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Difference between revisions of "Mock AIME 6 2006-2007 Problems/Problem 5"
Line 61: Line 61:
 
NOTE... case 4 is wrong.  Need to rewrite it
 
NOTE... case 4 is wrong.  Need to rewrite it
  
−
'''Case 4:''' <math>2120 \le n \le 2199</math>
+
'''Case 4:''' <math>2110 \le n \le 2199</math>
  
−
Let <math>2 \le k \le 9</math> be the 3rd digit of <math>n</math>
+
Let <math>1 \le k \le 9</math> be the 3rd digit of <math>n</math>
  
 
<math>2100+10k \le n \le 2109+10k</math>, and <math>2^2++1+k^2 \le S(n) \le 2^2+1+k^2+9^2</math>
 
<math>2100+10k \le n \le 2109+10k</math>, and <math>2^2++1+k^2 \le S(n) \le 2^2+1+k^2+9^2</math>
Line 69: Line 69:
 
<math>(k-1)10+93 \le n-2007 \le (k-1)10+102</math>, and <math>5+k^2 \le S(n) \le 86+k^2</math>
 
<math>(k-1)10+93 \le n-2007 \le (k-1)10+102</math>, and <math>5+k^2 \le S(n) \le 86+k^2</math>
  
−
At <math>k=2</math>, <math>10(k-1)+3=193>85+k^2>89</math>.
+
At <math>k=1</math>, <math>10(k-1)+93=93>86+k^2>87</math>.
  
−
At <math>k=3</math>, <math>10(k-1)+3=293>85+k^2>94</math>.
+
At <math>k=2</math>, <math>10(k-1)+93=103>86+k^2>90</math>.
  
−
At <math>k=4</math>, <math>10(k-1)+3=393>85+k^2>101</math>.
+
At <math>k=3</math>, <math>10(k-1)+93=113>86+k^2>95</math>.
  
−
At <math>k=5</math>, <math>10(k-1)+3=493>85+k^2>110</math>.
+
At <math>k=4</math>, <math>10(k-1)+93=123>86+k^2>102</math>.
  
−
At <math>k=6</math>, <math>10(k-1)+3=593>85+k^2>121</math>.
+
At <math>k=5</math>, <math>10(k-1)+93=133>86+k^2>111</math>.
  
−
At <math>k=7</math>, <math>10(k-1)+3=693>85+k^2>134</math>.
+
At <math>k=6</math>, <math>10(k-1)+93=143>86+k^2>122</math>.
  
−
At <math>k=8</math>, <math>10(k-1)+3=793>85+k^2>149</math>.
+
At <math>k=7</math>, <math>10(k-1)+93=153>86+k^2>135</math>.
  
−
At <math>k=9</math>, <math>10(k-1)+3=893>85+k^2>166</math>.
+
At <math>k=8</math>, <math>10(k-1)+93=163>86+k^2>150</math>.
  
−
Since <math>10(k-1)+3 > 85+k^2</math>, for <math>2 \le k \le 9</math>, then <math>n-2007\not\le S(n)</math> and there is '''no possible <math>n</math>''' when <math>n \ge 2120</math> when combined with the previous cases.
+
At <math>k=9</math>, <math>10(k-1)+93=173>86+k^2>167</math>.
  
 +
Since <math>10(k-1)+93 > 85+k^2</math>, for <math>1 \le k \le 9</math>, then <math>n-2007\not\le S(n)</math> and there is '''no possible <math>n</math>''' when <math>n \ge 2110</math> when combined with the previous cases.
 +
 +
'''Case 5:'''
  
 
...ongoing writing of solution...
 
...ongoing writing of solution...
  
 
~Tomas Diaz. orders@tomasdiaz.com
 
~Tomas Diaz. orders@tomasdiaz.com

Revision as of 16:06, 24 November 2023

Problem

Let $S(n)$ be the sum of the squares of the digits of $n$. How many positive integers $n>2007$ satisfy the inequality $n-S(n)\le 2007$?

