left over. | ||
+ | |||
+ | Find the sum of the distinct prime divisors of the least possible value of ''N'' satisfying the above conditions. | ||
+ | |||
+ | [[2013 AIME I Problems/Problem 11|Solution]] | ||
== Problem 12 == | == Problem 12 == | ||
− | + | Let <math>\bigtriangleup PQR</math> be a triangle with <math>\angle P = 75^o</math> and <math>\angle Q = 60^o</math>. A regular hexagon <math>ABCDEF</math> with side length 1 is drawn inside <math>\triangle PQR</math> so that side <math>\overline{AB}</math> lies on <math>\overline{PQ}</math>, side <math>\overline{CD}</math> lies on <math>\overline{QR}</math>, and one of the remaining vertices lies on <math>\overline{RP}</math>. There are positive integers <math>a, b, c, </math> and <math>d</math> such that the area of <math>\triangle PQR</math> can be expressed in the form <math>\frac{a+b\sqrt{c}}{d}</math>, where <math>a</math> and <math>d</math> are relatively prime, and c is not divisible by the square of any prime. Find <math>a+b+c+d</math>. | |
− | [[ | + | [[2013 AIME I Problems/Problem 12|Solution]] |
− | == Problem | + | ==Problem 11== |
− | + | Let <math>A = \{1, 2, 3, 4, 5, 6, 7\}</math>, and let <math>N</math> be the number of functions <math>f</math> from set <math>A</math> to set <math>A</math> such that <math>f(f(x))</math> is a constant function. Find the remainder when <math>N</math> is divided by <math>1000</math>. | |
+ | |||
+ | [[2013 AIME II Problems/Problem 11|Solution]] | ||
+ | ==Problem 11== | ||
+ | In <math>\triangle RED</math>, <math>\measuredangle DRE=75^{\circ}</math> and <math>\measuredangle RED=45^{\circ}</math>. <math> RD=1</math>. Let <math>M</math> be the midpoint of segment <math>\overline{RD}</math>. Point <math>C</math> lies on side <math>\overline{ED}</math> such that <math>\overline{RC}\perp\overline{EM}</math>. Extend segment <math>\overline{DE}</math> through <math>E</math> to point <math>A</math> such that <math>CA=AR</math>. Then <math>AE=\frac{a-\sqrt{b}}{c}</math>, where <math>a</math> and <math>c</math> are relatively prime positive integers, and <math>b</math> is a positive integer. Find <math>a+b+c</math>. | ||
+ | |||
+ | [[2014 AIME II Problems/Problem 11|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>. | ||
− | [[ | + | [[2013 AIME I Problems/Problem 15|Solution]] |
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− | |||
− | [[ | + | {{AIME box|year=2013|n=I|before=[[2012 AIME II Problems]]|after=[[2013 AIME II Problems]]}} |
− | |||
− | |||
− | + | {{MAA Notice}} | |
− | == Problem 14 == | + | ==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>. | |
− | [[ | + | [[2013 AIME II Problems/Problem 14|Solution]] |
− | == Problem 15 == | + | ==Problem 15== |
− | Let <math> | + | Let <math>A,B,C</math> be angles of an acute triangle with |
+ | <cmath> \begin{align*} | ||
+ | \cos^2 A + \cos^2 B + 2 \sin A \sin B \cos C &= \frac{15}{8} \text{ and} \\ | ||
+ | \cos^2 B + \cos^2 C + 2 \sin B \sin C \cos A &= \frac{14}{9} | ||
+ | \end{align*} </cmath> | ||
+ | There are positive integers <math>p</math>, <math>q</math>, <math>r</math>, and <math>s</math> for which <cmath> \cos^2 C + \cos^2 A + 2 \sin C \sin A \cos B = \frac{p-q\sqrt{r}}{s}, </cmath> where <math>p+q</math> and <math>s</math> are relatively prime and <math>r</math> is not divisible by the square of any prime. Find <math>p+q+r+s</math>. | ||
− | [[ | + | [[2013 AIME II Problems/Problem 15|Solution]] |
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− | |||
− | [[ | + | {{AIME box|year=2013|n=II|before=[[2013 AIME I Problems]]|after=[[2014 AIME I Problems]]}} |
− | |||
− | |||
− | + | {{MAA Notice}} | |
− | == Problem | + | ==Problem 13== |
− | + | On square <math>ABCD</math>, points <math>E,F,G</math>, and <math>H</math> lie on sides <math>\overline{AB},\overline{BC},\overline{CD},</math> and <math>\overline{DA},</math> respectively, so that <math>\overline{EG} \perp \overline{FH}</math> and <math>EG=FH = 34</math>. Segments <math>\overline{EG}</math> and <math>\overline{FH}</math> intersect at a point <math>P</math>, and the areas of the quadrilaterals <math>AEPH, BFPE, CGPF,</math> and <math>DHPG</math> are in the ratio <math>269:275:405:411.</math> Find the area of square <math>ABCD</math>. | |
− | < | + | <asy> |
− | </ | + | pair A = (0,sqrt(850)); |
+ | pair B = (0,0); | ||
+ | pair C = (sqrt(850),0); | ||
+ | pair D = (sqrt(850),sqrt(850)); | ||
+ | draw(A--B--C--D--cycle); | ||
+ | dotfactor = 3; | ||
+ | dot("$A$",A,dir(135)); | ||
+ | dot("$B$",B,dir(215)); | ||
+ | dot("$C$",C,dir(305)); | ||
+ | dot("$D$",D,dir(45)); | ||
+ | pair H = ((2sqrt(850)-sqrt(306))/6,sqrt(850)); | ||
+ | pair F = ((2sqrt(850)+sqrt(306)+7)/6,0); | ||
+ | dot("$H$",H,dir(90)); | ||
+ | dot("$F$",F,dir(270)); | ||
+ | draw(H--F); | ||
+ | pair E = (0,(sqrt(850)-6)/2); | ||
+ | pair G = (sqrt(850),(sqrt(850)+sqrt(100))/2); | ||
+ | dot("$E$",E,dir(180)); | ||
+ | dot("$G$",G,dir(0)); | ||
+ | draw(E--G); | ||
+ | pair P = extension(H,F,E,G); | ||
+ | dot("$P$",P,dir(60)); | ||
+ | label("$w$", intersectionpoint( A--P, E--H )); | ||
+ | label("$x$", intersectionpoint( B--P, E--F )); | ||
