We solve problems

Problem #PRU-115397

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

13–15 3.0

The sequence \(a_1, a_2, \dots\) is such that \(a_1 \in (1,2)\) and \(a_{k + 1} = a_k + \frac{k}{a_k}\) for any positive integer \(k\). Prove that it cannot contain more than one pair of terms with an integer sum.The sequence \(a_1, a_2, \dots\) is such that \(a_1 \in (1,2)\) and \(a_{k + 1} = a_k + \frac{k}{a_k}\) for any positive integer \(k\). Prove that it cannot contain more than one pair of terms with an integer sum.

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-116256

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

Mark has 1000 identical cubes, each of which has one pair of opposite faces which are coloured white, another pair which are blue and a third pair that are red. He made a large \(10 \times 10 \times 10\) cube from them, joining cubes to one another which have the same coloured faces. Prove that the large cube has a face which is solidly one colour.

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-116775

Problems Methods Examples and counterexamples. Constructive proofs Pigeonhole principle Pigeonhole principle (other) Set theory and logic Theory of algotithms Theory of algorithms (other)

13–16 4.0

We are given a polynomial \(P(x)\) and numbers \(a_1\), \(a_2\), \(a_3\), \(b_1\), \(b_2\), \(b_3\) such that \(a_1a_2a_3 \ne 0\). It turned out that \(P (a_1x + b_1) + P (a_2x + b_2) = P (a_3x + b_3)\) for any real \(x\). Prove that \(P (x)\) has at least one real root.

Problem #PRU-116817

Problems Methods Algebraic methods Counting in two ways Set theory and logic Mathematical logic Mathematical logic (other)

13–14 3.0

There is a group of 5 people: Alex, Beatrice, Victor, Gregory and Deborah. Each of them has one of the following codenames: V, W, X, Y, Z. We know that:

Alex is 1 year older than V,

Beatrice is 2 years older than W,

Victor is 3 years older than X,

Gregory is 4 years older than Y.

Who is older and by how much: Deborah or Z?

Problem #PRU-30395

Problems Number Theory Number theory. Divisibility Equations in integer numbers

13–15 3.0

Prove there are no natural numbers \(a\) and \(b\), such as \(a^2 - 3b^2 = 8\).

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).