We solve problems

Problems Combinatorics Geometry on grid paper

Cut the shape (see the figure) into two identical pieces (coinciding when placed on top of one another).

Problems Set theory and logic Algorithm Theory Algorithm Theory

Gabby is standing on a river bank. She has two clay jars: one – for 5 litres, and about the second Gabby remembers only that it holds either 3 or 4 litres. Help Gabby determine the capacity of the second jar. (Looking into the jar, you cannot figure out how much water is in it.)

Problem #PRU-104074

Problems Set theory and logic Set theory Inclusion-exclusion principle

11–13 2.0

In the garden of Sandra and Lewis 2006 rose bushes were growing. Lewis watered half of all the bushes, and Sandra watered half of all the bushes. At the same time, it turned out that exactly three bushes, the most beautiful, were watered by both Sandra and Lewis. How many rose bushes have not been watered?

Problem #PRU-104078

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

In a burrow there is a family of 24 mice. Every night exactly four of them are sent to the warehouse for cheese.

Could it occur that at some point in time each mouse went to the warehouse with every other mouse exactly one time?

Problem #PRU-104083

Problems Set theory and logic Algorithm Theory Balance puzzles

11–13 2.0

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?

Problem #PRU-104084

Problems Algebra Absolute value Equations containing absoute values

12–14 2.0

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

Problem #PRU-104089

Problems Combinatorics Geometry on grid paper Set theory and logic Algorithm Theory Game Theory Winning and loosing positions

11–14 3.0

There is a \(5\times 9\) rectangle drawn on squared paper. In the lower left corner of the rectangle is a button. Kevin and Sophie take turns moving the button any number of squares either to the right or up. Kevin goes first. The winner is the one who places the button in upper right corner. Who would win, Kevin or Sophie, by using the right strategy?

Problem #PRU-104090

Problems Algebra Rational functions Rational equations

12–14 2.0

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

Problem #PRU-104103

Problems Algebra Word Problems Percentages and ratios

12–14 2.0

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?

Problem #PRU-104104

Problems Calculus Functions of one variable. Continuity Certain properties of a function and recurrence relations. Algebra Polynomials Quadratic polynomials Quadratic polynomilal (other)

12–14 2.0

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