Mathematics and Algorithmic Thinking High School competition

I think I should start by explaining what this is -- the Mathematics and Algorithmic Thinking High School competition is a competition (yeah i just made it) for high schoolers, targeting mathematical and algorithmic thinking. Yes, I know that that's obvious from the title. What this entails is using mathematical concepts, in conjunction with algorithmic thinking, to answer the questions in the competition.

Rules of competition

No registration or anything is required; if you want to join wait until the competition starts and email in an answer to wen00@live.com.au. The period for accepting emails will last 24hrs although the questions will stay up indefinitely. You may use any resources available (in fact research is highly recommended) except for expicitly asking the question on any online forum. No proofs are required in your answer. Competitions will be held monthly; the maximum points per round will be 100 split between any number of questions

Competitions

One last note -- questions generally have some preamble; if you're familiar with mathematical terminology then it's unnecessary to read the preamble to each question -- just read the text after the [ point count ]


Round 1 Ends: [09/09/2018 10:00am] Round 2 Ends: [30/09/2018 10:00am]

Style of question

Here are some example questions of the style of question that'll be asked (highlight black for the answer):

  1. What are the last three digits of \(2018^{2018^{2018}}\)? 976
  2. The coordinates of a triangle are (-5,18), (-4,11) and (0,3). Find the coordinates of the circumcenter and answer with x and y coordinates concatenated (e.g if your coordinates are (123,45) then your answer would be 12345). 2018
  3. How many positive integers \(N\) between 2 and 1000000 inclusive have the property that \(k!\) (k factorial) for \(k>1\) never has exactly \(k\) digits in base \(N\)?
    For instance, if N=3 then when k=5, 5!=111103 has 5 digits in base 3 and thus N=3 is discounted from the total tally. 0
  4. Suppose that A(n) represents the point \( (n,n^2) \). Consider the convex pentagon with vertices \[A(2016^{2016}),A(2017^{2017}),A(2018^{2018}),A(2019^{2019}),A(2020^{2020})\] Count the number of lattice points inside or on the boundary of this pentagon. What are the first five digits of the answer? 24470