Difference between revisions of "User:Rowechen"
Line 1: | Line 1: | ||
Here's the AIME compilation I will be doing: | Here's the AIME compilation I will be doing: | ||
− | |||
− | |||
− | [[ | + | == 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]] |
− | |||
− | |||
− | [[ | + | {{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]] |
− | |||
− | |||
− | |||
− | [[ | + | {{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 17:38, 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.
![[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]](http://latex.artofproblemsolving.com/3/3/a/33abde80848746d240bfc392b8997b80cf3bccd5.png)
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.