2018-08-13 Basic Practice Problems Online

Pam normally weighs \(\SI{60}{\kilo\gram},\) but in an elevator, her weight measured \(\SI{70}{\kilo\gram}.\)

What could have been happening at the instant her weight was measured?

\[\begin{array}{cccc} & \color{red}A & \color{red}A & \color{red}A \\ \times & & & \color{red}A \\ \hline 3 & \color{blue}B & \color{blue}B & \color{red}A \end{array}\]

What is \({\color{blue}B}?\)

Is there a prime number \(p\) that satisfies this inequality?

\[2018!+2 \le p \le 2018!+2018\]

Note: \(2018!\) is the factorial of 2018. That is, \(2018!=1 \times 2 \times 3 \times \cdots \times 2016 \times 2017 \times 2018.\)

This \(4\times 4\) grid of unit squares is tiled with L-shaped pieces of area 3, with one square blacked out.

Now, the side length \(4=2^2\) of this square grid is extended to a larger power of 2, as illustrated below, and again one of the squares is blacked out.

Can you always tile this grid with L-shaped pieces of area 3, no matter where the black square is?

The grid can have side length 8, 16, 32, or any power of 2. One square is blacked out in each grid.

How many quadrilaterals are drawn in this figure?

Hint: Be sure to count both convex and concave quadrilaterals.

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