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Difference between revisions of "2022 SSMO Relay Round 3 Problems"

(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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[[2022 SSMO Relay Round 3 Problems/Problem 1|Solution]]
 
[[2022 SSMO Relay Round 3 Problems/Problem 1|Solution]]
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==Problem 2==
 
==Problem 2==
  
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Let <math>T=</math> TNYWR. In cyclic quadrilateral <math>ABCD,</math> <math>\angle{BAD}=60^{\circ},</math> and <math>BC=CD=T.</math> If <math>AB</math> is a positive integer, find twice the median of all (not necessarily distinct) possible values of <math>AB</math>.
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Let <math>T=TNYWR</math>. In cyclic quadrilateral <math>ABCD,</math> <math>\angle{BAD}=60^{\circ},</math> and <math>BC=CD=T.</math> If <math>AB</math> is a positive integer, find twice the median of all (not necessarily distinct) possible values of <math>AB</math>.
  
 
[[2022 SSMO Relay Round 3 Problems/Problem 2|Solution]]
 
[[2022 SSMO Relay Round 3 Problems/Problem 2|Solution]]
 +
 
==Problem 3==
 
==Problem 3==
  
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Let <math>T=</math> TNYWR. Let <math>f(x)</math> be a polynomial of degree 10, such that <math>f(i)=i</math> for all <math>i=1,2,\dots,10</math> and <math>f(11) =T</math>. Find the remainder when <math>f(13)</math> is divided by <math>1000</math>.
+
Let <math>T=TNYWR</math>. Let <math>f(x)</math> be a polynomial of degree 10, such that <math>f(i)=i</math> for all <math>i=1,2,\dots,10</math> and <math>f(11) =T</math>. Find the remainder when <math>f(13)</math> is divided by <math>1000</math>.
  
 
[[2022 SSMO Relay Round 3 Problems/Problem 3|Solution]]
 
[[2022 SSMO Relay Round 3 Problems/Problem 3|Solution]]

Latest revision as of 19:18, 2 May 2025

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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