Problems – We Solve Problem
Filter Problems
Showing 1 to 20 of 86 entries

#### Integer and fractional parts. Archimedean property , Pigeonhole principle (angles and lengths)

a$)$ Give an example of a positive number a such that {a} + {1 / a} = 1.
$\\$
b$)$ Can such an a be a rational number?

#### Integer and fractional parts. Archimedean property , Pigeonhole principle (angles and lengths)

At the cat show, 10 male cats and 19 female cats sit in a row where next to each female cat sits a fatter male cat. Prove that next to each male cat is a female cat, which is thinner than it.

#### Integer and fractional parts. Archimedean property , Pigeonhole principle (angles and lengths)

The numbers a and b are such that the first equation of the system

$cos x = ax + b$

$sin x + a = 0$

has exactly two solutions. Prove that the system has at least one solution.

#### Integer and fractional parts. Archimedean property , Pigeonhole principle (angles and lengths)

The numbers a and b are such that the first equation of the system

$sin x + a = bx$

$cos x = b$

has exactly two solutions. Prove that the system has at least one solution.

#### Integer and fractional parts. Archimedean property , Pigeonhole principle (angles and lengths)

Is there a line on the coordinate plane relative to which the graph of the function $y = 2^x$ is symmetric?

#### Integer and fractional parts. Archimedean property , Pigeonhole principle (angles and lengths)

Numbers 1, 2, 3, …, 101 are written out in a row in some order. Prove that one can cross out 90 of them so that the remaining 11 will be arranged in their magnitude $($either increasing or decreasing$)$.

#### Integer and fractional parts. Archimedean property , Pigeonhole principle (angles and lengths)

Find the general formula for the coefficients of the series
$(1 – 4x)^{ ½} = 1 + 2x + 6x^2 + 20x^3 + … + a_nx^n + …$

#### Integer and fractional parts. Archimedean property , Pigeonhole principle (angles and lengths)

Let p and q be positive numbers, where 1 / p + 1 / q = 1. Prove that The values of the variables are considered positive.

#### Integer and fractional parts. Archimedean property , Pigeonhole principle (angles and lengths)

Prove that for any $x_1$, …, $x_n$ $\in$ [0; $\pi$] the following inequality holds:
sin $(\frac{x_1+…+x_n}{n}) \geq \frac{sinx_1+…+sinx_n}{n}$

#### Integer and fractional parts. Archimedean property , Pigeonhole principle (angles and lengths)

We are given rational positive numbers p, q, where 1/p + 1/q = 1. Prove that for positive a and b, the following inequality holds: $ab ≤ ap/p + bq/q$.

#### Integer and fractional parts. Archimedean property , Pigeonhole principle (angles and lengths)

For what values of n does the polynomial $(x+1)^n$ – $x^n$ – 1 divide by:

a$)$ $x^2$ + x + 1; b$)$ $(x^2 + x + 1)^2$; c$)$ $(x^2 + x + 1)^3$?

My Problem Set reset
No Problems selected