Difference between revisions of "2016 AMC 10A Problems/Problem 23"
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− | ==Problem== | + | == Problem == |
A binary operation <math>\diamondsuit</math> has the properties that <math>a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c</math> and that <math>a\,\diamondsuit \,a=1</math> for all nonzero real numbers <math>a, b,</math> and <math>c</math>. (Here <math>\cdot</math> represents multiplication). The solution to the equation <math>2016 \,\diamondsuit\, (6\,\diamondsuit\, x)=100</math> can be written as <math>\tfrac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. What is <math>p+q?</math> | A binary operation <math>\diamondsuit</math> has the properties that <math>a\,\diamondsuit\, (b\,\diamondsuit \,c) = (a\,\diamondsuit \,b)\cdot c</math> and that <math>a\,\diamondsuit \,a=1</math> for all nonzero real numbers <math>a, b,</math> and <math>c</math>. (Here <math>\cdot</math> represents multiplication). The solution to the equation <math>2016 \,\diamondsuit\, (6\,\diamondsuit\, x)=100</math> can be written as <math>\tfrac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. What is <math>p+q?</math> | ||
<math>\textbf{(A) }109\qquad\textbf{(B) }201\qquad\textbf{(C) }301\qquad\textbf{(D) }3049\qquad\textbf{(E) }33,601</math> | <math>\textbf{(A) }109\qquad\textbf{(B) }201\qquad\textbf{(C) }301\qquad\textbf{(D) }3049\qquad\textbf{(E) }33,601</math> | ||
− | ==Solution== | + | == Solution 1 == |
− | === | + | We see that <math>a \, \diamondsuit \, a = 1</math>, and think of division. Testing, we see that the first condition <math>a \, \diamondsuit \, (b \, \diamondsuit \, c) = (a \, \diamondsuit \, b) \cdot c</math> is satisfied, because <math>\frac{a}{\frac{b}{c}} = \frac{a}{b} \cdot c</math>. Therefore, division can be the operation <math>\diamondsuit</math>. Solving the equation, <cmath>\frac{2016}{\frac{6}{x}} = \frac{2016}{6} \cdot x = 336x = 100\implies x=\frac{100}{336} = \frac{25}{84},</cmath> so the answer is <math>25 + 84 = \boxed{\textbf{(A) }109}</math>. |
− | + | == Solution 2 (Proving that <math>\diamondsuit</math> is division) == | |
+ | If the given conditions hold for all nonzero numbers <math>a, b,</math> and <math>c</math>, | ||
− | == | + | Let <math>a=b=c.</math> From the first two givens, this implies that |
− | |||
− | + | <cmath>a\diamondsuit\, (a\diamondsuit\, {a})=(a\diamondsuit\, a)\cdot{a}.</cmath> | |
+ | |||
+ | |||
+ | From <math>a\diamondsuit\,{a}=1,</math> this equation simply becomes <math>a\diamondsuit\,{1}=a.</math> | ||
+ | |||
+ | |||
+ | Let <math>c=b.</math> Substituting this into the first two conditions, we see that | ||
+ | |||
+ | <cmath>a\diamondsuit\, (b\diamondsuit\, {c})=(a\diamondsuit\, {b})\cdot{c} \implies a\diamondsuit\, (b\diamondsuit\, {b})=(a\diamondsuit\, {b})\cdot{b}.</cmath> | ||
+ | |||
+ | Substituting <math>b\diamondsuit\, {b} =1</math>, the second equation becomes | ||
+ | |||
+ | <cmath>a\diamondsuit\, {1}=(a\diamondsuit\, {b})\cdot{b} \implies a=(a\diamondsuit\,{b})\cdot{b}.</cmath> | ||
+ | |||
+ | Since <math>a, b</math> and <math>c</math> are nonzero, we can divide by <math>b</math> which yields, | ||
+ | |||
+ | <cmath>\frac{a}{b}=(a\diamondsuit\, {b}).</cmath> | ||
+ | |||
+ | Now we can find the value of <math>x</math> straightforwardly: | ||
+ | |||
+ | <cmath>\frac{2016}{(\frac{6}{x})}=100 \implies 2016=\frac{600}{x} \implies x=\frac{600}{2016} = \frac{25}{84}.</cmath> | ||
+ | |||
+ | Therefore, <math>a+b=25+84=\boxed{\textbf{A)} 109}</math> | ||
+ | |||
+ | -Benedict T (countmath1) | ||
− | + | Note: We only really cared about what <math>a\diamondsuit\,{b}</math> was, so we used the existence of <math>c</math> to get an expression in terms of just <math>a</math> and <math>b</math>. | |
+ | == Solution 3 == | ||
