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Difference between revisions of "2017 AIME I Problems"

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==Problem 3==
 
==Problem 3==
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For a positive integer <math>n</math>, let <math>d_n</math> be the units digit of <math>1 + 2 + \dots + n</math>. Find the remainder when
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<cmath>\sum_{n=1}^{2017} d_n</cmath>is divided by <math>1000</math>.
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[[2017 AIME I Problems/Problem 3 | Solution]]
 
[[2017 AIME I Problems/Problem 3 | Solution]]
  

Revision as of 14:30, 8 March 2017

2017 AIME I (Answer Key)
Printable version | AoPS Contest Collections • PDF

Instructions

  1. This is a 15-question, 3-hour examination. All answers are integers ranging from $000$ to $999$, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.
  2. No aids other than scratch paper, graph paper, ruler, compass, and protractor are permitted. In particular, calculators and computers are not permitted.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

Problem 1

Fifteen distinct points are designated on $\triangle ABC$: the 3 vertices $A$, $B$, and $C$; $3$ other points on side $\overline{AB}$; $4$ other points on side $\overline{BC}$; and $5$ other points on side $\overline{CA}$. Find the number of triangles with positive area whose vertices are among these $15$ points.

Solution

Problem 2

When each of 702, 787, and 855 is divided by the positive integer $m$, the remainder is always the positive integer $r$. When each of 412, 722, and 815 is divided by the positive integer $n$, the remainder is always the positive integer $s \neq r$. Fine $m+n+r+s$.

Solution

Problem 3

For a positive integer $n$, let $d_n$ be the units digit of $1 + 2 + \dots + n$. Find the remainder when \[\sum_{n=1}^{2017} d_n\]is divided by $1000$.

Solution

Problem 4

Solution

Problem 5

Solution

Problem 6

Solution

Problem 7

Solution

Problem 8

Solution

Problem 9

Solution

Problem 10

Solution

Problem 11

Solution

Problem 12

Solution

Problem 13

Solution

Problem 14

Solution

Problem 15

Solution

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

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