Difference between revisions of "2002 AIME I Problems/Problem 9"

Latest revision as of 18:44, 25 April 2024

Problem

Harold, Tanya, and Ulysses paint a very long picket fence.

Harold starts with the first picket and paints every $h$ th picket;
Tanya starts with the second picket and paints every $t$ th picket; and
Ulysses starts with the third picket and paints every $u$ th picket.

Call the positive integer $100h+10t+u$ paintable when the triple $(h,t,u)$ of positive integers results in every picket being painted exactly once. Find the sum of all the paintable integers.

Solution

Solution 1

Note that it is impossible for any of $h,t,u$ to be $1$ , since then each picket will have been painted one time, and then some will be painted more than once.

$h$ cannot be $2$ , or that will result in painting the third picket twice. If $h=3$ , then $t$ may not equal anything not divisible by $3$ , and the same for $u$ . Now for fourth and fifth pickets to be painted, $t$ and $u$ must be $3$ as well. This configuration works, so $333$ is paintable.

If $h$ is $4$ , then $t$ must be even. The same for $u$ , except that it can't be $2 \mod 4$ . Thus $u$ is $0 \mod 4$ and $t$ is $2 \mod 4$ . Since this is all $\mod 4$ , $t$ must be $2$ and $u$ must be $4$ , in order for $5,6$ to be paint-able. Thus $424$ is paintable.

$h$ cannot be greater than $4$ , since if that were the case then the answer would be greater than $999$ , which would be impossible for the AIME.

Thus the sum of all paintable numbers is $\boxed{757}$ .

Solution 2

Again, note that $h,t,u \neq 1$ . The three conditions state that no picket number $n$ may satisfy any two of the conditions: $n \equiv 1 \pmod{h},\ n \equiv 2 \pmod{t},\ n \equiv 3 \pmod{u}$ . By the Chinese Remainder Theorem, the greatest common divisor of any pair of the three numbers $\{h,t,u\}$ cannot be $1$ (since otherwise without loss of generality consider $\text{gcd}\,(h,t) = 1$ ; then there will be a common solution $\pmod{h \times t}$ ).

Now for $4$ to be paint-able, we require either $h = 3$ or $t=2$ , but not both.

In the former condition, since $\text{gcd}\,(h,t),\ \text{gcd}\,(h,u) \neq 1$ , it follows that $3|t,u$ . For $5$ and $6$ to be paint-able, we require $t = u = 3$ , and it is easy to see that $333$ works.
In the latter condition, similarly we require that $2|h,u$ . All even numbers are painted. We now renumber the remaining odd pickets to become the set of all positive integers ( $1,3,5, \ldots \rightarrow 1',2',3', \ldots$ , where $n' = \frac{n+1}{2}$ ), which requires the transformation $h' = h/2,\ u' = u/2$ , and with the painting starting respectively at $1',2'$ . Our new number system retains the same conditions as above, except without $t$ . We thus need $\text{gcd}\, (h',u') \neq 1, h',u' \neq 1$ . Then for $3',4'$ to be painted, we require $h' = u' = 2$ . This translates to $424$ , which we see works.

Thus the answer is $333+424 = \boxed{757}$ .

Solution 3

The three conditions state that no picket number $n$ may satisfy any two of the conditions: $n \equiv 1 \pmod{h},\ n \equiv 2 \pmod{t},\ n \equiv 3 \pmod{u}$ . Note that the smallest number, $min \{ h,t,u \},$ divides the other $2$ , and the next smallest divide the largest number, otherwise there is a common solution by the Chinese Remainder Theorem. It is also a necessary condition so that it paints exactly once. Note that the smallest number can't be at least $5$ , otherwise not all picket will be painted. We are left with few cases (we could also exclude $1$ as the possibility) which could be done quickly. Thus the answer is $333+424 = \boxed{757}$ .

