Term 1 Week 3 Junior Problems
Problem 1
Cities $A$, $B$, $C$, $D$, and $E$ are connected by roads $\widetilde{AB}$, $\widetilde{AD}$, $\widetilde{AE}$, $\widetilde{BC}$, $\widetilde{BD}$, $\widetilde{CD}$, and $\widetilde{DE}$. How many different routes are there from $A$ to $B$ that use each road exactly once? (Such a route will necessarily visit some cities more than once.)
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Solution
Note that cities $C$ and $E$ can be removed when counting paths because if a path goes in to $C$ or $E$, there is only one possible path to take out of cities $C$ or $E$. So the diagram is as follows:
Now we proceed with casework. Remember that there are two ways to travel from $A$ to $D$, $D$ to $A$, $B$ to $D$ and $D$ to $B$:
Case 1
$A \Rightarrow D$: From $D$, if the path returns to $A$, then the next path must go to $B\Rightarrow D \Rightarrow B$. There are $2 \cdot 1 \cdot 2 = 4$ possibilities of the path $ADABDB$. If the path goes to $D$ from $B$, then the path must continue with either $BDAB$ or $BADB$. There are $2 \cdot 2 \cdot 2 = 8$ possibilities. So, this case gives $4+8=12$ different possibilities.Case 2
$A \Rightarrow B$: The path must continue with $BDADB$. There are $2 \cdot 2 = 4$ possibilities for this case.Putting the two cases together gives $12+4 = \boxed{\textbf{(D)} \ 16}$
Problem 2
Danica, Jessica, Marina, Neo and Samantha went to a popular pet store at different times and bought different pets. Marina is the second person who went inside the store. Danica went in before the person who bought a cat but went inside after the person who bought a hamster. Jessica went inside after Neo. The last person who went inside bought a rabbit. Marina did not buy a cat, and Samantha bought a dog. Who bought a fish?
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Solution
Problem 3
Hannah, Ivan and Jefferson are collecting shells on Barbados island. Hannah collected $\frac{11}{19}$ of the total shells. the rest was shared by Ivan and Jefferson in a ratio of $13:11$. if Hannah collected 44 more shells, then Jefferson what is the total number of shells they collected altogether?
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Solution
Problem 4
Simplify $\sqrt{53 - 10\sqrt{6}}$
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Solution