One way to eliminate the <math>\diamondsuit</math> in this equation is to make <math>a = b</math> so that <math>a\,\diamondsuit\, (b\,\diamondsuit \,c) = c</math>. In this case, we can make <math>b = 2016</math>. | One way to eliminate the <math>\diamondsuit</math> in this equation is to make <math>a = b</math> so that <math>a\,\diamondsuit\, (b\,\diamondsuit \,c) = c</math>. In this case, we can make <math>b = 2016</math>. | ||
− | <cmath>2016 \,\diamondsuit\, (6\,\diamondsuit\, x)=100 | + | <cmath>2016 \,\diamondsuit\, (6\,\diamondsuit\, x)=100\implies |
(2016\, \diamondsuit\, 6) \cdot x = 100</cmath> | (2016\, \diamondsuit\, 6) \cdot x = 100</cmath> | ||
By multiplying both sides by <math>\frac{6}{x}</math>, we get: | By multiplying both sides by <math>\frac{6}{x}</math>, we get: | ||
− | <cmath>(2016\, \diamondsuit\, 6) \cdot 6 = \frac{600}{x} | + | <cmath>(2016\, \diamondsuit\, 6) \cdot 6 = \frac{600}{x}\implies |
2016 \, \diamondsuit\, (6\, \diamondsuit\, 6) = \frac{600}{x}</cmath> | 2016 \, \diamondsuit\, (6\, \diamondsuit\, 6) = \frac{600}{x}</cmath> | ||
Because <math>6\, \diamondsuit\, 6 = 2016\, \diamondsuit\, 2016 = 1:</math> | Because <math>6\, \diamondsuit\, 6 = 2016\, \diamondsuit\, 2016 = 1:</math> | ||
− | <cmath>2016 \, \diamondsuit\, (2016\, \diamondsuit\, 2016) = \frac{600}{x} | + | <cmath>2016 \, \diamondsuit\, (2016\, \diamondsuit\, 2016) = \frac{600}{x}\implies |
− | (2016\, \diamondsuit\, 2016) \cdot 2016 = \frac{600}{x} | + | (2016\, \diamondsuit\, 2016) \cdot 2016 = \frac{600}{x}\implies |
2016 = \frac{600}{x}</cmath> | 2016 = \frac{600}{x}</cmath> | ||
Therefore, <math>x = \frac{600}{2016} = \frac{25}{84}</math>, so the answer is <math>25 + 84 = \boxed{\textbf{(A) }109.}</math> | Therefore, <math>x = \frac{600}{2016} = \frac{25}{84}</math>, so the answer is <math>25 + 84 = \boxed{\textbf{(A) }109.}</math> | ||
− | ==See Also== | + | ==Solution 4== |
+ | |||
+ | We can manipulate the given identities to arrive at a conclusion about the binary operator <math>\diamondsuit</math>. Substituting <math>b = c</math> into the first identity yields <cmath>( a\ \diamondsuit\ b) \cdot b = a\ \diamondsuit\ (b\ \diamondsuit\ b) = a\ \diamondsuit\ 1 = a\ \diamondsuit\ ( a\ \diamondsuit\ a) = ( a\ \diamondsuit\ a) \cdot a = a.</cmath> Hence, <math>( a\ \diamondsuit\ b) \cdot b = a,</math> or, dividing both sides of the equation by <math>b,</math> <math>( a\ \diamondsuit\ b) = \frac{a}{b}.</math> | ||
+ | |||
+ | Hence, the given equation becomes <math>\frac{2016}{\frac{6}{x}} = 100</math>. Solving yields <math>x=\frac{100}{336} = \frac{25}{84},</math> so the answer is <math>25 + 84 = \boxed{\textbf{(A) }109.}</math> | ||
+ | |||
+ | ==Solution 5== | ||
+ | |||
+ | <math>2016 \diamondsuit (6 \diamondsuit x) = (2016 \diamondsuit 6) \cdot x = 100</math> | ||
+ | |||
+ | <math>2016 \diamondsuit (2016 \diamondsuit 1) = (2016 \diamondsuit 2016) \cdot 1 = 1 \cdot 1 = 1</math> | ||
+ | |||
+ | <math>2016 \diamondsuit 2016 = 1</math>, <math>2016 \diamondsuit (2016 \diamondsuit 1) = 1</math>, so <math>2016 \diamondsuit 1 = 2016</math> | ||
+ | |||
+ | <math>2016 \diamondsuit 1 = (2016 \diamondsuit 6) \cdot 6</math>, <math>2016 \diamondsuit 6 = \frac{2016 \diamondsuit 1}{6} = 336</math> | ||
+ | |||
+ | <math>x = \frac{100}{2016 \diamondsuit 6} = \frac{100}{336} = \frac{25}{84}</math>, <math>24 + 85 = \boxed{\textbf{(A) }109}</math> | ||
+ | |||
+ | ~[https://artofproblemsolving.com/wiki/index.php/User:Isabelchen isabelchen] | ||
+ | |||
+ | ==Solution 6 (Fast)== | ||
+ | Notice that <math>2016 \diamondsuit (6 \diamondsuit 6)=(2016 \diamondsuit 6) \cdot 6 = 2016</math>. Hence, <math>2016 \diamondsuit 6 = 336</math>. Thus, <math>2016 \diamondsuit (6 \diamondsuit x)=100 \implies (2016 \diamondsuit 6) \cdot x = 100 \implies 336x=100 \implies x=\frac{25}{84}</math>. Therefore, the answer is <math>\boxed{\textbf{(A) }109}</math> | ||
+ | |||
+ | ~Mathmagicops | ||
+ | |||
+ | ==Solution 7== | ||
+ | We can rewrite <math>2016\diamondsuit (6\diamondsuit x)=100</math> as <math>2016\diamondsuit 6=\dfrac{100}x.</math> We do a series of algebraic manipulations: | ||
