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2022 SSMO Relay Round 3 Problems

Revision as of 21:57, 31 May 2023 by Pinkpig (talk | contribs) (Created page with "==Problem 1== Let <math>f:\mathbb Z\rightarrow\mathbb Z</math> be a function such that <math>f(0)=0</math> and <math>f\left(|x^2-4|\right)=0</math> if <math>f(x)=0</math>. Mo...")
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Problem 1

Let $f:\mathbb Z\rightarrow\mathbb Z$ be a function such that $f(0)=0$ and $f\left(|x^2-4|\right)=0$ if $f(x)=0$. Moreover, $|f(x+1)-f(x)|=1$ for all $x\in \mathbb Z$. Let $N$ be the number of possible sequences $\{f(1),f(2),\dots,f(21)\}$. Find the remainder when $N$ is divided by 1000.

Solution

Problem 2

Let $T=$ TNYWR. In cyclic quadrilateral $ABCD,$ $\angle{BAD}=60^{\circ},$ and $BC=CD=T.$ If $AB$ is a positive integer, find twice the median of all (not necessarily distinct) possible values of $AB$.

Solution

Problem 3

Let $T=$ TNYWR. Let $f(x)$ be a polynomial of degree 10, such that $f(i)=i$ for all $i=1,2,\dots,10$ and $f(11) =T$. Find the remainder when $f(13)$ is divided by $1000$.

Solution

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