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2017-11-27 Intermediate

What percentage of the square's area is colored blue?

15 can be written as the difference between two positive perfect squares: $15 = 16-1= 4^2 - 1^2.$ Which of the following numbers cannot be written as the difference of two positive perfect squares?

A factory manufactures wooden squares whose side length should be one meter. However, the side length of the squares is not always $\SI{1}{\meter}$ but varies uniformly between $\SI{1}{\meter}$ and $\SI{1.1}{\meter}.$ Note that the height and the breadth of each square piece remain equal to each other.

Two inspectors, Alice and Bob, want to calculate the average area of a square.

• Alice claims that the average area is $\SI{1.1025}{\meter\squared}$ because the average side length is $\SI{1.05}{\meter}$ and $1.05 \times 1.05 = 1.1025.$
• Bob claims that the average area is $\SI{1.1050}{\meter\squared}$ because the the area is between $(\SI{1}{\meter} \times \SI{1}{\meter})$ and $(\SI{1.1}{\meter} \times \SI{1.1}{\meter}),$ and $\frac{1 \times 1 + 1.1 \times 1.1}{2} = 1.105$.

Who is correct?

An accelerometer is made using tubular sections, as shown below. The accelerometer is placed in a rocket that travels in the $\vec{x}$ direction, and the liquid levels are steady at $\SI{20}{\centi\meter}$ and $\SI{10}{\centi\meter}.$

Find the magnitude of the rocket’s acceleration (in $\text{m/s}^2$). Submit your answer to 3 decimal places.


Details and Assumptions:

• The diameter of the tube is much smaller than the heights of the liquid in the two arms.
• $g = 10 \text{ m/s}^{2}.$

A ladder is placed on a stairway (as shown). What is the length of the shorter leg?


Details and Assumptions:

• Each step has the same horizontal run of $15 \text{ cm}$ and vertical run of $10 \text{ cm}.$
• Each leg of the ladder rests on the horizontal midpoint of a step.
• The angles that the legs make with the steps are congruent.
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