Difference between revisions of "2012 AIME I Problems"
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{{AIME Problems|year=2012|n=I}} | {{AIME Problems|year=2012|n=I}} | ||
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== Problem 1 == | == Problem 1 == | ||
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== See also == | == See also == | ||
− | {{AIME box|year=2012|n=I|before=[[2011 AIME II]]|after=[[2012 AIME II]]}} | + | {{AIME box|year=2012|n=I|before=[[2011 AIME II Problems]]|after=[[2012 AIME II Problems]]}} |
* [[American Invitational Mathematics Examination]] | * [[American Invitational Mathematics Examination]] |
Latest revision as of 14:49, 2 June 2023
2012 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
Problem 1
Find the number of positive integers with three not necessarily distinct digits, , with
and
such that both
and
are multiples of
.
Problem 2
The terms of an arithmetic sequence add to . The first term of the sequence is increased by
, the second term is increased by
, the third term is increased by
, and in general, the
th term is increased by the
th odd positive integer. The terms of the new sequence add to
. Find the sum of the first, last, and middle term of the original sequence.
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
Butch and Sundance need to get out of Dodge. To travel as quickly as possible, each alternates walking and riding their only horse, Sparky, as follows. Butch begins by walking while Sundance rides. When Sundance reaches the first of the hitching posts that are conveniently located at one-mile intervals along their route, he ties Sparky to the post and begins walking. When Butch reaches Sparky, he rides until he passes Sundance, then leaves Sparky at the next hitching post and resumes walking, and they continue in this manner. Sparky, Butch, and Sundance walk at
and
miles per hour, respectively. The first time Butch and Sundance meet at a milepost, they are
miles from Dodge, and they have been traveling for
minutes. Find
.
Problem 5
Let be the set of all binary integers that can be written using exactly
zeros and
ones where leading zeros are allowed. If all possible subtractions are performed in which one element of
is subtracted from another, find the number of times the answer
is obtained.
Problem 6
The complex numbers and
satisfy
and the imaginary part of
is
, for relatively prime positive integers
and
with
Find
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 8
Cube labeled as shown below, has edge length
and is cut by a plane passing through vertex
and the midpoints
and
of
and
respectively. The plane divides the cube into two solids. The volume of the larger of the two solids can be written in the form
where
and
are relatively prime positive integers. Find
![[asy]import cse5; unitsize(10mm); pathpen=black; dotfactor=3; pair A = (0,0), B = (3.8,0), C = (5.876,1.564), D = (2.076,1.564), E = (0,3.8), F = (3.8,3.8), G = (5.876,5.364), H = (2.076,5.364), M = (1.9,0), N = (5.876,3.465); pair[] dotted = {A,B,C,D,E,F,G,H,M,N}; D(A--B--C--G--H--E--A); D(E--F--B); D(F--G); pathpen=dashed; D(A--D--H); D(D--C); dot(dotted); label("$A$",A,SW); label("$B$",B,S); label("$C$",C,SE); label("$D$",D,NW); label("$E$",E,W); label("$F$",F,SE); label("$G$",G,NE); label("$H$",H,NW); label("$M$",M,S); label("$N$",N,NE); [/asy]](http://latex.artofproblemsolving.com/a/f/1/af154db9d25256bd90239e9f4e37253b234659e0.png)
Problem 9
Let
and
be positive real numbers that satisfy
The value of
can be expressed in the form
where
and
are relatively prime positive integers. Find
Problem 10
Let be the set of all perfect squares whose rightmost three digits in base
are
. Let
be the set of all numbers of the form
, where
is in
. In other words,
is the set of numbers that result when the last three digits of each number in
are truncated. Find the remainder when the tenth smallest element of
is divided by
.
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
Problem 12
Let be a right triangle with right angle at
Let
and
be points on
with
between
and
such that
and
trisect
If
then
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 13
Three concentric circles have radii
and
An equilateral triangle with one vertex on each circle has side length
The largest possible area of the triangle can be written as
where
and
are positive integers,
and
are relatively prime, and
is not divisible by the square of any prime. Find
Problem 14
Complex numbers
and
are zeros of a polynomial
and
The points corresponding to
and
in the complex plane are the vertices of a right triangle with hypotenuse
Find
Problem 15
There are mathematicians seated around a circular table with
seats numbered
in clockwise order. After a break they again sit around the table. The mathematicians note that there is a positive integer
such that
-
(
![$1$](http://latex.artofproblemsolving.com/d/c/e/dce34f4dfb2406144304ad0d6106c5382ddd1446.png)
![$k,$](http://latex.artofproblemsolving.com/c/9/9/c99a6aaebc17474bd09ea71d84d4e769a0662b55.png)
![$k$](http://latex.artofproblemsolving.com/8/c/3/8c325612684d41304b9751c175df7bcc0f61f64f.png)
![$ka$](http://latex.artofproblemsolving.com/4/0/8/408730e185e0b41fcc93fd334b48e79f1c8e799d.png)
![$i + n$](http://latex.artofproblemsolving.com/d/4/9/d49c43f56bface4f34de7ad1d3e6a46a70fc1b63.png)
![$i$](http://latex.artofproblemsolving.com/3/4/8/34857b3ba74ce5cd8607f3ebd23e9015908ada71.png)
-
(
![$2$](http://latex.artofproblemsolving.com/4/1/c/41c544263a265ff15498ee45f7392c5f86c6d151.png)
Find the number of possible values of with
See also
2012 AIME I (Problems • Answer Key • Resources) | ||
Preceded by 2011 AIME II Problems |
Followed by 2012 AIME II Problems | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
- American Invitational Mathematics Examination
- AIME Problems and Solutions
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.