Problems – We Solve Problems
Filter Problems
Showing 1 to 12 of 12 entries

#### Extremal principle (other) , Mathematical induction (other) , Polynomial remainder theorem (little Bezout's theorem). Factorisation. , Polynomials with integer coefficients and integer values , Proof by contradiction , Rational and irrational numbers

Prove that if the irreducible rational fraction p/q is a root of the polynomial $P (x)$ with integer coefficients, then $P (x) = (qx – p) Q (x)$, where the polynomial $Q (x)$ also has integer coefficients.

#### Extremal principle (other) , Mathematical induction (other) , Partitions into pairs and groups bijections , Pigeonhole principle (other)

We are given $n+1$ different natural numbers, which are less than $2n$ $(n>1)$. Prove that among them there will always be three numbers, where the sum of two of them is equal to the third.

#### Iterations , Mathematical induction (other) , Processes and operations , Rational and irrational numbers , Theory of algorithms (other)

With a non-zero number, the following operations are allowed: $x \rightarrow \frac{1+x}{x}, x \rightarrow \frac{1-x}{x}$. Is it true that from every non-zero rational number one can obtain each rational number with the help of a finite number of such operations?

My Problem Set reset
No Problems selected