The "Series" Series - Problem 3

Algebra Level 5

A positive integer will be called awesome if it has the following properties:

  1. It ranges between 10000 10000 and 100000 100000 , not inclusive. Therefore an awesome integer must contain 5 5 digits.

  2. The ten thousands digit is less than the thousands digit.

  3. The thousands digit is less than the hundreds digit.

  4. The hundreds digit is less than the tens digit.

  5. The tens digit is less than the ones digit.

Robert writes ALL possible awesome integers on a piece of paper. He then arranges them in ascending order, putting them in a sequence a n a_{n} in which a 1 a_{1} has the smallest value, and a n a_{n} has the biggest value. Find n + a 100 n + a_{100}


The answer is 24915.

Sanchayapol Lewgasamsarn by Sanchayapol Lewgasamsarn , 20, Thailand

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1 solution

T h e a w e s o m e i n t e g e r s p q r s t ‾ s t a r t s f r o m a 1 = 12345 ‾ a n d e n d a t a n = 56789 ‾ . C o n s t r a i n s : − 1 ≤ p ≤ 5. p + 1 ≤ q ≤ 6. q + 1 ≤ r ≤ 7. r + 1 ≤ s ≤ 8. s + 1 ≤ t ≤ 9. A W E S O M E S F O R A L L P O S S I B L E S T A R T O F r T I L L r E N D S A T r = 7. r s t a w e s o m e s H o w m a n y 3 4 5 , 6 , 7 , 8 , 9 12345 , 12346 , 12347 , 12348 , 12349 5 5 6 , 7 , 8 , 9 12356 , 12357 , 12358 , 12359 4 6 7 , 8 , 9 12367 , 12368 , 12369 3 7 8 , 9 12378 , 12379 2 8 9 12389 1 ‾ a 1 = 12345 t o a 15 = 12389 15 r s t a w e s o m e s H o w m a n y 4 5 6 , 7 , 8 , 9 12456 , 12457 , 12458 , 12459 4 6 7 , 8 , 9 12467 , 12468 , 12469 3 7 8 , 9 12478 , 12479 2 8 9 12489 1 ‾ a 16 = 12467 t o a 25 = 12489 10 r s t a w e s o m e s H o w m a n y 5 6 7 , 8 , 9 12567 , 12568 , 12569 3 7 8 , 9 12578 , 12579 2 8 9 12589 1 ‾ a 26 = 12567 t o a 31 = 12589 6 r s t a w e s o m e s H o w m a n y 6 7 8 , 9 12678 , 12679 2 8 9 12689 1 ‾ a 32 = 12678 t o a 34 = 12689 3 r s t a w e s o m e s H o w m a n y 7 8 9 12789 1 ‾ a 35 = 12789 1 A l l r e n d i n r = 7. T A B L E f o r p = 1. W h e n p q ‾ i s S t a r t o f r a w e s o m e s f r o m t o T o t a l a w e s o m e s . 12 ‾ 3 a 1 = 12345 a 35 = 12789 15 + 10 + 6 + 3 + 1 = 35 13 ‾ 4 a 36 = 13456 a 55 = 13789 10 + 6 + 3 + 1 = 20 14 ‾ 5 a 56 = 14567 a 65 = 14789 6 + 3 + 1 = 10 15 ‾ 6 a 66 = 15678 a 69 = 15789 3 + 1 = 4 16 ‾ 7 a 70 = 16789 1 = 1 . S o f o r p = 1 w e h a v e 35 + 20 + 10 + 4 + 1 = 70 a w e s o m e s T A B L E f o r p = 2. W h e n p q ‾ i s S t a r t o f r a w e s o m e s f r o m t o T o t a l a w e s o m e s . o v e r l i n e 23 4 a 71 = 23456 a 90 = 23789 10 + 6 + 3 + 1 = 20 24 ‾ 5 a 91 = 24567 a 100 = 24789 6 + 3 + 1 = 10 25 ‾ 6 a 101 = 25678 a 104 = 25789 3 + 1 = 4 26 ‾ 7 a 105 = 26789 1 = 1 . S o f o r p = 2 w e h a v e 20 + 10 + 4 + 1 = 35 a w e s o m e s . T A B L E f o r p = 3. W h e n p q ‾ i s S t a r t o f r a w e s o m e s f r o m t o T o t a l a w e s o m e s . 