Problems

In a physics club, the teacher created the following experiment. He spread out 16 weights of weight 1, 2, 3, ..., 16 grams onto weighing scales, so that one of the bowls outweighed the other. Fifteen students in turn left the classroom and took with them one weight each, and after each student’s departure, the scales changed their position and outweighed the opposite bowl of the scales. What weight could remain on the scales?

Solve the equation: \(|x-2005| + |2005-x|=2006\).

Solve the equation: \[x + \frac{x}{x} + \frac{x}{x+\frac{x}{x}} = 1\]

Between them, Jennifer and Alex shared the money they made from running a lemonade stand. Jennifer thought: “If I took \(40\%\) more money then Alex’s share would decrease by \(60\%\)”. How would Alex’s share of the profits change if Jennifer took \(50\%\) more money for herself?

Find all functions \(f (x)\) such that \(f (2x + 1) = 4x^2 + 14x + 7\).

In a board, 20 pins are placed (see the picture). The distance between any adjacent pins is 1 inch. Pull a string of length 19 inches from the first pin to the second one, so that it goes through all the pins.

Henry did not manage to get into the elevator on the first floor of the building and decided to go up the stairs. It takes 2 minutes to rise to the third floor. How long does it take to rise to the ninth floor?

The grasshopper jumps on the interval \([0,1]\). On one jump, he can get from the point \(x\) either to the point \(x/3^{1/2}\), or to the point \(x/3^{1/2} + (1- (1/3^{1/2}))\). On the interval \([0,1]\) the point \(a\) is chosen.

Prove that starting from any point, the grasshopper can be, after a few jumps, at a distance less than \(1/100\) from point \(a\).

Two different numbers \(x\) and \(y\) (not necessarily integers) are such that \(x^2-2000x=y^2-2000y\). Find the sum of \(x\) and \(y\).

All of the sweets of different sorts in stock are arranged in \(n\) boxes, for which prices are set at \(1, 2, \dots , n\), respectively. It is required to buy such \(k\) of these boxes of the least total value, which contain at least \(k/n\) of the mass of all of the sweets. It is known that the mass of sweets in each box does not exceed the mass of sweets in any more expensive box.

a) What boxes should I buy when \(n = 10\) and \(k = 3\)?

b) The same question for arbitrary natural numbers \(n \geq k\).