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2017-02-06 Intermediate

         

A positive integer \(n\) leaves the same remainder of 35 when divided by both 2009 and 2010.

What is the remainder when \(n\) is divided by 42?

A faulty beam balance has pans of different masses, and its beam remains horizontal when the pans are empty. On such a beam balance, a mass of \(m\) is put on one pan and it is balanced by a weight of 9 grams. When the same mass is put on the other pan, it can be balanced by a weight of 16 grams.

Find the mass \(m\) in grams.

The following is my attempt at proving that \(1 < -1\). In which of these steps did I first make a mistake by using flawed logic?

Step 1: Let \(i = \sqrt{-1} \), then \(i^2 = -1\) and \(i^4 = \big(i^2\big)^2 = (-1)^2 = 1\).

Step 2: Hence, we have \(i^2 < i^4 \).

Step 3: We divide both sides by \(i\) to get \(i < i^3\).

Step 4: We divide both sides by \(i\) again to get \(1 < i^2 \) or, equivalently, \(1 < -1 \).

Andrew and Benjamin are playing a game on an \( 8 \times 8 \) chessboard. Each turn, they place a knight in a position that isn't threatened by other knights that are already on the board. The first person who is unable to place a knight loses the game.

As an explicit example, the board above shows a possible sequence of 5 turns, where all the squares that are threatened by other knights are marked with red X's.

If Andrew goes first, who will win this game?

\[\large \sum_{n=1}^{\infty} \frac{2^{2^{n}}-1}{2^{2^{n+1}}} = \frac ab \]

If the equation above holds true for coprime positive integers \(a\) and \(b\), find the value of \(a+b\).

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