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Problem #PRU-109704

Problems Algebra Algebraic inequalities and systems of inequalities Algebraic inequalities (other) Methods Algebraic methods Counting in two ways Calculus Real numbers Integer and fractional parts. Archimedean property

13–15 4.0

Prove that for any positive integer \(n\) the inequality

is true.

Problem #PRU-109707

Problems Functions of one variable. Continuity Certain properties of a function and recurrence relations. Calculus

14–16 4.0

Find all the functions \(f\colon \mathbb {R} \rightarrow \mathbb {R}\) which satisfy the inequality \(f (x + y) + f (y + z) + f (z + x) \geq 3f (x + 2y + 3z)\) for all \(x, y, z\).

Problem #PRU-109715

Problems Algebra Sequences Progressions Geometric progression Sums of number sequences and difference series Real numbers Integer and fractional parts. Archimedean property Calculus

13–15 4.0

Find the sum \(1/3 + 2/3 + 2^2/3 + 2^3/3 + \dots + 2^{1000}/3\).

Problem #PRU-109748

Problems Geometry Plane geometry Convex and non-convex figures Convex polygons Discrete Mathematics Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Polygons Polygons (other)

13–15 4.0

We are given a convex 200-sided polygon in which no three diagonals intersect at the same point. Each of the diagonals is coloured in one of 999 colours. Prove that there is some triangle inside the polygon whose sides lie some of the diagonals, so that all 3 sides are the same colour. The vertices of the triangle do not necessarily have to be the vertices of the polygon.

Problem #PRU-109766

Problems Fun Problems Word Problems Tables and tournaments Methods Number tables and its properties Pigeonhole principle Pigeonhole principle (other)

12–14 4.0

Is it possible to arrange natural numbers from 1 to \(2002^2\) in the cells of a \(2002\times2002\) table so that for each cell of this table one could choose a triplet of numbers, from a row or column, where one of the numbers is equal to the product of the other two?

Problem #PRU-109780

Problems Real numbers Discrete Mathematics Set theory and logic Set theory Calculus

14–16 4.0

A number set \(M\) contains \(2003\) distinct positive numbers, such that for any three distinct elements \(a, b, c\) in \(M\), the number \(a^2 + bc\) is rational. Prove that we can choose a natural number \(n\) such that for any \(a\) in \(M\) the number \(a\sqrt{n}\) is rational.

Problem #PRU-109787

Problems Algebra Polynomials Real numbers Rational and irrational numbers Algebraic identities for polynomials Factoring polynomials Calculus

14–16 4.0

A numeric set \(M\) containing 2003 distinct numbers is such that for every two distinct elements \(a, b\) in \(M\), the number \(a^2+ b\sqrt 2\) is rational. Prove that for any \(a\) in \(M\) the number \(q\sqrt 2\) is rational.

Problem #PRU-109819

Problems Functions of one variable. Continuity Monotonicity, boundedness Certain properties of a function and recurrence relations. Calculus

14–16 4.0

Is there a bounded function \(f\colon \mathbb{R} \rightarrow \mathbb{R}\) such that \(f (1)> 0\) and \(f (x)\) satisfies the inequality \(f^2 (x + y) \geq f^2 (x) + 2f (xy) + f^2 (y)\) for all \(x, y \in \mathbb{R}\)?

Problem #PRU-109912

Problems Functions of one variable. Continuity Methods Certain properties of a function and recurrence relations. Methods for solving problems with parameter Calculus

15–16 4.0

For which \(\alpha\) does there exist a function \(f\colon \mathbb{R} \rightarrow \mathbb{R}\) that is not a constant, such that \(f (\alpha (x + y)) = f (x) + f (y)\)?

Problem #PRU-109937

Problems Methods Invariants and semi-invariants Algebraic methods Invariants Proof by exhaustion Processes and operations Pigeonhole principle (finite number of poits, lines etc.) Pigeonhole principle

14–16 4.0

We are given a table of size \(n \times n\). \(n-1\) of the cells in the table contain the number \(1\). The remainder contain the number \(0\). We are allowed to carry out the following operation on the table:

1. Pick a cell.

2. Subtract 1 from the number in that cell.

3. Add 1 to every other cell in the same row or column as the chosen cell.

Is it possible, using only this operation, to create a table in which all the cells contain the same number?