+ | label("$y$", intersectionpoint( C--P, G--F )); | ||
+ | label("$z$", intersectionpoint( D--P, G--H ));</asy> | ||
− | and | + | [[2014 AIME I Problems/Problem 13|Solution]] |
+ | ==Problem 15== | ||
+ | In <math>\triangle ABC, AB = 3, BC = 4,</math> and <math>CA = 5</math>. Circle <math>\omega</math> intersects <math>\overline{AB}</math> at <math>E</math> and <math>B, \overline{BC}</math> at <math>B</math> and <math>D,</math> and <math>\overline{AC}</math> at <math>F</math> and <math>G</math>. Given that <math>EF=DF</math> and <math>\frac{DG}{EG} = \frac{3}{4},</math> length <math>DE=\frac{a\sqrt{b}}{c},</math> where <math>a</math> and <math>c</math> are relatively prime positive integers, and <math>b</math> is a positive integer not divisible by the square of any prime. Find <math>a+b+c</math>. | ||
− | + | [[2014 AIME I Problems/Problem 15|Solution]] | |
− | |||
− | + | {{AIME box|year=2014|n=I|before=[[2013 AIME II Problems]]|after=[[2014 AIME II Problems]]}} | |
− | + | {{MAA Notice}} |
Revision as of 16:38, 29 May 2020
Here's the AIME compilation I will be doing:
Contents
Problem 3
Nine people sit down for dinner where there are three choices of meals. Three people order the beef meal, three order the chicken meal, and three order the fish meal. The waiter serves the nine meals in random order. Find the number of ways in which the waiter could serve the meal types to the nine people so that exactly one person receives the type of meal ordered by that person.
Problem 4
In equiangular octagon , and . The self-intersecting octagon enclosed six non-overlapping triangular regions. Let be the area enclosed by , that is, the total area of the six triangular regions. Then , where and are relatively prime positive integers. Find .
Problem 5
Suppose that , , and are complex numbers such that , , and , where . Then there are real numbers and such that . Find .
Problem 7
Let be the set of all integers such that . For example, is the set . How many of the sets do not contain a perfect square?
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 .
Note: represents the number of elements in the set .
Problem 7
At each of the sixteen circles in the network below stands a student. A total of coins are distributed among the sixteen students. All at once, all students give away all their coins by passing an equal number of coins to each of their neighbors in the network. After the trade, all students have the same number of coins as they started with. Find the number of coins the student standing at the center circle had originally.
Problem 11
Ms. Math's kindergarten class has 16 registered students. The classroom has a very large number, N, of play blocks which satisfies the conditions:
(a) If 16, 15, or 14 students are present in the class, then in each case all the blocks can be distributed in equal numbers to each student, and
(b) There are three integers such that when , , or students are present and the blocks are distributed in equal numbers to each student, there are exactly three blocks left over.
Find the sum of the distinct prime divisors of the least possible value of N satisfying the above conditions.
Problem 12
Let be a triangle with and . A regular hexagon with side length 1 is drawn inside so that side lies on , side lies on , and one of the remaining vertices lies on . There are positive integers and such that the area of can be expressed in the form , where and are relatively prime, and c is not divisible by the square of any prime. Find .
Problem 11
Let , and let be the number of functions from set to set such that is a constant function. Find the remainder when is divided by .
Problem 11
In , and . . Let be the midpoint of segment . Point lies on side such that . Extend segment through to point such that . Then , where and are relatively prime positive integers, and is a positive integer. 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 .
2013 AIME I (Problems • Answer Key • Resources) | ||
Preceded by 2012 AIME II Problems |
Followed by 2013 AIME II Problems | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
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 15
Let be angles of an acute triangle with There are positive integers , , , and for which where and are relatively prime and is not divisible by the square of any prime. Find .
2013 AIME II (Problems • Answer Key • Resources) | ||
Preceded by 2013 AIME I Problems |
Followed by 2014 AIME I Problems | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
Problem 13
On square , points , and lie on sides and respectively, so that and . Segments and intersect at a point , and the areas of the quadrilaterals and are in the ratio Find the area of square .
Problem 15
In and . Circle intersects at and at and and at and . Given that and length where and are relatively prime positive integers, and is a positive integer not divisible by the square of any prime. Find .
2014 AIME I (Problems • Answer Key • Resources) | ||
Preceded by 2013 AIME II Problems |
Followed by 2014 AIME II Problems | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.