@@ Line 2: / Line 2: @@
 Harold, Tanya, and Ulysses paint a very long picket fence.
-Harold starts with the first picket and paints every <math>h</math> th picket;
+* Harold starts with the first picket and paints every <math>h</math>th picket;
+* Tanya starts with the second picket and paints every <math>t</math>th picket; and
+* Ulysses starts with the third picket and paints every <math>u</math>th picket.
-Tanya starts with the second picket and paints every <math>t</math> th picket; and
+Call the positive integer <math>100h+10t+u</math> paintable when the triple <math>(h,t,u)</math> of positive integers results in every picket being painted exactly once. Find the sum of all the paintable integers.
+== Solution ==
+=== Solution 1 ===
+Note that it is impossible for any of <math>h,t,u</math> to be <math>1</math>, since then each picket will have been painted one time, and then some will be painted more than once.
-Ulysses starts with the third picket and paints every <math>u</math> th picket.
+<math>h</math> cannot be <math>2</math>, or that will result in painting the third picket twice. If <math>h=3</math>, then <math>t</math> may not equal anything not divisible by <math>3</math>, and the same for <math>u</math>. Now for fourth and fifth pickets to be painted, <math>t</math> and <math>u</math> must be <math>3</math> as well. This configuration works, so <math>333</math> is paintable.
-Call the positive integer <math>100h+10t+u</math> paintable when the triple <math>(h,t,u)</math> of positive integers results in every picket being painted exactly once. Find the sum of all the paintable integers.
+If <math>h</math> is <math>4</math>, then <math>t</math> must be even. The same for <math>u</math>, except that it can't be <math>2 \mod 4</math>. Thus <math>u</math> is <math>0 \mod 4</math> and <math>t</math> is <math>2 \mod 4</math>. Since this is all <math>\mod 4</math>, <math>t</math> must be <math>2</math> and <math>u</math> must be <math>4</math>, in order for <math>5,6</math> to be paint-able. Thus <math>424</math> is paintable.
+<math>h</math> cannot be greater than <math>4</math>, since if that were the case then the answer would be greater than <math>999</math>, which would be impossible for the AIME.
-== Solution ==
+Thus the sum of all paintable numbers is <math>\boxed{757}</math>.
-<math>h</math> cannot be 1 or 2, or that will result in painting the third picket twice. If <math>h=3</math>, then <math>t</math> may not equal anything not divisible by 3, and the same for <math>u</math>. Now for every picket to be painted, <math>t</math> and <math>u</math> must be 3 as well. So <math>333</math> is paintable.
-If <math>h</math> is 4, then <math>t</math> can't be 1 or 3 mod 4, but can be 2 or 0 mod 4. The same for <math>u</math>, except that it can't be 2 mod 4. Thus u is 0 mod 4 and t is 2 mod 4. Since this is all mod 4, t must be 2 and u must be 4. Thus 424 is paintable.
+=== Solution 2 ===
+Again, note that <math>h,t,u \neq 1</math>. The three conditions state that no picket number <math>n</math> may satisfy any two of the conditions: <math>n \equiv 1 \pmod{h},\ n \equiv 2 \pmod{t},\ n \equiv 3 \pmod{u}</math>. By the [[Chinese Remainder Theorem]], the [[greatest common divisor]] of any pair of the three numbers <math>\{h,t,u\}</math> cannot be <math>1</math> (since otherwise [[without loss of generality]] consider <math>\text{gcd}\,(h,t) = 1</math>; then there will be a common solution <math>\pmod{h \times t}</math>).
-Since answers on the [[AIME]] can't be greater that <math>999</math> and the sum of the paintable integers found so far is <math>333+424=757</math>, any remaining paintable integer must be less than or equal to <math>999-757=242</math>. However, this implies that <math>h \le 2</math> in any remaining paintable integers that exist, which results in the third picket being painted twice. Thus, there are no other paintable numbers, so the sum of all paintable numbers is <math>\boxed{757}</math>.
+Now for <math>4</math> to be paint-able, we require either <math>h = 3</math> or <math>t=2</math>, but not both.
+*In the former condition, since <math>\text{gcd}\,(h,t),\ \text{gcd}\,(h,u) \neq 1</math>, it follows that <math>3|t,u</math>. For <math>5</math> and <math>6</math> to be paint-able, we require <math>t = u = 3</math>, and it is easy to see that <math>333</math> works.
+*In the latter condition, similarly we require that <math>2|h,u</math>. All even numbers are painted. We now renumber the remaining odd pickets to become the set of all positive integers (<math>1,3,5, \ldots \rightarrow 1',2',3', \ldots</math>, where <math>n' = \frac{n+1}{2}</math>), which requires the transformation <math>h' = h/2,\ u' = u/2</math>, and with the painting starting respectively at <math>1',2'</math>. Our new number system retains the same conditions as above, except without <math>t</math>. We thus need <math>\text{gcd}\, (h',u') \neq 1, h',u' \neq 1</math>. Then for <math>3',4'</math> to be painted, we require <math>h' = u' = 2</math>. This translates to <math>424</math>, which we see works.
-{{incomplete|solution}}
+Thus the answer is <math>333+424 = \boxed{757}</math>.
+=== Solution 3 ===
+The three conditions state that no picket number <math>n</math> may satisfy any two of the conditions: <math>n \equiv 1 \pmod{h},\ n \equiv 2 \pmod{t},\ n \equiv 3 \pmod{u}</math>. Note that the smallest number, <math>min \{ h,t,u \}, </math> divides the other <math>2</math>, and the next smallest divide the largest number, otherwise there is a common solution by the [[Chinese Remainder Theorem]]. It is also a necessary condition so that it paints exactly once. Note that the smallest number can't be at least <math>5</math>, otherwise not all picket will be painted. We are left with few cases (we could also exclude <math>1</math> as the possibility) which could be done quickly. Thus the answer is <math>333+424 = \boxed{757}</math>.
 == See also ==
 {{AIME box|year=2002|n=I|num-b=8|num-a=10}}
+[[Category:Intermediate Number Theory Problems]]
+{{MAA Notice}}

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