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Basic Arithmetic

Please Excuse My Dear Aunt Sally (PEMDAS), she has the hardest time remembering how to make sense of these arithmetic rules...

Level 2

         

\[ \large \color{red}3^{\color{blue}{1506}} - \color{red}3^{\color{blue}{1505}} = \ ? \]

A boy is pushing a ring of mass 2 kg and radius 0.5 m with a stick as shown in the figure. The stick applies a force of 2N on the ring and rolls it without slipping with an acceleration of 0.3 \(m/s^{2}\). The coefficient of friction between the ground and the ring is large enough that rolling always occurs and the coefficient of friction between the stick and the ring is \[\frac{p}{10}\]. Then the value of p is?

Evaluate \[ \left( 20142014 \times 20142014-20142004 \times 20142024 \right).\]

\[\Large \color{blue}{n}^{\color{red}{200}} < \color{green}{5}^{\color{brown}{300}}\]

Find the largest integer \(n\) that satisfies the above inequality.

Fill the empty squares with integers from 1 to 9, using each integer exactly once, to get the given results. Calculations are done from left to right, and from top to bottom.

What is the sum of the sums of numbers on the 2 diagonals?

Note that the center square is added twice.

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