We solve problems

Problems Number Theory Numeral systems Decimal number system Methods Trial and error

Let \(x\) be a 2 digit number. Let \(A\), \(B\) be the first (tens) and second (units) digits of \(x\), respectively. Suppose \(A\) is twice as large as \(B\). If we add the square of \(A\) to \(x\) then we get the square of a certain whole number. Find the value of \(x\).

Problem #PRU-104083

Problems Discrete Mathematics 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 Discrete Mathematics Combinatorics Geometry on grid paper 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 Fun Problems 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

12–14 2.0

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

Problem #PRU-104117

Problems Geometry Plane geometry Visual geometry

12–13 2.0

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.

Problem #PRU-104122

Problems Algebra and arithmetic Arithmetics. Mental maths Fun Problems Joke Problems

12–13 2.0

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?

Problem #PRU-104879

Problems Discrete Mathematics Set theory and logic Mathematical logic Mathematical logic (other)

11–13 3.0

There are \(12\) aliens in the High Council of the planet of liars and truth tellers. “There is no-one honest here,” said the first member of the council. “There is at most one honest person here,” said the second person. The third person said that there are at most \(2\) honest members, the fourth person said there are at most \(3\) honest aliens, and so on until the twelfth person, who said there are at most \(11\) honest aliens. How many honest members are in the High Council?