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2017-02-06 Intermediate

         

A positive integer nn leaves the same remainder of 35 when divided by both 2009 and 2010.

What is the remainder when nn 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 mm 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 mm in grams.

The following is my attempt at proving that 1<11 < -1. In which of these steps did I first make a mistake by using flawed logic?

Step 1: Let i=1i = \sqrt{-1} , then i2=1i^2 = -1 and i4=(i2)2=(1)2=1i^4 = \big(i^2\big)^2 = (-1)^2 = 1.

Step 2: Hence, we have i2<i4i^2 < i^4 .

Step 3: We divide both sides by ii to get i<i3i < i^3.

Step 4: We divide both sides by ii again to get 1<i21 < i^2 or, equivalently, 1<11 < -1 .

Andrew and Benjamin are playing a game on an 8×8 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?

n=122n122n+1=ab\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 aa and bb, find the value of a+ba+b.

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