Waste less time on Facebook — follow Brilliant.
×

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?

View solution
Djordje Veljkovic by Djordje Veljkovic

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.

View solution
Rohit Gupta by Rohit Gupta

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, \(1 < -1 \).

View solution
Pi Han Goh by Pi Han Goh

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?

View solution
Chung Kevin by Chung Kevin

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

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

View solution
Freddie Hand by Freddie Hand
×

Problem Loading...

Note Loading...

Set Loading...