We solve problems

Problems Algebra Rational functions Rational equations

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

Problems Fun Problems Word Problems Percentages and ratios

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?

Problem #PRU-105052

Problems Methods Pigeonhole principle Pigeonhole principle (other) Proof by contradiction Fun Problems Word Problems Tables and tournaments Tables and tournaments (other)

11–13 3.0

In a chess tournament, each participant played two games with each of the other participants: one with white pieces, the other with black. At the end of the tournament, it turned out that all of the participants scored the same number of points (1 point for a victory, \(\frac{1}{2}\) a point for a draw and 0 points for a loss). Prove that there are two participants who have won the same number of games using white pieces.

Problem #PRU-105063

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

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 \(a\).

Problem #PRU-105072

Problems Algebra Polynomials Algebraic identities for polynomials Identical transformations

12–14 2.0

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

Problem #PRU-105204

Problems Methods Pigeonhole principle Pigeonhole principle (other) Proof by contradiction

12–14 3.0

In a mathematical olympiad, \(m>1\) candidates solved \(n>1\) problems. Each candidate solved a different number of problems to all the others. Each problem was solved by a different number of candidates to all the others. Prove that one of the candidates solved exactly one problem.