We solve problems

Problems Algebra and arithmetic Rational functions Rational equations

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

Problems Algebra and arithmetic 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-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.

Problem #PRU-105208

Problems Algebra and arithmetic Word problems Tables and tournaments Number tables and its properties Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 4.0

A teacher filled the squares of a chequered table with \(5\times5\) different integers and gave one copy of it to Janine and one to Zahara. Janine selects the largest number in the table, then she deletes the row and column containing this number, and then she selects the largest number of the remaining integers, then she deletes the row and column containing this number, etc. Zahara performs similar operations, each time choosing the smallest numbers. Can the teacher fill up the table in such a way that the sum of the five numbers chosen by Zahara is greater than the sum of the five numbers chosen by Janine?

Problem #PRU-107698

Problems Algebra and arithmetic Mean values Methods Proof by contradiction

11–14 2.0

Is it possible for the mean of some 35 whole numbers to equal \(6.35\)?

Problem #PRU-107702

Problems Geometry Solid geometry Parallelepipeds Special cases of parallelepipeds Cube Combinatorics Dissections, partitions, covers and tilings Features of dissection pieces Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

13–16 3.0

The surface of a \(3\times 3\times 3\) Rubik’s Cube contains 54 squares. What is the maximum number of squares we can mark, so that no marked squares share a vertex or are directly adjacent to another marked square?

Problem #PRU-107749

Problems Algebra and arithmetic Number systems Decimal number system Algebraic equations and systems of equations Diophantine equations

13–15 3.0

The student did not notice the multiplication sign between two three-digit numbers and wrote one six-digit number, which turned out to be seven times bigger than their product. Determine these numbers.

Problem #PRU-107760

Algebraic equations and systems of equations Diophantine equations Problems Algebra and arithmetic Number systems Decimal number system

13–15 3.0

The student did not notice the multiplication sign between two seven-digit numbers and wrote one fourteen-digit number, which turned out to be three times bigger than their product. Determine these numbers.