Difference between revisions of "2003 AIME I Problems/Problem 9"
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Call the number <math>\overline{abcd}</math>. Then <math>a+b=c+d</math>. Set <math>a+b=x</math>. | Call the number <math>\overline{abcd}</math>. Then <math>a+b=c+d</math>. Set <math>a+b=x</math>. | ||
− | Clearly, <math>2\ | + | Clearly, <math>2\le x \le18</math>. |
If <math>x=1</math>: The only case is <math>1001</math> or <math>1010</math>. 2 choices. | If <math>x=1</math>: The only case is <math>1001</math> or <math>1010</math>. 2 choices. | ||
− | If <math>x=2</math>: then since <math>a\neq0</math>, <math>a=1=b</math> or <math>a=2, b=0</math>. There are 3 choices for <math>(c,d)</math>: <math>(2,0), (0, 2), (1, 1). < | + | If <math>x=2</math>: then since <math>a\neq0</math>, <math>a=1=b</math> or <math>a=2, b=0</math>. There are 3 choices for <math>(c,d)</math>: <math>(2,0), (0, 2), (1, 1)</math>. <math>2*3=6</math> here. |
− | If < | + | If <math>x=3</math>: Clearly, <math>a\neq b</math> because if so, the sum will be even, not odd. Counting <math>(a,b)=(3,0)</math>, we have <math>4</math> choices. Subtracting that, we have <math>3</math> choices. Since it doesn't matter whether <math>c=0</math> or <math>d=0</math>, we have 4 choices for <math>(c,d)</math>. So <math>3*4=12</math> here. |
− | If < | + | If <math>x=4</math>: Continue as above. <math>4</math> choices for <math>(a,b)</math>. <math>5</math> choices for <math>(c,d)</math>. <math>4*5=20</math> here. |
− | If < | + | If <math>x=5</math>: You get the point. <math>5*6=30</math>. |
− | If < | + | If <math>x=6</math>: <math>6*7=42</math>. |
− | If < | + | If <math>x=7</math>: <math>7*8=56</math>. |
− | If < | + | If <math>x=8</math>: <math>8*9=72</math>. |
− | If < | + | If <math>x=9</math>: <math>9*10=90</math>. |
− | Now we need to be careful because if < | + | Now we need to be careful because if <math>x=10</math>, <math>(c,d)=(0,10)</math> is not valid. However, we don't have to worry about <math>a\neq0</math>. |
− | If < | + | If <math>x=10</math>: <math>(a,b)=(1,9), (2, 8), ..., (9, 1)</math>. Same thing for <math>(c,d)</math>. <math>9*9=81</math>. |
− | If < | + | If <math>x=11</math>: We start at <math>(a,b)= (2,9)</math>. So <math>8*8</math>. |
− | Continue this pattern until < | + | Continue this pattern until <math>x=18: 1*1=1</math>. Add everything up: we have <math>\boxed{615}</math>. |
~hastapasta | ~hastapasta |
Revision as of 18:35, 29 April 2022
Problem
An integer between and
, inclusive, is called balanced if the sum of its two leftmost digits equals the sum of its two rightmost digits. How many balanced integers are there?
Solution
If the common sum of the first two and last two digits is , such that
, there are
choices for the first two digits and
choices for the second two digits (since zero may not be the first digit). This gives
balanced numbers. If the common sum of the first two and last two digits is
, such that
, there are
choices for both pairs. This gives
balanced numbers. Thus, there are in total
balanced numbers.
Both summations may be calculated using the formula for the sum of consecutive squares, namely .
Solution 2 (Painful Casework)
Call the number . Then
. Set
.
Clearly, .
If : The only case is
or
. 2 choices.
If : then since
,
or
. There are 3 choices for
:
.
here.
If : Clearly,
because if so, the sum will be even, not odd. Counting
, we have
choices. Subtracting that, we have
choices. Since it doesn't matter whether
or
, we have 4 choices for
. So
here.
If : Continue as above.
choices for
.
choices for
.
here.
If : You get the point.
.
If :
.
If :
.
If :
.
If :
.
Now we need to be careful because if ,
is not valid. However, we don't have to worry about
.
If :
. Same thing for
.
.
If : We start at
. So
.
Continue this pattern until . Add everything up: we have
.
~hastapasta
See also
2003 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 8 |
Followed by Problem 10 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.