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Problem #PRU-115447

Problems Calculus Real numbers Integer and fractional parts. Archimedean property Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

13–15 3.0

Prove that if the expression

takes a rational value, then the expression

also takes on a rational value.

Problem #PRU-116558

Problems Calculus Real numbers Integer and fractional parts. Archimedean property Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

13–15 3.0

The nonzero numbers \(a\), \(b\), \(c\) are such that every two of the three equations \(ax^{11} + bx^4 + c = 0\), \(bx^{11} + cx^4 + a = 0\), \(cx^{11} + ax^4 + b = 0\) have a common root. Prove that all three equations have a common root.

Problem #PRU-116704

Problems Calculus Real numbers Integer and fractional parts. Archimedean property Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

13–15 3.0

The teacher wrote on the board in alphabetical order all possible \(2^n\) words consisting of \(n\) letters A or B. Then he replaced each word with a product of \(n\) factors, correcting each letter A by \(x\), and each letter B by \((1 - x)\), and added several of the first of these polynomials in \(x\). Prove that the resulting polynomial is either a constant or increasing function in \(x\) on the interval \([0, 1]\).

Problem #PRU-30428

Problems Algebra and arithmetic Algebraic equations and systems of equations Higher order equations. Palindromic polynomial equations Equations of higher order (other) Calculus Real numbers Integer and fractional parts. Archimedean property

12–14 2.0

Prove that a graph with \(n\) vertices, the degree of each of which is at least \(\frac{n-1}{2}\), is connected.

Problem #PRU-30429

Problems Algebra and arithmetic Algebraic equations and systems of equations Higher order equations. Palindromic polynomial equations Equations of higher order (other) Calculus Real numbers Integer and fractional parts. Archimedean property

12–14 2.0

In the Far East, the only type of transport is a carpet-plane. From the capital there are 21 carpet-planes, from the city of Dalny there is one carpet-plane, and from all of the other cities there are 20. Prove that you can fly from the capital to Dalny (possibly with interchanges).

Problem #PRU-30782

Problems Algebra and arithmetic Algebraic equations and systems of equations Higher order equations. Palindromic polynomial equations Equations of higher order (other) Calculus Real numbers Integer and fractional parts. Archimedean property

12–14 2.0

a) they have 10 vertices, the degree of each of which is equal to 9?

b) they have 8 vertices, the degree of each of which is equal to 3?

c) are they connected, without cycles and contain 6 edges?

Problem #PRU-30808

Problems Algebra and arithmetic Algebraic equations and systems of equations Higher order equations. Palindromic polynomial equations Equations of higher order (other) Calculus Real numbers Integer and fractional parts. Archimedean property

12–14 2.0

On the plane 100 circles are given, which make up a connected figure (that is, not falling apart into pieces). Prove that this figure can be drawn without taking the pencil off of the paper and going over any line twice.

Problem #PRU-30820

Problems Algebra and arithmetic Algebraic equations and systems of equations Higher order equations. Palindromic polynomial equations Equations of higher order (other) Calculus Real numbers Integer and fractional parts. Archimedean property

12–14 2.0

In some country there is a capital and another 100 cities. Some cities (including the capital) are connected by one-way roads. From each non-capital city 20 roads emerge, and 21 roads enter each such city. Prove that you cannot travel to the capital from any city.

Problem #PRU-30822

Problems Algebra and arithmetic Algebraic equations and systems of equations Higher order equations. Palindromic polynomial equations Equations of higher order (other) Calculus Real numbers Integer and fractional parts. Archimedean property

12–14 2.0

Prove that on the edges of a connected graph one can arrange arrows so that from some vertex one can reach any other vertex along the arrows.

Problem #PRU-31072

Problems Algebra and arithmetic Algebraic equations and systems of equations Higher order equations. Palindromic polynomial equations Equations of higher order (other) Calculus Real numbers Integer and fractional parts. Archimedean property

12–14 2.0

In some country 89 roads emerge from the capital, from the city of Dalny – one road, from the remaining 1988 cities – 20 roads (in each).

Prove that from the capital you can drive to Dalny.