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2017-03-20 Basic

         

Within the orange square, there are 4 circles each with radius 1. What is the area of the green region in the center?


Hint: Connect the centers of the circles.

An ant climbing down an ant-hill can choose to go either down and to the right or down and to the left at each level until it reaches the bottom of the hill. And at each junction, the ant collects a number of sugar grains indicated by the numbers in the circle.

Being greedy, this ant believes that if it always moves down towards the larger of the two values directly below, it will be able to collect the largest number of sugar grains overall. With the hill as shown above, is this greedy strategy optimal for collecting sugar?

A double-decker bus wants to take a turn safely, without tipping over. Which of the following will decrease the chance of the bus toppling?

Using a single (possibly zig-zagging) line, can we cut the \( 4 \times 6 \) rectangle into 2 identical pieces, and rearrange them into a \( 5 \times 5\) square with a \( 1 \times 1 \) hole in the center?


Note: The cut may not be straight.

As an explicit example, this is how to get a \( 6 \times 5 \) square with a \( 2 \times 3 \) hole in the center:

I captured a snapshot of lightning as it hit the ground. What is the sum of all 15 marked angles?

Assume that the base of the cloud and the ground to be parallel lines.

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