Difference between revisions of "2018 AIME II Problems"
Icematrix2 (talk | contribs) (Tag: Undo) |
|||
(10 intermediate revisions by 5 users not shown) | |||
Line 9: | Line 9: | ||
==Problem 2== | ==Problem 2== | ||
− | Let <math>a_{0} = 2</math>, <math>a_{1} = 5</math>, and <math>a_{2} = 8</math>, and for <math>n > 2</math> define <math>a_{n}</math> recursively to be the remainder when <math>4 | + | Let <math>a_{0} = 2</math>, <math>a_{1} = 5</math>, and <math>a_{2} = 8</math>, and for <math>n > 2</math> define <math>a_{n}</math> recursively to be the remainder when <math>4(a_{n-1} + a_{n-2} + a_{n-3})</math> is divided by <math>11</math>. Find <math>a_{2018} \cdot a_{2020} \cdot a_{2022}</math>. |
[[2018 AIME II Problems/Problem 2 | Solution]] | [[2018 AIME II Problems/Problem 2 | Solution]] | ||
Line 21: | Line 21: | ||
==Problem 4== | ==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 = | + | 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]] | [[2018 AIME II Problems/Problem 4 | Solution]] | ||
Line 35: | Line 35: | ||
A real number <math>a</math> is chosen randomly and uniformly from the interval <math>[-20, 18]</math>. The probability that the roots of the polynomial | A real number <math>a</math> is chosen randomly and uniformly from the interval <math>[-20, 18]</math>. The probability that the roots of the polynomial | ||
− | < | + | <cmath>x^4 + 2ax^3 + (2a - 2)x^2 + (-4a + 3)x - 2</cmath> |
are all real can be written in the form <math>\dfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. | are all real can be written in the form <math>\dfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</math>. | ||
Line 93: | Line 93: | ||
Find the number of functions <math>f</math> from <math>\{0, 1, 2, 3, 4, 5, 6\}</math> to the integers such that <math>f(0) = 0</math>, <math>f(6) = 12</math>, and | Find the number of functions <math>f</math> from <math>\{0, 1, 2, 3, 4, 5, 6\}</math> to the integers such that <math>f(0) = 0</math>, <math>f(6) = 12</math>, and | ||
− | < | + | <cmath>|x - y| \leq |f(x) - f(y)| \leq 3|x - y|</cmath> |
for all <math>x</math> and <math>y</math> in <math>\{0, 1, 2, 3, 4, 5, 6\}</math>. | for all <math>x</math> and <math>y</math> in <math>\{0, 1, 2, 3, 4, 5, 6\}</math>. | ||
Line 99: | Line 99: | ||
[[2018 AIME II Problems/Problem 15 | Solution]] | [[2018 AIME II Problems/Problem 15 | Solution]] | ||
− | {{AIME box|year=2018|n=II|before=[[2018 AIME I]]|after=[[2019 AIME I]]}} | + | {{AIME box|year=2018|n=II|before=[[2018 AIME I Problems]]|after=[[2019 AIME I Problems]]}} |
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 18:25, 9 October 2020
2018 AIME II (Answer Key) | AoPS Contest Collections | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
Problem 1
Points , , and lie in that order along a straight path where the distance from to is meters. Ina runs twice as fast as Eve, and Paul runs twice as fast as Ina. The three runners start running at the same time with Ina starting at and running toward , Paul starting at and running toward , and Eve starting at and running toward . When Paul meets Eve, he turns around and runs toward . Paul and Ina both arrive at at the same time. Find the number of meters from to .
Problem 2
Let , , and , and for define recursively to be the remainder when is divided by . Find .
Problem 3
Find the sum of all positive integers such that the base- integer is a perfect square and the base- integer is a perfect cube.
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 6
A real number is chosen randomly and uniformly from the interval . The probability that the roots of the polynomial
are all real can be written in the form , where and are relatively prime positive integers. Find .
Problem 7
Triangle has side lengths , , and . Points are on segment with between and for , and points are on segment with between and for . Furthermore, each segment , , is parallel to . The segments cut the triangle into regions, consisting of trapezoids and triangle. Each of the regions has the same area. Find the number of segments , , that have rational length.
Problem 8
A frog is positioned at the origin of the coordinate plane. From the point , the frog can jump to any of the points , , , or . Find the number of distinct sequences of jumps in which the frog begins at and ends at .
Problem 9
Octagon with side lengths and is formed by removing 6-8-10 triangles from the corners of a rectangle with side on a short side of the rectangle, as shown. Let be the midpoint of , and partition the octagon into 7 triangles by drawing segments , , , , , and . Find the area of the convex polygon whose vertices are the centroids of these 7 triangles.
Problem 10
Find the number of functions from to that satisfy for all in .
Problem 11
Find the number of permutations of such that for each with , at least one of the first terms of the permutation is greater than .
Problem 12
Let be a convex quadrilateral with , , and . Assume that the diagonals of intersect at point , and that the sum of the areas of triangles and equals the sum of the areas of triangles and . Find the area of quadrilateral .
Problem 13
Misha rolls a standard, fair six-sided die until she rolls 1-2-3 in that order on three consecutive rolls. The probability that she will roll the die an odd number of times is where and are relatively prime positive integers. Find .
Problem 14
The incircle of triangle is tangent to at . Let be the other intersection of with . Points and lie on and , respectively, so that is tangent to at . Assume that , , , and , where and are relatively prime positive integers. Find .
Problem 15
Find the number of functions from to the integers such that , , and
for all and in .
2018 AIME II (Problems • Answer Key • Resources) | ||
Preceded by 2018 AIME I Problems |
Followed by 2019 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.