Difference between revisions of "2013 AIME I Problems"
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== Problem 11 == | == Problem 11 == | ||
− | Ms. Math's kindergarten class has 16 registered students. The classroom has a very large number, | + | Ms. Math's kindergarten class has <math>16</math> registered students. The classroom has a very large number, <math>N</math>, 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 | + | (a) If <math>16</math>, <math>15</math>, or <math>14</math> 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. | (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 | + | Find the sum of the distinct prime divisors of the least possible value of <math>N</math> satisfying the above conditions. |
[[2013 AIME I Problems/Problem 11|Solution]] | [[2013 AIME I Problems/Problem 11|Solution]] | ||
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==Problem 15== | ==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>. | + | 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]] | [[2013 AIME I Problems/Problem 15|Solution]] |
Latest revision as of 13:07, 24 February 2024
2013 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
The AIME Triathlon consists of a half-mile swim, a 30-mile bicycle ride, and an eight-mile run. Tom swims, bicycles, and runs at constant rates. He runs fives times as fast as he swims, and he bicycles twice as fast as he runs. Tom completes the AIME Triathlon in four and a quarter hours. How many minutes does he spend bicycling?
Problem 2
Find the number of five-digit positive integers, , that satisfy the following conditions:
-
(a) the number
![$n$](http://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png)
![$5,$](http://latex.artofproblemsolving.com/2/1/4/21415a2eb558eb7c199a0382fcd50cbeda9299fb.png)
-
(b) the first and last digits of
![$n$](http://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png)
-
(c) the sum of the digits of
![$n$](http://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png)
![$5.$](http://latex.artofproblemsolving.com/b/8/3/b835f2c6c592edc4583ce996f86bcc0d07ca8da5.png)
Problem 3
Let be a square, and let
and
be points on
and
respectively. The line through
parallel to
and the line through
parallel to
divide
into two squares and two nonsquare rectangles. The sum of the areas of the two squares is
of the area of square
Find
Problem 4
In the array of squares shown below,
squares are colored red, and the remaining
squares are colored blue. If one of all possible such colorings is chosen at random, the probability that the chosen colored array appears the same when rotated
around the central square is
, where
is a positive integer. Find
.
Problem 5
The real root of the equation can be written in the form
, where
,
, and
are positive integers. Find
.
Problem 6
Melinda has three empty boxes and textbooks, three of which are mathematics textbooks. One box will hold any three of her textbooks, one will hold any four of her textbooks, and one will hold any five of her textbooks. If Melinda packs her textbooks into these boxes in random order, the probability that all three mathematics textbooks end up in the same box can be written as
, where
and
are relatively prime positive integers. Find
.
Problem 7
A rectangular box has width inches, length
inches, and height
inches, where
and
are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle with an area of
square inches. 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
.
Problem 9
A paper equilateral triangle has side length
. The paper triangle is folded so that vertex
touches a point on side
a distance
from point
. The length of the line segment along which the triangle is folded 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 10
There are nonzero integers ,
,
, and
such that the complex number
is a zero of the polynomial
. For each possible combination of
and
, let
be the sum of the zeros of
. Find the sum of the
's for all possible combinations of
and
.
Problem 11
Ms. Math's kindergarten class has registered students. The classroom has a very large number,
, of play blocks which satisfies the conditions:
(a) If ,
, or
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 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 13
Triangle has side lengths
,
, and
. For each positive integer
, points
and
are located on
and
, respectively, creating three similar triangles
. The area of the union of all triangles
for
can be expressed as
, where
and
are relatively prime positive integers. Find
.
Problem 14
For , let
and
so that . Then
where
and
are relatively prime positive integers. 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.