34 ‾ 5 a 106 = 34567 a 115 = 34789 6 + 3 + 1 = 10 35 ‾ 6 a 116 = 35678 a 119 = 35789 3 + 1 = 4 36 ‾ 7 a 120 = 36789 1 = 1 . S o f o r p = 3 w e h a v e 10 + 4 + 1 = 15 a w e s o m e s . T A B L E f o r p = 4. W h e n p q ‾ i s S t a r t o f r a w e s o m e s f r o m t o T o t a l a w e s o m e s . 45 ‾ 6 a 121 = 45678 a 124 = 45789 3 + 1 = 4 46 ‾ 7 a 125 = 46789 1 = 1 . S o f o r p = 4 w e h a v e 4 + 1 = 5 a w e s o m e s . T A B L E f o r p = 5. W h e n p q ‾ i s S t a r t o f r a w e s o m e T o t a l a w e s o m e s . 56 ‾ 7 a n = a 126 = 56789 1 = 1 . S o f o r p = 5 w e h a v e 1 a w e s o m e . S o n = 126 a n d n 100 = 24789. n + n 100 = 126 + 24789 = 24915. The~ awesome~integers~\overline{pqrst}~starts~from~~~a_1= \overline{12345}~~and~~end~~~at~~a_n=\overline{56789}.\\ Constrains:- \quad\quad 1~ \leq~p~ \leq~5. \quad\quad\quad\quad~p+1~ \leq~q~ \leq~6. \quad\quad\quad\quad~q+1~ \leq~r~ \leq~7. \quad\quad\quad\quad~r+1~ \leq~s~ \leq~8. \quad\quad\quad\quad~s+1~ \leq~t~ \leq~9.\\ ~~~~~~\\ AWESOMES~FOR~ALL~POSSIBLE~START~OF~~r~~TILL~~r~~ENDS~~AT~~r=7.\\~~~\\ \begin{array}{|c |c | c |c |c| } \hline r & s & t & awesomes & How~many \\ ~~~\\ \hline 3 & 4 & 5,~6,~7,~8,~9 & ~12345, ~12346, ~12347 , ~12348, ~12349~ & 5 \\ ~ & 5 & 6,~7,~8,~9 &~12356,~12357,~12358,~12359 & 4 \\ ~ & 6 & 7,~8,~9&~12367,~12368,~12369&3\\ ~ & 7 & ~8,~9~&12378,~12379&2 \\ ~ & 8 & 9&12389 & \underline{ ~~~1~~~}\\ & ~&~&a_1=12345~~~to~~~a_{15}=12389 &~ \huge 15 \\ \hline \end{array} \\ ~~~~\\ \begin{array}{|c |c | c |c |c| } \hline r & s & t & awesomes & How~many \\ ~~~\\ \hline 4 ~ & 5 ~~~& 6,~7,~8,~9 &~12456,~12457,~12458,~12459 & 4 \\ ~ & 6 & 7,~8,~9 & ~12467,~12468,~12469 & 3\\ ~ & 7 & ~8,~9 & 12478,~12479 & 2 \\ ~ & 8 & 9& 12489 & \underline{ ~~~1~~~}\\ & ~&~&~a_{16}=12467~~~to~~~a_{25}=12489 &~ \huge 10 \\ \hline \end{array} \\ ~~~~\\ \begin{array}{|c |c | c |c |c| } \hline r & s & t & awesomes & How~many \\ ~~~\\ \hline 5 & 6 & 7,~8,~9 & ~12567,~12568,~12569 & 3 \\ ~ & 7 & ~8,~9~& 12578,~12579 & 2 \\ ~ & 8 & 9 & 12589 & \underline{ ~~~1~~~}\\ & ~&~&~a_{26}=12567~~~to~~~a_{31}=12589 &~ \huge 6 \\ \hline \end{array} \\ ~~~~\\ \begin{array}{|c |c | c |c |c| } \hline r & s & t & awesomes & How~many \\ ~~~\\ \hline 6 & 7 & ~8,~9~ & 12678,~12679 & 2 \\ ~ & 8 & 9 & 12689 & \underline{ ~~~1~~~}\\ & ~&~&~a_{32}=12678~~~to~~~a_{34}=12689 &~ \huge 3 \\ \hline \end{array} \\ ~~~~\\ \begin{array}{|c |c | c |c |c| } \hline r & s & t & awesomes & How~many \\ ~~~\\ \hline 7 & 8 & 9 & 12789 & \underline{ ~~~1~~~}\\ & ~&~&~a_{35}=12789 &~ \huge 1 \\ \hline \end{array} \\ ~~~~\\ ~~~~\\~~~~\\ \Large~ \color{#20A900}{All~~r~~end~~in~~r=7.