Difference between revisions of "2021 JMPSC Accuracy Problems/Problem 15"
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We can easily find that <math>f(n+1)=10f(n)+25.</math> Using our inductive assumption, we obtain <cmath>\frac{f(n+1)}{25}=10 \cdot (19\underbrace{111 \cdots 1}_{n-1 \text{ ones}})+1=19 \cdot \underbrace{111 \cdots 1}_{n-1 \text{ ones}},</cmath> as desired. <math>\mathbb{Q.E.D.}</math> | We can easily find that <math>f(n+1)=10f(n)+25.</math> Using our inductive assumption, we obtain <cmath>\frac{f(n+1)}{25}=10 \cdot (19\underbrace{111 \cdots 1}_{n-1 \text{ ones}})+1=19 \cdot \underbrace{111 \cdots 1}_{n-1 \text{ ones}},</cmath> as desired. <math>\mathbb{Q.E.D.}</math> | ||
− | ~ | + | ~Solution by pinkpig, <math>\LaTeX</math>/wording fixes by samrocksnature |
==Solution 2 (More Algebraic)== | ==Solution 2 (More Algebraic)== | ||
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<math>\linebreak</math> | <math>\linebreak</math> | ||
~Geometry285 | ~Geometry285 | ||
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==See also== | ==See also== | ||
− | #[[2021 JMPSC | + | #[[2021 JMPSC Accuracy Problems|Other 2021 JMPSC Accuracy Problems]] |
− | #[[2021 JMPSC | + | #[[2021 JMPSC Accuracy Answer Key|2021 JMPSC Accuracy Answer Key]] |
#[[JMPSC Problems and Solutions|All JMPSC Problems and Solutions]] | #[[JMPSC Problems and Solutions|All JMPSC Problems and Solutions]] | ||
{{JMPSC Notice}} | {{JMPSC Notice}} |
Latest revision as of 00:19, 12 July 2021
Problem
For all positive integers define the function
to output
For example,
,
, and
Find the last three digits of
Solution
We can easily find that and so on. Thus, we claim
that
Now, we find we can easily find that
We proceed by induction. Our base case is
Our inductive assumption is
and we wish to prove that this pattern holds for
We can easily find that Using our inductive assumption, we obtain
as desired.
~Solution by pinkpig, /wording fixes by samrocksnature
Solution 2 (More Algebraic)
We only care about the last
digits, so we evaluate
. Note the expression is simply
, so factoring a
we have
. Now, we can divide by
to get
Evaluate the last
digits to get
~Geometry285
See also
The problems on this page are copyrighted by the Junior Mathematicians' Problem Solving Competition.