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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?
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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.
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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=−1, then i2=−1 and i4=(i2)2=(−1)2=1.
Step 2: Hence, we have i2<i4.
Step 3: We divide both sides by i to get i<i3.
Step 4: We divide both sides by i again to get 1<i2 or, equivalently, 1<−1.
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Andrew and Benjamin are playing a game on an 8×8 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?
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n=1∑∞22n+122n−1=ba
If the equation above holds true for coprime positive integers a and b, find the value of a+b.
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