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Difference between revisions of "1973 USAMO Problems/Problem 2"

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<center> <math>X_0=1</math>, <math>X_1=1</math>, <math>X_{n+1}=X_n+2X_{n-1}</math> <math>(n=1,2,3,\dots),</math></center>
 
<center> <math>X_0=1</math>, <math>X_1=1</math>, <math>X_{n+1}=X_n+2X_{n-1}</math> <math>(n=1,2,3,\dots),</math></center>
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<center><math>Y_0=1</math>, <math>Y_1=1</math>, <math>Y_{n+1}=2Y_n+3Y_{n-1}</math> <math>(n=1,2,3,\dots)</math>.</center>
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<center><math>Y_0=1</math>, <math>Y_1=7</math>, <math>Y_{n+1}=2Y_n+3Y_{n-1}</math> <math>(n=1,2,3,\dots)</math>.</center>
  
 
Thus, the first few terms of the sequences are:
 
Thus, the first few terms of the sequences are:
  
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<center> <math>1</math>, <math>1</math>, <math>3</math>, <math>5</math>, <math>11</math>, <math>21</math>, <math>\dots</math>,</center>
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<center> <math>X:1, 1, 3, 5, 11, 21, \dots</math>,</center>
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<center> <math>1</math>, <math>7</math>, <math>17</math>, <math>55</math>, <math>161</math>, <math>487</math>, <math>\dots</math>.</center>
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<center> <math>Y:1, 7, 17, 55, 161, 487, \dots</math>.</center>
  
 
Prove that, except for the "1", there is no term which occurs in both sequences.
 
Prove that, except for the "1", there is no term which occurs in both sequences.

Revision as of 14:18, 30 December 2008

Problem

Let $\{X_n\}$ and $\{Y_n\}$ denote two sequences of integers defined as follows:

$X_0=1$, $X_1=1$, $X_{n+1}=X_n+2X_{n-1}$ $(n=1,2,3,\dots),$
$Y_0=1$, $Y_1=7$, $Y_{n+1}=2Y_n+3Y_{n-1}$ $(n=1,2,3,\dots)$.

Thus, the first few terms of the sequences are:

$X:1, 1, 3, 5, 11, 21, \dots$,
$Y:1, 7, 17, 55, 161, 487, \dots$.

Prove that, except for the "1", there is no term which occurs in both sequences.

Solution

We can look at each sequence $\bmod{8}$:

$X$: $1$, $1$, $3$, $5$, $3$, $5$, $\dots$,
$Y$: $1$, $7$, $1$, $7$, $1$, $7$, $\dots$.
  • Proof that $X$ repeats $\bmod{8}$:

The third and fourth terms are $3$ and $5$ $\bmod{8}$. Plugging into the formula, we see that the next term is $11\equiv 3\bmod{8}$, and plugging $5$ and $3$, we get that the next term is $13\equiv 5\bmod{8}$. Thus the sequence $X$ repeats, and the pattern is $3,5,3,5,\dots$.

  • Proof that $Y$ repeats $\bmod{8}$:

The first and second terms are $1$ and $7$ $\bmod{8}$. Plugging into the formula, we see that the next term is $17\equiv 1\bmod{8}$, and plugging $7$ and $1$, we get that the next term is $23\equiv 7\bmod{8}$. Thus the sequence $Y$ repeats, and the pattern is $1,7,1,7,\dots$.


Combining both results, we see that $X_i$ and $Y_j$ are not congruent $\bmod{8}$ when $i\geq 3$ and $j\geq 2$. Thus after the "1", the terms of each sequence are not equal.

See also

1973 USAMO (Problems • Resources)
Preceded by
Problem 1 		Followed by
Problem 3
1 • 2 • 3 • 4 • 5
All USAMO Problems and Solutions
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