Difference between revisions of "2017 AIME I Problems"
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==Problem 9== | ==Problem 9== | ||
+ | Let <math>a_{10} = 10</math>, and for each integer <math>n >10</math> let <math>a_n = 100a_{n - 1} + n</math>. Find the least <math>n > 10</math> such that <math>a_n</math> is a multiple of <math>99</math>. | ||
+ | |||
[[2017 AIME I Problems/Problem 9 | Solution]] | [[2017 AIME I Problems/Problem 9 | Solution]] | ||
Revision as of 14:32, 8 March 2017
2017 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
[hide]Problem 1
Fifteen distinct points are designated on : the 3 vertices
,
, and
;
other points on side
;
other points on side
; and
other points on side
. Find the number of triangles with positive area whose vertices are among these
points.
Problem 2
When each of 702, 787, and 855 is divided by the positive integer , the remainder is always the positive integer
. When each of 412, 722, and 815 is divided by the positive integer
, the remainder is always the positive integer
. Fine
.
Problem 3
For a positive integer , let
be the units digit of
. Find the remainder when
is divided by
.
Problem 4
A pyramid has a triangular base with side lengths ,
, and
. The three edges of the pyramid from the three corners of the base to the fourth vertex of the pyramid all have length
. The volume of the pyramid is
, where
and
are positive integers, and
is not divisible by the square of any prime. Find
.
Problem 5
A rational number written in base eight is , where all digits are nonzero. The same number in base twelve is
. Find the base-ten number
.
Problem 6
A circle is circumscribed around an isosceles triangle whose two congruent angles have degree measure . Two points are chosen independently and uniformly at random on the circle, and a chord is drawn between them. The probability that the chord intersects the triangle is
. Find the difference between the largest and smallest possible values of
.
Problem 7
For nonnegative integers and
with
, let
. Let
denote the sum of all
, where
and
are nonnegative integers with
. Find the remainder when
is divided by
.
Problem 8
Two real numbers and
are chosen independently and uniformly at random from the interval
. Let
and
be two points on the plane with
. Let
and
be on the same side of line
such that the degree measures of
and
are
and
respectively, and
and
are both right angles. The probability that
is equal to
, where
and
are relatively prime positive integers. Find
.
Problem 9
Let , and for each integer
let
. Find the least
such that
is a multiple of
.
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
2017 AIME I (Problems • Answer Key • Resources) | ||
Preceded by 2016 AIME II |
Followed by 2017 AIME II | |
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.