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?

ABCDABCD is a square and AE=BF=CG=DH.AE=BF=CG=DH.

What is the area of the orange quadrilateral DHMG?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 1st1^\text{st} student said, "The number is divisible by 2."
  • The 2nd2^\text{nd} student said, "The number is divisible by 3."
  • The 3rd3^\text{rd} student said, "The number is divisible by 4."
  • \quad \quad \vdots
  • The 30th30^\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 mthm^\text{th} and (m+1)th(m+1)^\text{th} students spoke wrongly, then enter m.m.

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

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