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2017-06-05 Basic

$\begin{array} { rrrr } & & A & B & C \\ & & B & C & A \\ &+ & C & A & B \\ \hline & & &\phantom0 & \end{array}$

In the cryptogram above, $$A,B,$$ and $$C$$ are each single-digit integers that are not necessarily all distinct. Which of the following is a possible value of the final sum?

The hexagon below is regular. Which of the two colored areas is larger?

A ceiling fan is hung on a hook through a spring balance, which reads $$M$$ kg when the fan is off.

If the fan is switched on and rotates at a constant speed, then what will be the reading on the balance?


Details and Assumptions:

• Ignore the effects of torque.
• The fan blows the air downwards.

You are offered two cupcakes. One is poisoned and the other is safe to eat.

You happen to be in a village full of knights (who always tell the truth) and knaves (who always lie), but you can't tell which is which by their appearance.

You ask one of them, "Which cupcake is safe to eat?" To this he makes the following two statements,

 1 2 "If I were a knave, I'd say the one on the right." "But I'd say the one on the left, if I were a knight." 

Which cupcake is safe to eat?

Assumption: The person you ask knows which cupcake is which.


The above is one possible way to arrange 5 unit squares into a polygon, and this polygon has a perimeter of 12. What is the minimum possible perimeter of a polygon that can be constructed with 5 unit squares?


Details and Assumptions:

• No overlapping is allowed.
• A vertex can only touch a vertex--not anywhere on the side--of other squares.
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