Problems of the Week

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2018-08-27 Intermediate

         

When sunlight passes through water droplets in the atmosphere, some of the refracted light is reflected. Because different colors bend by different angles, we see a circular arc of different colors—a rainbow.

At what time of day does a rainbow appear the tallest when viewed from the ground?

\(ABCD\) is a square and \(AE=BF=CG=DH.\)

What is the area of the orange quadrilateral \(DHMG?\)

A teacher wrote a number on the board and asked the students to tell about the divisors of the number one by one.

  • The \(1^\text{st}\) student said, "The number is divisible by 2."
  • The \(2^\text{nd}\) student said, "The number is divisible by 3."
  • The \(3^\text{rd}\) student said, "The number is divisible by 4."
  • \(\quad \quad \vdots\)
  • The \(30^\text{th}\) student said, "The number is divisible by 31."

The teacher then said that exactly two consecutive students were incorrect.

Which two students were incorrect?

If the \(m^\text{th}\) and \((m+1)^\text{th}\) students spoke wrongly, then enter \(m.\)

\( a !+ b !+c != d !,\) where \( a, b, c, d\) are all positive integers.

What is the number of possible solutions \((a, b, c, d)?\)

Suppose you inscribe a smaller triangle in the triangle below such that its three vertices are on different sides of the given triangle.

What is the smallest possible perimeter of the inscribed triangle?

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