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My friend sent me a question from an olympiad and im not sure that we have followed the right method, we both did the same thing:

The age of a man was 2/61 of the year in which he died. How old would he have been if he lived until 1992?

Surly then he dies in 1992 and then his age is 2/61 times 1992 rounded to the nearest integer? I am unsure though, this seems too simple.

simplton
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If you let $a$ be the age of the man in the year in which he died and $d$ be the year of his death, then $a=\frac {2d}{61}$.

Since $61$ is prime and $a$ is an integer we have to have $d=61k$, in which case $a=2k$.

So if, for example, $k=32$ then $d=1952$ and $a=64$, which would give an age of [ ] in $1992$.

If $k=31$ then $d=1891$ and $a=62$ implying age [ ] in $1992$. To exclude this possibility (and others) you have to use a nonmathematical assumption, for example that human beings do not live beyond age $120$.

Mark Bennet
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  • what is age []? – simplton May 16 '13 at 20:24
  • @simplton I left some work for you to do - if he was 64 in 1952, how old would he have been in 1992? – Mark Bennet May 16 '13 at 20:25
  • Why $a=\frac{2d}{61}$? Must be $a=\left[\frac{2d}{61}\right]$ – Gaston Burrull May 16 '13 at 20:30
  • Note also that we could try $k=33$ with $d=2013$ and $a=66$. So the date of the question could be material to the answer - if this were the solution,last year it wouldn't have been possible to say that he had died. However the wording "if he had lived ..." excludes this possibility - he clearly died before 1992. – Mark Bennet May 16 '13 at 20:30
  • and that would be 83, correct? – simplton May 16 '13 at 20:31
  • @GastónBurrull There is nothing in the question to suggest any rounding to the nearest integer, or taking the integer part, or anything like that. The fact that 61 is prime restricts the possible years to multiples of 61 - and 61 is a large enough number to make only one solution "realistic". – Mark Bennet May 16 '13 at 20:36
  • @simplton - how many years from 1952 to 1992? – Mark Bennet May 16 '13 at 20:37
  • 1992-1952+1, so its should be 64 +41 = 105? – simplton May 16 '13 at 20:39
  • I don't understand you, why is $d=61k$? This is an absurd since $d=1992$ and $1992$ is not divisible by $61$ – Gaston Burrull May 16 '13 at 20:43
  • @GastónBurrull But the man did not die in 1992 - he died some time before that, in an unknown year I called $d$. The information given in the question is not $d=1992$ but $d \lt 1992$. – Mark Bennet May 16 '13 at 20:46
  • @GastónBurrull The problem says that he was not alive in $1992$, so $d<1992$, not $d=1992$. – Mario Carneiro May 16 '13 at 20:46
  • @simplton - If he had died in 1953, how old would he have been? You don't need an inclusive count here. – Mark Bennet May 16 '13 at 20:47
  • @MarioCarneiro If he lived until 1992 means that he dies in 1992. Right? – Gaston Burrull May 16 '13 at 20:48
  • he would have been 64 giving 104 as his age in 1992 – simplton May 16 '13 at 20:49
  • @GastónBurrull It says "if he had lived" as a counterfactual. Thus he did not live until 1992, and we are considering this idea as an alternative to the reality. Here $a$ represents the age he actually lived to, and [] is the age he would have lived to if he had not met his untimely demise in year $d$ and instead lived until 1992. (Note: "If he lived to 1992" and "If he had lived to 1992" mean two different things; this may be a grammar confusion.) – Mario Carneiro May 16 '13 at 20:52
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    Oops, it doesn't actually say "if he had lived" in the OP. Well, it should have, since it does include the phrase "how old would he have been", which also indicates a counterfactual. – Mario Carneiro May 16 '13 at 20:56
  • It doesn't say "how old would he have been in 1992?" but "how old would he have been if he lived until 1992?" Though this is a little ambiguous without the "had", there is a difference - the "if" suggest he "might have, but didn't". However he still couldn't have died in 1992, because the year of his death is specified to be divisible by 61. – Mark Bennet May 16 '13 at 21:02
  • I take the blame for introducing "had" in my comment - it isn't there. But I would still exclude date of death 2013, because the "if" suggests might have been alive in 1992, but wasn't. – Mark Bennet May 16 '13 at 21:04