Filter Problems

Showing 1 to 20 of 72 entries

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.

Does there exist a number h such that for any natural number n the number [$h \times 1969^n$] is not divisible by [$h \times 1969^{n-1}$]?

Gary drew an empty table of $50 \times 50$ and wrote on top of each column and to the left of each row a number. It turned out that all 100 written numbers are different, and 50 of them are rational, and the remaining 50 are irrational. Then, in each cell of the table, he wrote down a product of numbers written at the top of its column and to the left of the row $($ the “multiplication table” $)$. What is the largest number of products in this table which could be rational numbers?

In the Republic of mathematicians, the number α $>$ 2 was chosen and coins were issued with denominations of 1 pound, as well as in $α^k$ pounds for every natural k. In this case α was chosen so that the value of all the coins, except for the smallest, was irrational. Could it be that any amount of a natural number of pounds can be made with these coins, using coins of each denomination no more than 6 times?

The equations $ax^2$ + bx + c = 0 $(1)$ and – $ax^2$ + bx + c $(2)$ are given. Prove that if $x_1$ and $x_2$ are, respectively, any roots of the equations $(1)$ and $(2)$, then there is a root $x_3$ of the equation ½ $ax^2$ + bx + c such that either $x_1$ $≤$ $x_3$ $≤$ $x_2$ or $x_1$ $≥$ $x_3$ $≥$ $x_2$.

The triangle $C_1C_2O$ is given. Within it the bisector $C_2C_3$ is drawn, then in the triangle $C_2C_3O$ – bisector $C_3C_4$ and so on. Prove that the sequence of angles $γ_n$ = $C_{n + 1}C_nO$ tends to a limit, and find this limit if $C_1OC_2$ = α.

Author: V.A. Popov

On the interval [0; 1] a function f is given. This function is non-negative at all points, f $(1)$ = 1 and, finally, for any two non-negative numbers $x_1$ and $x_2$ whose sum does not exceed 1, the quantity $f (x_1 + x_2)$ does not exceed the sum of $f (x_1)$ and $f (x_2 )$.

a) Prove that for any number x on the interval [0; 1], the inequality $f (x_2) ≤ 2x$ holds.

b) Prove that for any number x on the interval [0; 1], the $f (x_2) ≤ 1.9x$ must be true?

The Newton method $($see Problem 61328$)$ does not always allow us to approach the root of the equation f $($x$)$ = 0. Find the initial condition $x_0$ for the polynomial f $(x)$ = x $(x – 1)$ $(x + 1)$ such that f $(x_0)$ $\neq$

$x_0$ and $x_2$ = $x_0$.

Let p and q be nonzero real numbers and $p^2$ – 4q $>$ 0. Prove that the following sequences converge:

a$)$ $y_0$ = 0, $y_{n + 1} = \frac{q}{p-y_n} (n \geq 0)$;

b$)$ $z_0$ = 0, $z_{n + 1}$ = p – q/$z_n$ $( n \geq 0 )$.

Establish a connection between the limiting values of these sequences y *, z * and the roots of the equation $x^2$ – px + q = 0.

Cut the interval [-1, 1] into black and white segments so that the integrals of any a$)$ linear function; b$)$ a square trinomial in white and black segments are equal.

Old calculator II. The derivative of the function ln x for x = 1 is 1. Hence

$\lim\limits_{x\to0}$

= $\lim\limits_{x\to0}$

= 1.

Use this fact to approximate the natural logarithm of the number N. As in Problem 61302, the standard arithmetic operations and the square root extraction operation are allowed.

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 $\sqrt{2}$.

Let f $(x)$ be a polynomial of degree n with roots $α_1$, …, $α_n$. We define the polygon M as the convex hull of the points $α_1$, …, $α_n$ on the complex plane. Prove that the roots of the derivative of this polynomial lie inside the polygon M.

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.

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.

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

Prove that for any infinite continued fraction [$a_0$; $a_1$, …, $a_n$, …] there exists a limit of its suitable fractions – an irrational number α. Explain why if this number α is decomposed into an infinite continued fraction by the algorithm of task 60606, then an infinite continued fraction is obtained.

$x_1$ is the real root of the equation $x^2$ + ax + b = 0, $x_2$ is the real root of the equation $x^2$ – ax – b = 0.

Prove that the equation $x^2$ + 2ax + 2b = 0 has a real root, enclosed between $x_1$ and $x_2$. $($ a and b are real numbers $)$.

Prove that for any natural number $a_1> 1$ there exists an increasing sequence of natural numbers $a_1, a_2, a_3$, …, for which $a_1^2+ a_2^2 +…+ a_k^2$ is divisible by $a_1+ a_2+…+ a_k$ for all k ≥ 1.