+ | |||
+ | <cmath>(2016\diamondsuit 6)\cdot6=\dfrac{600}x</cmath> | ||
+ | <cmath>2016\diamondsuit (6\diamondsuit 6)=\dfrac{600}x</cmath> | ||
+ | <cmath>2016\diamondsuit 1=\dfrac{600}x.</cmath> | ||
+ | |||
+ | Let <math>a=2016\diamondsuit 1.</math> | ||
+ | |||
+ | <cmath>2016a=(2016\diamondsuit 1)\cdot 2016</cmath> | ||
+ | <cmath>2016a=2016\diamondsuit (1\diamondsuit 2016)</cmath> | ||
+ | |||
+ | We let <math>b=1\diamondsuit 2016.</math> | ||
+ | |||
+ | <cmath>2016b=(1\diamondsuit 2016)\cdot2016</cmath> | ||
+ | <cmath>2016b=1\diamondsuit (2016\diamondsuit 2016)</cmath> | ||
+ | <cmath>2016b=1\diamondsuit 1=1</cmath> | ||
+ | <cmath>b=\dfrac1{2016}</cmath> | ||
+ | |||
+ | We notice that <math>1\diamondsuit 2016=\dfrac1{2016},</math> which makes <math>a\diamondsuit b</math> look suspiciously like <math>\dfrac ab.</math> Sure enough, when we try the given condition on <math>a\diamondsuit b=\dfrac ab,</math> we see that it works. We evaluate <math>2016\diamondsuit (6\diamondsuit x)=100,</math> and get <math>x=\dfrac{25}{84},</math> and therefore <math>p+q=25+84=109.</math> | ||
+ | |||
+ | ~Technodoggo | ||
+ | |||
+ | == Video Solution 1== | ||
+ | https://www.youtube.com/watch?v=8GULAMwu5oE | ||
+ | |||
+ | == Video Solution 2(Meta-Solving Technique) == | ||
+ | https://youtu.be/GmUWIXXf_uk?t=1632 | ||
+ | |||
+ | ~ pi_is_3.14 | ||
+ | |||
+ | == See Also == | ||
{{AMC10 box|year=2016|ab=A|num-b=22|num-a=24}} | {{AMC10 box|year=2016|ab=A|num-b=22|num-a=24}} | ||
{{AMC12 box|year=2016|ab=A|num-b=19|num-a=21}} | {{AMC12 box|year=2016|ab=A|num-b=19|num-a=21}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 21:43, 14 September 2023
Contents
Problem
A binary operation has the properties that and that for all nonzero real numbers and . (Here represents multiplication). The solution to the equation can be written as , where and are relatively prime positive integers. What is
Solution 1
We see that , and think of division. Testing, we see that the first condition is satisfied, because . Therefore, division can be the operation . Solving the equation, so the answer is .
Solution 2 (Proving that is division)
If the given conditions hold for all nonzero numbers and ,
Let From the first two givens, this implies that
From this equation simply becomes
Let Substituting this into the first two conditions, we see that
Substituting , the second equation becomes
Since and are nonzero, we can divide by which yields,
Now we can find the value of straightforwardly:
Therefore,
-Benedict T (countmath1)
Note: We only really cared about what was, so we used the existence of to get an expression in terms of just and .
Solution 3
One way to eliminate the in this equation is to make so that . In this case, we can make .
By multiplying both sides by , we get:
Because
Therefore, , so the answer is
Solution 4
We can manipulate the given identities to arrive at a conclusion about the binary operator . Substituting into the first identity yields Hence, or, dividing both sides of the equation by
Hence, the given equation becomes . Solving yields so the answer is
Solution 5
, , so
,
,
Solution 6 (Fast)
Notice that . Hence, . Thus, . Therefore, the answer is
~Mathmagicops
Solution 7
We can rewrite as We do a series of algebraic manipulations:
Let
We let
We notice that which makes look suspiciously like Sure enough, when we try the given condition on we see that it works. We evaluate and get and therefore
~Technodoggo
Video Solution 1
https://www.youtube.com/watch?v=8GULAMwu5oE
Video Solution 2(Meta-Solving Technique)
https://youtu.be/GmUWIXXf_uk?t=1632
~ pi_is_3.14
See Also
2016 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 22 |
Followed by Problem 24 | |
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 10 Problems and Solutions |
2016 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 19 |
Followed by Problem 21 |
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.