Solution

We start by rearranging the inequality the following way:

$n-2007\le S(n)$ and compare the possible values for the left hand side and the right hand side of this inequality.

Case 1: $n$ has 5 digits or more.

Let $d$ = number of digits of n.

Then as a function of d,

$10^d \le n < 10^{d+1}-1$, and $1 \le S(n) \le 9^2d$

$10^d - 2007 \le n-2007 < 10^{d+1}-2008$, and $1 \le S(n) \le 81d$

when $d \ge 5$,

$10^d - 2007 \ge 10^5 -2007$

$10^d - 2007 \ge 10^5 -2007 > 81d$

Since $10^d - 2007 > 81d$ for $d \ge 5$, then $n-2007\not\le S(n)$ and there is no possible $n$ when $n$ has 5 or more digits.

Case 2: $n$ has 4 digits and $n \ge 3000$

$3000 \le n \le 9999$, and $3^2 \le S(n) \le 3^2+3 \times 9^2$

$993 \le n-2007 \le 7992$, and $9 \le S(n) \le 252$

Since $993 > 252$, then $n-2007\not\le S(n)$ and there is no possible $n$ when $n$ has 4 digits and $n \ge 3000$.

Case 3: $2200 \le n \le 2999$

Let $2 \le k \le 9$ be the 2nd digit of $n$

$2000+100k \le n \le 2099+100k$, and $2^2+k^2 \le S(n) \le 2^2+k^2+2 \times 9^2$

$(k-1)100+93 \le n-2007 \le (k-1)100+92$, and $4+k^2 \le S(n) \le 166+k^2$

At $k=2$, $100(k-1)+93=193>166+k^2>170$.

At $k=3$, $100(k-1)+93=293>166+k^2>175$.

At $k=4$, $100(k-1)+93=393>166+k^2>182$.

At $k=5$, $100(k-1)+93=493>166+k^2>191$.

At $k=6$, $100(k-1)+93=593>166+k^2>202$.

At $k=7$, $100(k-1)+93=693>166+k^2>215$.

At $k=8$, $100(k-1)+93=793>166+k^2>230$.

At $k=9$, $100(k-1)+93=893>166+k^2>247$.

Since $100(k-1)+93 > 166+k^2$, for $2 \le k \le 9$, then $n-2007\not\le S(n)$ and there is no possible $n$ when $n \ge 2200$ when combined with the previous cases.

NOTE... case 4 is wrong. Need to rewrite it

Case 4: $2110 \le n \le 2199$

Let $1 \le k \le 9$ be the 3rd digit of $n$

$2100+10k \le n \le 2109+10k$, and $2^2++1+k^2 \le S(n) \le 2^2+1+k^2+9^2$

$(k-1)10+93 \le n-2007 \le (k-1)10+102$, and $5+k^2 \le S(n) \le 86+k^2$

At $k=1$, $10(k-1)+93=93>86+k^2>87$.

At $k=2$, $10(k-1)+93=103>86+k^2>90$.

At $k=3$, $10(k-1)+93=113>86+k^2>95$.

At $k=4$, $10(k-1)+93=123>86+k^2>102$.

At $k=5$, $10(k-1)+93=133>86+k^2>111$.

At $k=6$, $10(k-1)+93=143>86+k^2>122$.

At $k=7$, $10(k-1)+93=153>86+k^2>135$.

At $k=8$, $10(k-1)+93=163>86+k^2>150$.

At $k=9$, $10(k-1)+93=173>86+k^2>167$.

Since $10(k-1)+93 > 85+k^2$, for $1 \le k \le 9$, then $n-2007\not\le S(n)$ and there is no possible $n$ when $n \ge 2110$ when combined with the previous cases.

Case 5:

...ongoing writing of solution...

~Tomas Diaz. orders@tomasdiaz.com

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