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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?

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Pranshu Gaba by Pranshu Gaba

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

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

Chan Lye Lee by Chan Lye Lee

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 \(m.\)

Harsh Shrivastava by Harsh Shrivastava

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

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

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Tan Peng by Tan Peng

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?

Matthieu Courbaize by Matthieu Courbaize
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