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Some NMTC sub-junior level Problems

Well.. Many people don't know but I was a part of STEM's Horizon ( Now, I have left them due to boards etc.) BTW STEM's Horizons is really great! And anyone interested in Olympiad math should join it! More info about it in below (make sure to check it out!).

So here are some problems I sent to them. They are fairly easy, and most of them are repetitive ideas. But they are my first sets of problems ( I know UMO was there but still..) The solutions will be posted in another blog posts. You guys can type out sols in the comments sections too :)

Problems:-

1. What is maximum possible number dividing  $x^2+x+1$ and $x^5 +x^4 +x^3 + 3x^2 +2 x +4$ for all $x\in \Bbb{N}$


2. Let $P$ be the sum of all $x$ and $y$ satisfying $45^x-2^x=2021^y.$ What is the last two digits of $p^2+p+1.$


3. What is the greatest value of $r$ such that $3^r$ is factor of $10^{2022}-8^{674}$.


4. Find all possible tuples $ (x,y,l)$ such that $\frac{x}{100}=\frac{20}{y}=\frac{5}{l}.$


5. Consider the following sequence $D(1),D(2),\dots D(999999998),$ where $D(i)$ is the sum of the digits modulo $9.$ Then what is the sum of all residues which appeared lesser than the other residues in the sequence $D(1),D(2),\dots D(999999998)$?


6.Define $f(k)=f(k-1)+(k-1)^2.$ What is $f(225)'s$ last two digits?


7. Find the sum of all possible Numbers $N$ such that when $N$ taken in base $4$, $N_4=ABC, $ and $\frac{1}{A}+\frac{1}{B}+\frac{1}{C}=\frac{1}{A+B+C}+\frac{1}{8}.$


8. Given a cyclic quad $PQRS $ , with $PQ=27, RS=18.$ Define $A=PQ\cap RS $( such that $ A-R-S$ in this order and $A-P-Q$ in this order , if $AS=36$, then what is$AQ$?


9.Consider a rectangle $3\times 6$ rectangle whose perimeter( boundary ) is covered by $22 (1\times 1)$ squares tiles  , These $22$ squares can be coloured by $4$ colours (Red,Black,Purple,Green), so that the colouring of the  boundary is symmetrical by the perpendicaular bisectors of the sides. In how many ways can you do that ?


10. Given that $9$ is a root of $x^3-21x^2+143x-315$, find the sum of the squares of the other two roots.


11. Given that P(x) is a 4 degree polynomial with $P(0)=1,P(2)=3, P(5)=9,P(10)=0,P(7)=0.$ Find the sum of all positive integer cfs of the polynomial.


12. In a school of $25000$ students, a survey noted that only  30%  were muggles, and then only 25% of the muggles played basketball. How many muggles played basketball?


13. What is the smallest $5$ digit number such that , the number is leaves remainder $1$ when divided by $3$, leaves remainder $ 2$ when divided by $ 5$, leaves remainder $4$ when divided by  $7.$


14. Given there are $10$ distinct socks, and you have $5$ different dyes  such that each dye can colour atmost $2$ socks. In how many can one colour the socks?


15.Milk decided to count the total number of $1$'s written in all the pg no's of the whole book, when she counted she realised there were $15$ $1'$s , and total no of $5'$s were $7.$ What is the last page number of her book?

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Here's the pdf compilation link NMTC (Google drive link). Anyways, I think from now on the blog will start having less math post for like 2 months, you guys have to bear it with me :(.

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STEM Horizons is a stem community founded by some motivated high schoolers from around the world. Sharing this wonderful initiative with all of you. Anyone can become part of this, it doesn't matter if you're a middle schooler, high schooler, college students, and beyond. We will work to create wonderful opportunity in STEM field. You can join the Discord server from this link https://discord.gg/Az7DsaSyYf . The link to the website is https://stemhorizons.wordpress.com/



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Sunaina💜

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Comments

  1. AnonymousApril 4, 2021 at 7:55 PM

    Nice blog!

    ReplyDelete
  2. Nirjhar NathApril 4, 2021 at 9:49 PM

    A question of same standard or probably less:
    Find all positive integer pairs (x,y) satisfying x³-(y!)²=2021

    ReplyDelete
    Replies
    1. Sunaina Pati April 4, 2021 at 11:25 PM

      Awww, cute! We take mod 9. For $y\ge 3$ it's not possible since the $x^3\equiv 5 \mod 9$ but cubes are always $0,1,-1 \mod 9.$ So $y=1,2$ Checking we get that no solutions exist.

      Delete
    2. Nirjhar NathApril 5, 2021 at 6:27 AM

      👌

      Delete
  3. Amit April 5, 2021 at 5:27 PM

    Q 1 solution:
    Let d be a common divisor of both the polynomials.So d divides $x^5+x^4+x^3+3x^2+2x+4-(x^3+3)(x^2+x+1)=-x+1$.So d divides $x^2+x+1+x(-x+1)=2x+1$.Finally we get d divides $2x+1+2(-x+1)=3$.So maximum possible value is 3 which is certainly obtained when x=1

    ReplyDelete
    Replies
    1. Sunaina Pati April 5, 2021 at 7:00 PM

      Well, it says for all x, so answer must be 1, right? since for x=3, clearly 3 doesn't divide x^2+x+1. So the answer is 1. I think the statement wasn't clear though, so my bad.

      Delete
    2. Amit April 5, 2021 at 9:30 PM

      Well I misinterpreted as well

      Delete
  4. ASNApril 19, 2021 at 5:13 PM

    Wow, these problems were yours? Nice!

    ReplyDelete
    Replies
    1. Sunaina Pati April 25, 2021 at 9:28 AM

      yep! Sadly we didn't had a nmtc stage 1 mock :(

      Delete
  5. Abishek Thiyagarajan March 11, 2022 at 7:12 PM

    Congrats for apmo!!

    ReplyDelete
    Replies
    1. Sunaina Pati April 7, 2022 at 6:37 AM

      Loll thenku! One simply needs to be an inmo awardee to give it though :|

      Delete
  6. RedBeastofHell2021March 16, 2025 at 1:14 AM

    Hello, I just wanted to say that problem 2 can be made more fun; instead of having 2 unknowns to have 3 unknowns like "45^x-2^y=2021^z" where x,y,z are natural (or nonnegative) numbers. Also problem 2 looks very similar to that of N5 from the JBMO 2019 Shortlist...
    Nice problems! I hope that u will post more in the future

    ReplyDelete
  7. RedBeastofHell2021March 16, 2025 at 1:15 AM

    Hello, wanted to say that problem 2 looks like N5 from JBMO 2019 Shortlist

    ReplyDelete

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