}\\ \quad\quad\quad\quad\quad\Large TABLE~~ for~~ p=1.\\~~\\ \begin{array}{|c |c | c |c |c| } \hline When~\overline{pq}~is & Start~of~r & awesomes~from~ & to & Total~~awesomes. \\ ~~~\\ \hline \overline{12} & 3 & ~~~a_1=12345~~~ &~~~a_{35}=12789 & 15+10+6+3+1={\Huge 35} \\ \overline{13} & 4 &a_{36}=13456~~~ &~~~a_{55}=13789 & 10+6+3+1={\Huge 20}\\ \overline{14} & 5 &a_{56}=14567~~~ &~~~a_{65}=14789 & 6+3+1={\Huge 10}\\ \overline{15} & 6 &a_{66}=15678~~~ &~~~a_{69}=15789 & 3+1={\Huge 4}\\ \overline{16} & 7 &a_{70}=16789 & &1={\Huge 1~}\\ \hline \end{array}.\\ So~for~p=1~~we~have~35+20+10+4+1 ={\Huge \color{#3D99F6}{70}}~~awesomes \\ ~~~~\\ \quad\quad\quad\quad\quad\Large TABLE~~ for~~ p=2.\\~~\\ \begin{array}{|c |c | c |c |c| } \hline When~\overline{pq}~is & Start~of~r & awesomes~from~ & to & Total~~awesomes. \\ ~~~\\ \hline overline{23} & 4 &a_{71}=23456~~~ &~~~a_{90}=23789 & 10+6+3+1={\Huge 20}\\ \overline{24} & 5 &a_{91}=24567~~~ &~~~{\color{#D61F06}{a_{100}=24789} } & 6+3+1={\Huge 10}\\ \overline{25} & 6 &a_{101}=25678~~~ &~~~a_{104}=25789 & 3+1={\Huge 4}\\ \overline{26} & 7 &a_{105}=26789 & &1={\Huge 1~}\\ \hline \end{array} .~~~So~ for~ p=2~ we~ have~~ 20+10+4+1={\Huge \color{#3D99F6}{35}}~ awesomes. \\ ~~~ \\ \quad\quad\quad\quad\quad\Large TABLE~~ for~~ p=3.\\~~\\ \begin{array}{|c |c | c |c |c| } \hline When~\overline{pq}~is & Start~of~r & awesomes~from~ & to & Total~~awesomes. \\ ~~~\\ \hline \overline{34} & 5 &a_{106}=34567~~~ &~~~a_{115}=34789 & 6+3+1={\Huge 10}\\ \overline{35} & 6 &a_{116}=35678~~~ &~~~a_{119}=35789 & 3+1={\Huge 4}\\ \overline{36} & 7 &a_{120}=36789 & &1={\Huge 1~}\\ \hline \end{array}.~~~~~~~~So~ for~ p=3~ we~ have~~ 10+4+1={\Huge \color{#3D99F6}{15}}~~ awesomes. \\ \quad\quad\quad\quad\quad\Large TABLE~~ for~~ p=4.\\~~\\ \begin{array}{|c |c | c |c |c| } \hline When~\overline{pq}~is & Start~of~r & awesomes~from~ & to & Total~~awesomes. \\ ~~~\\ \hline \overline{45} & 6 & a_{121}=45678~~~ &~~~a_{124}=45789 & 3+1={\Huge 4}\\ \overline{46} & 7 & a_{125}=46789 & &1={\Huge 1~}\\ \hline \end{array}.~~~~~~~~So~ for~ p=4~ we~ have~~ 4+1={\Huge \color{#3D99F6}{5}}~~ awesomes. \\ \quad\quad\quad\quad\quad\Large TABLE~~ for~~ p=5.\\~~\\ \begin{array}{|c |c | c |c |c| } \hline When~\overline{pq}~is & Start~of~r & awesome & Total~~awesomes. \\ ~~~\\ \hline \overline{56} & 7 & a_n=a_{\Large \color{#D61F06}{126}}=56789~~~ & 1={\Huge 1}\\ \hline \end{array}.~~~~~~~~So~ for~ p=5~ we~ have~~ {\Huge \color{#3D99F6}{1}}~ awesome. \\ ~~~~\\ So~n=126~~and~n_{100}=24789.\\ n+n_{100}=126+24789=\Huge~~~~ \color{#D61F06}{24915}.

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