We solve problems

Problems Methods Examples and counterexamples. Constructive proofs

Decipher the following puzzle. All the numbers indicated by the letter E, are even (not necessarily equal); all the numbers indicated by the letter O are odd (also not necessarily equal).

Problem #PRU-97766

Problems Algebra Algebraic equations and systems of equations Higher order equations. Palindromic polynomial equations Equations of higher order (other)

12–14 3.0

Determine all integer solutions of the equation \(yk = x^2 + x\). Where \(k\) is an integer greater than \(1\).

Problem #PRU-97832

Problems Calculus Functions of one variable. Continuity Continuous functions (general properties)

14–17 3.0

\(f(x)\) is an increasing function defined on the interval \([0, 1]\). It is known that the range of its values belongs to the interval \([0, 1]\). Prove that, for any natural \(N\), the graph of the function can be covered by \(N\) rectangles whose sides are parallel to the coordinate axes so that the area of each is \(1/N^2\). (In a rectangle we include its interior points and the points of its boundary).

Problem #PRU-97865

Problems Calculus Real numbers Integer and fractional parts. Archimedean property

14–16 3.0

a) Give an example of a positive number \(a\) such that \(\{a\} + \{1 / a\} = 1\).

b) Can such an \(a\) be a rational number?

Problem #PRU-97900

Problems Algebra Sequences

15–18 3.0

For which natural \(n\) does the number \(\frac{n^2}{1.001^n}\) reach its maximum value?

Problem #PRU-97903

Problems Algebra Functional equations

15–18 3.0

The function \(F\) is given on the whole real axis, and for each \(x\) the equality holds: \(F (x + 1) F (x) + F (x + 1) + 1 = 0\).

Prove that the function \(F\) can not be continuous.

Problem #PRU-97942

Problems Discrete Mathematics Algorithm Theory Game theory Game theory (other)

10–13 3.0

Two players in turn increase a natural number in such a way that at each increase the difference between the new and old values of the number is greater than zero, but less than the old value. The initial value of the number is 2. The winner is the one who can create the number 1987. Who wins with the correct strategy: the first player or his partner?

Problem #PRU-97984

Problems Number Theory Divisibility Division with remainders. Arithmetic of remainders Arithmetic of remainders Methods Examples and counterexamples. Constructive proofs

13–16 3.0

a) We are given two cogs, each with 14 teeth. They are placed on top of one another, so that their teeth are in line with one another and their projection looks like a single cog. After this 4 teeth are removed from each cog, the same 4 teeth on each one. Is it always then possible to rotate one of the cogs with respect to the other so that the projection of the two partially toothless cogs appears as a single complete cog? The cogs can be rotated in the same plane, but cannot be flipped over.

b) The same question, but this time two cogs of 13 teeth each from which 4 are again removed?

Problem #PRU-97991

Problems Fun Problems Word Problems Tables and tournaments Chessboards and chess pieces

10–15 3.0

What is the minimum number of squares that need to be marked on a chessboard, so that:

1) There are no horizontally, vertically, or diagonally adjacent marked squares.

2) Adding any single new marked square breaks rule 1.

Problem #PRU-98017

Problems Methods Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

A staircase has 100 steps. Vivian wants to go down the stairs, starting from the top, and she can only do so by jumping down and then up, down and then up, and so on. The jumps can be of three types – six steps (jumping over five to land on the sixth), seven steps or eight steps. Note that Vivian does not jump onto the same step twice. Will she be able to go down the stairs?