Difference between revisions of "2020 AMC 12A Problems/Problem 25"
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==Remarks of Solution 2 and Video Solution 3== | ==Remarks of Solution 2 and Video Solution 3== | ||
===Graph=== | ===Graph=== | ||
+ | Let <math>f(x)=\lfloor x \rfloor \cdot \{x\}</math> and <math>g(x)=a \cdot x^2.</math> We make the following table of values: | ||
+ | |||
+ | <cmath>\begin{array}{c|c|c|l} | ||
+ | \boldsymbol{x} & \boldsymbol{\lfloor x \rfloor} & \boldsymbol{f(x)} & \textbf{\ \ \ Equation} \\ [1.5ex] | ||
+ | \hline | ||
+ | & & & \\ [-1ex] | ||
+ | [0,1) & 0 & 0 & y=0 \\ [1.5ex] | ||
+ | [1,2) & 1 & [0,1) & y=x-1 \\ [1.5ex] | ||
+ | [2,3) & 2 & [0,2) & y=2x-4 \\ [1.5ex] | ||
+ | [3,4) & 3 & [0,3) & y=3x-9 \\ [1.5ex] | ||
+ | [4,5) & 4 & [0,4) & y=4x-16 \\ [1.5ex] | ||
+ | \cdots & \cdots & \cdots & \ \ \ \ \ \ \ \cdots \\ [1.5ex] | ||
+ | [m,m+1) & m & [0,m) & y=mx-m^2 | ||
+ | \end{array}</cmath> | ||
+ | |||
+ | We graph <math>f(x)</math> by branches: | ||
+ | |||
+ | [[File:2020 AMC 12A Problem 25.png|center]] | ||
+ | |||
+ | ~MRENTHUSIASM (Graph by Desmos: https://www.desmos.com/calculator/ouvaiqjdzj) | ||
− | |||
===Analysis=== | ===Analysis=== | ||
Revision as of 14:38, 7 March 2021
Contents
Problem
The number , where
and
are relatively prime positive integers, has the property that the sum of all real numbers
satisfying
is
, where
denotes the greatest integer less than or equal to
and
denotes the fractional part of
. What is
?
Solution 1
Let be the unique solution in this range. Note that
is also a solution as long as
, hence all our solutions are
for some
. This sum
must be between
and
, which gives
and
. Plugging this back in gives
.
Solution 2
First note that when
while
. Thus we only need to look at positive solutions (
doesn't affect the sum of the solutions).
Next, we breakdown
down for each interval
, where
is a positive integer. Assume
, then
. This means that when
,
. Setting this equal to
gives
We're looking at the solution with the positive
, which is
. Note that if
is the greatest
such that
has a solution, the sum of all these solutions is slightly over
, which is
when
, just under
. Checking this gives
~ktong
Video Solution 1 (Geometry)
This video shows how things like The Pythagorean Theorem and The Law of Sines work together to solve this seemingly algebraic problem: https://www.youtube.com/watch?v=6IJ7Jxa98zw&feature=youtu.be
Video Solution 2
https://www.youtube.com/watch?v=xex8TBSzKNE ~ MathEx
Video Solution 3 (by Art of Problem-Solving)
https://www.youtube.com/watch?v=7_mdreGBPvg&t=428s&ab_channel=ArtofProblemSolving
Created by Richard Rusczyk
Remarks of Solution 2 and Video Solution 3
Graph
Let and
We make the following table of values:
We graph by branches:
~MRENTHUSIASM (Graph by Desmos: https://www.desmos.com/calculator/ouvaiqjdzj)
Analysis
~MRENTHUSIASM
See Also
2020 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 24 |
Followed by Last Problem |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.