Difference between revisions of "2017 AIME I Problems"
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==Problem 7== | ==Problem 7== | ||
| + | For nonnegative integers <math>a</math> and <math>b</math> with <math>a + b \leq 6</math>, let <math>T(a, b) = \binom{6}{a} \binom{6}{b} \binom{6}{a + b}</math>. Let <math>S</math> denote the sum of all <math>T(a, b)</math>, where <math>a</math> and <math>b</math> are nonnegative integers with <math>a + b \leq 6</math>. Find the remainder when <math>S</math> is divided by <math>1000</math>. | ||
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[[2017 AIME I Problems/Problem 7 | Solution]] | [[2017 AIME I Problems/Problem 7 | Solution]] | ||
Revision as of 15:31, 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 | ||
to 
, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.Contents
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
![\[\sum_{n=1}^{2017} d_n\]](http://latex.artofproblemsolving.com/c/2/5/c25a797fd18625aea3a70ae1b6906027405232ff.png)
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
Problem 9
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. 
