Problems – Page 2 – We Solve Problems
Filter Problems
Showing 21 to 40 of 94 entries

Examples and counterexamples. Constructive proofs , Pigeonhole principle (other) , Theory of algorithms (other)

a$)$ Using geometric considerations, prove that the base and the side of an isosceles triangle with an angle of 36 $^{\circ}$ at the vertex are incommensurable.

b$)$ Invent a geometric proof of the irrationality of $\sqrt{2}$.

Examples and counterexamples. Constructive proofs , Pigeonhole principle (other) , Theory of algorithms (other)

Let f $(x)$ = $(x – a)$ $(x – b)$ $(x – c)$ be a polynomial of the third degree with complex roots a, b, c.
Prove that the roots of the derivative of this polynomial lie inside the triangle with vertices at the points a, b, c.

Examples and counterexamples. Constructive proofs , Pigeonhole principle (other) , Theory of algorithms (other)

Using the theorem on rational roots of a polynomial $($ see Problem 61013 $)$, prove that if p / q is rational and cos $(p / q)$ $^{\circ}$ ≠ 0, ± 1, ± 1, then
cos $(p / q)$ $^{\circ}$ is irrational.

Examples and counterexamples. Constructive proofs , Pigeonhole principle (other) , Theory of algorithms (other)

It is known that cos α$^{\circ}$ = 1/3. Is α a rational number?

Examples and counterexamples. Constructive proofs , Pigeonhole principle (other) , Theory of algorithms (other)

Prove that there are infinitely many composite numbers among the numbers [$2^k$ $\sqrt{2}$] $(k = 0, 1, …)$.

My Problem Set reset
No Problems selected