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2017-09-04 Intermediate


Are there distinct positive integers mm and nn such that mnmn evenly divides m2+n2?m^2+n^2?

9 identical, spherical balls are packed into a cube of edge length 100.

The 8 balls in corners are each tangent to the three faces making up its corner, and the 9th9^\text{th} ball is tangent to the other 8.

What is the radius of the spherical balls, to 1 decimal place?

Let nn be an integer randomly chosen in the interval [1,120][1, 120] , and consider the sum Sn=k=1nk.S_n = \displaystyle \sum_{k=1}^n k.

SnS_n is least likely to be divisible by __________.\text{\_\_\_\_\_\_\_\_\_\_}.

Point PP is inside square ABCDABCD such that AP=1,BP=2,CP=3.AP=1, BP=2, CP=3.

Find APB\angle APB in degrees.

A student randomly chooses mm distinct numbers from among the first 2017 positive integers and writes them all on a chalkboard.

She then chooses two of the numbers written on the chalkboard, erases them both, and writes down their least common multiple. She repeats this process until only one number remains on the chalkboard.

What is the smallest integer mm such that the final number is guaranteed to be a multiple of 128?

It may be helpful to note that 128=27128=2^{7} and 2017 is prime.

An example of the process with \(m=4.\) An example of the process with m=4.m=4.

Bonus: Can you come up with a simple formula for the smallest mm such that the final number is a multiple of nn, where nn is a positive integer?


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