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Difference between revisions of "2023 AIME I Problems"

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==Problem 10==
 
==Problem 10==
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These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.
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There exists a unique positive integer <math>a</math> such that the sum
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<cmath>
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U=\sum_{n=1}^{2023}\left\lfloor\frac{n^2-n a}{5}\right\rfloor
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</cmath>
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is an integer strictly between <math>-1000</math> and 1000 . For that unique <math>a</math>, find <math>a+U</math>. Here <math>\lfloor x\rfloor</math> is the greatest integer less than or equal to <math>x</math>.
  
 
[[2023 AIME I Problems/Problem 10|Solution]]
 
[[2023 AIME I Problems/Problem 10|Solution]]

Revision as of 14:27, 8 February 2023

2023 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, rulers and compasses are permitted. In particular, graph paper, protractors, calculators and computers are not permitted.
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15

Problem 1

Five men and nine women stand equally spaced around a circle in random order. The probability that every man stands diametrically opposite a woman is $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Solution

Problem 2

Positive real numbers $b \not= 1$ and $n$ satisfy the equations \[\sqrt{\log_b n} = \log_b \sqrt{n} \qquad \text{and} \qquad b \cdot \log_b n = \log_b (bn).\] The value of $n$ is $\frac{j}{k},$ where $j$ and $k$ are relatively prime positive integers. Find $j+k.$

Solution

Problem 3

A plane contains $40$ lines, no $2$ of which are parallel. Suppose that there are $3$ points where exactly $3$ lines intersect, $4$ points where exactly $4$ lines intersect, $5$ points where exactly $5$ lines intersect, $6$ points where exactly $6$ lines intersect, and no points where more than $6$ lines intersect. Find the number of points where exactly $2$ lines intersect.

Solution

Problem 4

These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.

Solution

Problem 5

Let $P$ be a point on the circle circumscribing square $ABCD$ that satisfies $PA\cdot PC=56$ and $PB\cdot PD=90$. Find the area of $ABCD$.

Solution

Problem 6

These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.

Solution

Problem 7

Call a positive integer $n$ extra-distinct if the remainders when $n$ is divided by $2, 3, 4, 5,$ and $6$ are distinct. Find the number of extra-distinct positive integers less than $1000$.

Solution

Problem 8

These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.

Solution

Problem 9

Find the number of cubic polynomials $p(x) = x^3 + ax^2 + bx + c$, where $a$, $b$, and $c$ are integers in $\{-20, -19,-18, \dots , 18, 19, 20\}$, such that there is a unique integer $m \neq 2$ with $p(m) = p(2)$.

Solution

Problem 10

There exists a unique positive integer $a$ such that the sum \[U=\sum_{n=1}^{2023}\left\lfloor\frac{n^2-n a}{5}\right\rfloor\] is an integer strictly between $-1000$ and 1000 . For that unique $a$, find $a+U$. Here $\lfloor x\rfloor$ is the greatest integer less than or equal to $x$.

Solution

Problem 11

These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM. Unofficial problem statement has been posted.

Solution

Problem 12

These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.

Solution

Problem 13

These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.

Solution

Problem 14

These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.

Solution

Problem 15

These problems will not be available until the 2023 AIME I is released on February 8th, 2023, at 12:00 AM.

Solution

See also

2023 AIME I (Problems • Answer Key • Resources)
Preceded by
2022 AIME II 		Followed by
2023 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. AMC logo.png

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