# Problems of the Week

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# 2017-11-20 Basic

A mirage is a frequent optical occurrence for desert travelers, which appears as a distant body of water. Which of the following best explains this phenomenon? Note: The mirage featured in this picture is known as an inferior mirage.

Alice arranges 3 coins in a row. You cannot see them, and need to guess the sequence—heads or tails—of the 3 coins. You make three guesses, and Alice responds as follows: Guess $\hspace{1.4cm}$ Response H H H "You got only 1 coin correct." T T H "You got only 1 coin correct." H T T "You got only 1 coin correct."

What is the correct sequence?

I'm thinking of a number.

• It is a two-digit positive integer.
• The sum of its digits is 10.
• Subtracting 72 from the number swaps its two digits.

What is the number?

10 identical circles are stacked in a pyramid shape, where $A$ and $B$ are centers of two of the circles. A line is drawn through $A$ and $B,$ dividing the figure into 2 areas: red on the left and and blue on the right.

What is the ratio of the red area to the blue area? A kite is divided into four regions by two line segments that connect the midpoints of opposite sides. Is the total blue area equal to the total orange area? ×