2014 National MathCounts Winning Question

Number Theory Level 2

The smallest integer of a set of consecutive integers is -32. If the sum of these integers is 67, how many integers are in the set?

Note: Taken from the 2014 National MathCounts Countdown Round. The winner(Swapnil Garg of CA) solved this question mentally in 12 seconds. Can you?

The answer is 67.

5 solutions

Tristan Shin
May 11, 2014

Because the set starts at -32 and sums to a positive number, the set must go up to at least 32. This is 65 integers with a sum of 0. The next two numbers, 33 and 34, sum to 67. This is $\boxed {67}$ integers that sum to 67.

Yeah, I DID THIS FASTER THAN SWAPNIL!!!!!!!! :D:D:D:D:D

David Lee - 7 years, 1 month ago

Same here! :DDDD

Sharky Kesa - 7 years, 1 month ago

Me too! The explanation can be extended to any number. The answer was hidden in the question. For any number $a$ , the sum of the consecutive integers starting with $-a$ and ending with $a+2$ is always equal to the number of consecutive integers.

Proof:

The sum of the terms boils down to $a+1+a+2=2a+3$ , because terms from $-a$ to $a$ get cancelled. Now, the number of terms which cancel out is $2a+1$ , and then the two terms $a+1, a+2$ are added. Thus, the number of terms is $2a+3=$ the sum of the terms.

Nanayaranaraknas Vahdam - 7 years, 1 month ago

Me too. And I was actually at Nationals. Sorry to creep you out @Daniel Liu , but I was sitting directly in front of you... I basically just said, "There's a bunch of numbers that sum to 67, and they have an average. Either there's 1 number, 67 (obviously not), or 67 numbers with an average of 1, which works. HA. I beat you, Swapnil! (I don't know if that's valid math, but nothing at National countdown is valid anyway.)

Frodo Baggins - 7 years ago

That's a cool way to look at it. I was stupid and bashed it out. :D

Finn Hulse - 7 years, 1 month ago

It's how Swapnil Garg solved it.

Daniel Liu - 7 years, 1 month ago

Its also how i solved it

ashutosh mahapatra - 7 years, 1 month ago

Definitely. So dude how did you and your team do?

Finn Hulse - 7 years, 1 month ago

@Finn Hulse – Is that to Daniel or Me? I didn't make Nationals this year...

Tristan Shin - 7 years, 1 month ago

@Tristan Shin – Oh I was talking to Daniel but have you ever made it to Nationals or States? What were your scores there? :D

Finn Hulse - 7 years, 1 month ago

good question !!! and great answer !!! what can i say is ...........

Rishabh Jain - 7 years, 1 month ago

Baidehi Chattopadhyay
May 11, 2014

The final equation comes as k^2-65k-134=0, solving this we get k=67. However, one may try to do it mentally because of the fact that the set starts at a negative number but ends at a positive number so the set must at least go up to +32. So, total number of integers is 65 with the total sum 0. So, next numbers add up to 67. So total number=65+2=67

First took me 25 seconds to think of then 4sec to solve it

ashutosh mahapatra - 7 years, 1 month ago

Noel Lo
Apr 22, 2015

I took about 20 seconds LOL.

Fox To-ong
Dec 28, 2014

from -32 to 32 ----------------- 65 integers including zero and the sum is 0 next is 33 and 34 the sum is 67 total integers -------65 + 2 = 67

Ashwin Upadhyay
Jul 5, 2014

did it in less than 10 seconds..... faster than swapnil

2014 National MathCounts Winning Question

The answer is 67.

5 solutions

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