Dylan's Math Sets

Number Theory Level 3

Dylan is writing solutions to his problems for some math sets. There are $25$ problems on his set, and while he is incredibly fast at solving problems, he is quite slow at writing up solutions. He takes $x$ minutes to solve the $xth$ problem, but to write it up, it takes him $x^{2}$ minutes.

If it takes him $6$ minutes to prepare his set (like writing his name, and the date etc.) and he plans to finish the set over $3$ days, how much time should he spend on it each day? Express your answer as: if it takes him $m$ hours and $n$ minutes, your answer should be $n + m$ .

For example, the $5th$ problem would take him $5$ minutes to solve, but $25$ ( $5^{2}$ ) minutes to write it up. Assume that he does an equal part each day. Also assume that Dylan does not live on Earth; he lives on a planet with 100 hour days.

The answer is 64.

1 solution

Vivek Bhupatiraju
Feb 28, 2014

We use two very cool formulas that are very easy to prove here:

The time it takes him to solve $x$ problems: $\frac{x(x+1)}{2}$ minutes

The time it takes him to write up $x$ problems: $\frac{x(x+1)(2x+1)}{6}$ minutes

This means it takes him $25 \times 13$ minutes, or $325$ minutes to solve, and $\frac{25 \times 26 \times 51}{6}$ , or $5525$ minutes to write up.

This is a total of $325 + 5525$ minutes, or $5850$ minutes. This is a total of $5856$ minutes (as we need the $6$ minutes to set up his set!), and $5856/3$ minutes per day, or $1952$ minutes.

This comes out to $32$ hours and $32$ minutes, which means the answer is $32+32$ minutes, or $\boxed{64}$ .

You say "that are very easy to prove"

Can you prove the sum of squares one then?

Nathan Ramesh - 7 years, 3 months ago

It's not hard. Use the identity: $(k+1)^{3} - k^{3} = 3k^{2} +3k +1$ , look around a little, and prove it. I would do it here, but the LATEX would take forever to type up.

You can also use geometry, and square pyramids.

Vivek Bhupatiraju - 7 years, 3 months ago

I don't mind no latex ;)

Please explain.

Nathan Ramesh - 7 years, 3 months ago

I'm still waiting for this proof, please.

Nathan Ramesh - 7 years, 3 months ago

@Nathan Ramesh – See this buddy: Proofs of The Formula for The Sum of Squares .

Tunk-Fey Ariawan - 7 years, 3 months ago

@Nathan Ramesh – Yes! The proof below is the same thing I would do.

Vivek Bhupatiraju - 7 years, 3 months ago

How does it make sense to work 32 hours a day??

Phalguni Shah - 7 years, 3 months ago

I was thinking the same thing... I didn't understand that at all

Nathan Ramesh - 7 years, 3 months ago

Oh my goodness......wow I didn't even see that. Well then. I'm going to change the question a little.

Vivek Bhupatiraju - 7 years, 3 months ago

I knew exactly the answer but I chose to not answer it since the answer didn't make sense.

Tunk-Fey Ariawan - 7 years, 3 months ago

who works 32 hrs a day

AYUSH ANAND - 7 years, 3 months ago

I fixed the problem.

Vivek Bhupatiraju - 7 years, 3 months ago

The first formula is for getting the sum of numbers from $1$ to $x$ , and the second is for getting the sum of numbers from $1^{2}$ to $x^{2}$ .

Vivek Bhupatiraju - 7 years, 3 months ago

It's not supposed to say 32 + 32 minutes :P

Vivek Bhupatiraju - 7 years, 3 months ago

Dylan's Math Sets

The answer is 64.

1 solution

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