Difference between revisions of "2013 AIME I Problems"
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== Problem 5 == | == Problem 5 == | ||
The real root of the equation <math>8x^3-3x^2-3x-1=0</math> can be written in the form <math>\frac{\sqrt[3]{a}+\sqrt[3]{b}+1}{c}</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are positive integers. Find <math>a+b+c</math>. | The real root of the equation <math>8x^3-3x^2-3x-1=0</math> can be written in the form <math>\frac{\sqrt[3]{a}+\sqrt[3]{b}+1}{c}</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are positive integers. Find <math>a+b+c</math>. | ||
| + | |||
| + | == 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. | ||
Revision as of 11:47, 16 March 2013
| 2013 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
|
Instructions
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| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
to 
, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.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
is divisible by 
-
(b) the first and last digits of
are equal, and
-
(c) the sum of the digits of
is divisible by 
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 13 squares shown below, 8 squares are colored red, and the remaining 5 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 90 degrees around the central square is 
, where n is a positive integer. Find n.
Problem 5
The real root of the equation 
can be written in the form ![$\frac{\sqrt[3]{a}+\sqrt[3]{b}+1}{c}$](http://latex.artofproblemsolving.com/f/8/6/f86e8d0df85bcd94247daf8f1f852a7ac5a55d6d.png)
, where 
, 
, and 
are positive integers. Find 
.
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.
