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Difference between revisions of "1963 AHSME Problems/Problem 13"

(Solution to Problem 13 — pretty tough!)
 
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==Solution==
 
==Solution==
  
−
Assume <math>c,d \ge 0</math>, and WLOG, assume <math>a<0</math> and <math>a \le b</math>.  This also takes into accout when <math>b</math> is negative.  That means
+
Assume <math>c,d \ge 0</math>, and WLOG, assume <math>a<0</math> and <math>a \le b</math>.  This also takes into account when <math>b</math> is negative.  That means
−
<cmath>\frac{1}{2^-a} + 2^b = 3^c + 3^d</cmath>
+
<cmath>\frac{1}{2^{-a}} + 2^b = 3^c + 3^d</cmath>
−
Multiply both sides by <math>2^-a</math> to get
+
Multiply both sides by <math>2^{-a}</math> to get
−
<cmath>1 + 2^{-a+b} = 2^-a (3^c + 3^d)</cmath>
+
<cmath>1 + 2^{-a+b} = 2^{-a} (3^c + 3^d)</cmath>
−
Note that both sides are integers.  If <math>a \ne b</math>, then the right side is even while the left side is odd, so equality can not happen.  If <math>a = b</math>, then <math>2^-a (3^c + 3^d) = 2</math>, and since  <math>a<0</math>, <math>a = -1</math> and <math>3^c + 3^d = 1</math>.  No nonnegative value of <math>c</math> and <math>d</math> works, so equality can not happen.  Thus, <math>a</math> and <math>b</math> can not be negative when <math>c,d \ge 0</math>.
+
Note that both sides are integers.  If <math>a \ne b</math>, then the right side is even while the left side is odd, so equality can not happen.  If <math>a = b</math>, then <math>2^{-a} (3^c + 3^d) = 2</math>, and since  <math>a<0</math>, <math>a = -1</math> and <math>3^c + 3^d = 1</math>.  No nonnegative value of <math>c</math> and <math>d</math> works, so equality can not happen.  Thus, <math>a</math> and <math>b</math> can not be negative when <math>c,d \ge 0</math>.
  
 
Assume <math>a,b \ge 0</math>, and WLOG, assume <math>c < 0</math> and <math>c \le d</math>.  This also takes into account when <math>d</math> is negative.  That means
 
Assume <math>a,b \ge 0</math>, and WLOG, assume <math>c < 0</math> and <math>c \le d</math>.  This also takes into account when <math>d</math> is negative.  That means
−
<cmath>2^a + 2^b = \frac{1}{3^-c} + 3^d</cmath>
+
<cmath>2^a + 2^b = \frac{1}{3^{-c}} + 3^d</cmath>
−
Multiply both sides by <math>3^-c</math> to get
+
Multiply both sides by <math>3^{-c}</math> to get
−
<cmath>3^-c (2^a + 2^b) = 1 + 3^{d-c}</cmath>
+
<cmath>3^{-c} (2^a + 2^b) = 1 + 3^{d-c}</cmath>
 
That makes both sides integers.  The left side is congruent to <math>0</math> modulo <math>3</math> while the right side is congruent to <math>1</math> or <math>2</math> modulo <math>3</math>, so equality can not happen.  Thus, <math>c</math> and <math>d</math> can not be negative when <math>a,b \ge 0</math>.
 
That makes both sides integers.  The left side is congruent to <math>0</math> modulo <math>3</math> while the right side is congruent to <math>1</math> or <math>2</math> modulo <math>3</math>, so equality can not happen.  Thus, <math>c</math> and <math>d</math> can not be negative when <math>a,b \ge 0</math>.
  
 
Assume <math>a,c < 0</math>, and WLOG, let <math>a \le b</math> and <math>c \le d</math>.  This also takes into account when <math>b</math> or <math>d</math> is negative.  That means
 
Assume <math>a,c < 0</math>, and WLOG, let <math>a \le b</math> and <math>c \le d</math>.  This also takes into account when <math>b</math> or <math>d</math> is negative.  That means
−
<cmath>\frac{1}{2^-a} + 2^b = \frac{1}{3^-c} + 3^d</cmath>
+
<cmath>\frac{1}{2^{-a}} + 2^b = \frac{1}{3^{-c}} + 3^d</cmath>
−
Multiply both sides by <math>2^-a \cdot 3^-c</math> to get
+
Multiply both sides by <math>2^{-a} \cdot 3^{-c}</math> to get
−
<cmath>3^-c (1 + 2^{b-a}) = 2^-a (1 + 3^{d-c})</cmath>
+
<cmath>3^{-c} (1 + 2^{b-a}) = 2^{-a} (1 + 3^{d-c})</cmath>
 
That makes both sides integers.  The left side is congruent to <math>0</math> modulo <math>3</math> while the right side is congruent to <math>1</math> or <math>2</math> modulo <math>3</math>, so equality can not happen.  Thus, <math>a</math> and <math>c</math> can not be negative.
 
That makes both sides integers.  The left side is congruent to <math>0</math> modulo <math>3</math> while the right side is congruent to <math>1</math> or <math>2</math> modulo <math>3</math>, so equality can not happen.  Thus, <math>a</math> and <math>c</math> can not be negative.
  

Latest revision as of 11:47, 14 August 2021

Problem

If $2^a+2^b=3^c+3^d$, the number of integers $a,b,c,d$ which can possibly be negative, is, at most:

$\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 1 \qquad \textbf{(E)}\ 0$

Solution

Assume $c,d \ge 0$, and WLOG, assume $a<0$ and $a \le b$. This also takes into account when $b$ is negative. That means \[\frac{1}{2^{-a}} + 2^b = 3^c + 3^d\] Multiply both sides by $2^{-a}$ to get \[1 + 2^{-a+b} = 2^{-a} (3^c + 3^d)\] Note that both sides are integers. If $a \ne b$, then the right side is even while the left side is odd, so equality can not happen. If $a = b$, then $2^{-a} (3^c + 3^d) = 2$, and since $a<0$, $a = -1$ and $3^c + 3^d = 1$. No nonnegative value of $c$ and $d$ works, so equality can not happen. Thus, $a$ and $b$ can not be negative when $c,d \ge 0$.

Assume $a,b \ge 0$, and WLOG, assume $c < 0$ and $c \le d$. This also takes into account when $d$ is negative. That means \[2^a + 2^b = \frac{1}{3^{-c}} + 3^d\] Multiply both sides by $3^{-c}$ to get \[3^{-c} (2^a + 2^b) = 1 + 3^{d-c}\] That makes both sides integers. The left side is congruent to $0$ modulo $3$ while the right side is congruent to $1$ or $2$ modulo $3$, so equality can not happen. Thus, $c$ and $d$ can not be negative when $a,b \ge 0$.

Assume $a,c < 0$, and WLOG, let $a \le b$ and $c \le d$. This also takes into account when $b$ or $d$ is negative. That means \[\frac{1}{2^{-a}} + 2^b = \frac{1}{3^{-c}} + 3^d\] Multiply both sides by $2^{-a} \cdot 3^{-c}$ to get \[3^{-c} (1 + 2^{b-a}) = 2^{-a} (1 + 3^{d-c})\] That makes both sides integers. The left side is congruent to $0$ modulo $3$ while the right side is congruent to $1$ or $2$ modulo $3$, so equality can not happen. Thus, $a$ and $c$ can not be negative.

Putting all the information together, none of $a,b,c,d$ can be negative, so the answer is $\boxed{\textbf{(E)}}$.

See Also

1963 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 12 		Followed by
Problem 14
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 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40
All AHSME Problems and Solutions

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