Difference between revisions of "2014 AIME I Problems"
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==Problem 7== | ==Problem 7== | ||
− | + | Let w and z be complex numbers such that |w| = 1 and |z| = 10. Let \theta = \arg(\frac{w-z}{z}). The maximum possible value of \tan^2 \theta can be written as \frac{p}{q}, where p and q are relatively proime positive integers. Find p+q. (Note that \arg(w), for w \neq 0, denotes the measure of the agle that the ray from 0 to w makes with the positive real axis in the complex plane. | |
[[2014 AIME I Problems/Problem 7|Solution]] | [[2014 AIME I Problems/Problem 7|Solution]] |
Revision as of 12:11, 14 March 2014
2014 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 |
Contents
[hide]Problem 1
Problem 2
An urn contains green balls and
blue balls. A second urn contains
green balls and
blue balls. A single ball is drawn at random from each urn. The probability that both balls are of the same color is 0.58. Find
.
Problem 3
Find the number of rational numbers
such that when
is written as a fraction in lowest terms, the numerator and the denominator have a sum of 1000.
Problem 4
Jon and Steve ride their bicycles along a path that parallels two side-by-side train tracks running the east/west direction. Jon rides east as 20 miles per hour, and Steve rides west at 20 miles per hour. Two trains of equal length, traveling in opposite directions at constant but different speeds each pass the two riders. Each train takes exactly 1 minute to go past Jon. The westbound train takes 10 times as long as the eastbound train to go past Steve. The length of each train is , where
and
are relatively prime positive integers. Find
.
Problem 5
Problem 6
The graphs and
have y-intercepts of 2013 and 2014, respectively, and each graph has two positive integer x-intercepts. Find
.
Problem 7
Let w and z be complex numbers such that |w| = 1 and |z| = 10. Let \theta = \arg(\frac{w-z}{z}). The maximum possible value of \tan^2 \theta can be written as \frac{p}{q}, where p and q are relatively proime positive integers. Find p+q. (Note that \arg(w), for w \neq 0, denotes the measure of the agle that the ray from 0 to w makes with the positive real axis in the complex plane.
Problem 8
The positive integers and
both end in the same sequence of four digits
when written in base 10, where digit a is not zero. Find the three-digit number
.
Problem 9
Let be the three real roots of the equation
. Find
.
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Let be the largest real solution to the equation
There are positive integers and
such that
. Find
.
Problem 15
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.