Found problems: 34
Find all real solutions of the equation: $$x=\frac{2z^2}{1+z^2}$$ $$y=\frac{2x^2}{1+x^2}$$ $$z=\frac{2y^2}{1+y^2}$$
Let $a$, $b$ and $c$ be real numbers such that $abc(a+b)(b+c)(c+a)\neq0$ and $(a+b+c)\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right)=\frac{1007}{1008}$
Prove that $\frac{ab}{(a+c)(b+c)}+\frac{bc}{(b+a)(c+a)}+\frac{ca}{(c+b)(a+b)}=2017$
Are there integers $m, n \geq 2$ such that the following property is always true?
$$``\text{For any real numbers } x, y, \text{ if } x^m + y^m \text{ and } x^n + y^n \text{ are integers, then } x + y \text{ is an integer}".$$
Define a sequence $(a_n)$ by $a_1 =1, a_2 =2,$ and $a_{k+2}=2a_{k+1}+a_k$ for all positive integers $k$. Determine all real numbers $\beta >0$ which satisfy the following conditions:
(A) There are infinitely pairs of positive integers $(p,q)$ such that $\left| \frac{p}{q}- \sqrt{2} \, \right| < \frac{\beta}{q^2 }.$
(B) There are only finitely many pairs of positive integers $(p,q)$ with $\left| \frac{p}{q}- \sqrt{2} \,\right| < \frac{\beta}{q^2 }$ for which there is no index $k$ with $q=a_k.$
Find all real solutions of the equation
\[17^x+2^x=11^x+2^{3x}.\]
Let $n\ge 3$ be a positive integer. Find the largest real number $t_n$ as a function of $n$ such that the inequality
\[\max\left(|a_1+a_2|, |a_2+a_3|, \dots ,|a_{n-1}+a_{n}| , |a_n+a_1|\right) \ge t_n \cdot \max(|a_1|,|a_2|, \dots ,|a_n|)\]
holds for all real numbers $a_1, a_2, \dots , a_n$ .
[i]Proposed by Rohan Goyal and Rijul Saini[/i]
Ana has an iron material of mass $20.2$ kg. She asks Bilyana to make $n$ weights to be used in a classical weighning scale with two plates. Bilyana agrees under the condition that each of the $n$ weights is at least $10$ g. Determine the smallest possible value of $n$ for which Ana would always be able to determine the mass of any material (the mass can be any real number between $0$ and $20.2$ kg) with an error of at most $10$ g.
Find the least positive integer $n$, such that there is a polynomial \[ P(x) = a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_1x+a_0 \] with real coefficients that satisfies both of the following properties:
- For $i=0,1,\dots,2n$ it is $2014 \leq a_i \leq 2015$.
- There is a real number $\xi$ with $P(\xi)=0$.
There are given two positive real number $a<b$. Show that there exist positive integers $p, \ q, \ r, \ s$ satisfying following conditions:
$1$. $a< \frac{p}{q} < \frac{r}{s} < b$.
$2.$ $p^2+q^2=r^2+s^2$.
Let $x, y, z$ be three real distinct numbers such that
$$\begin{cases} x^2-x=yz \\ y^2-y=zx \\ z^2-z=xy \end{cases}$$ Show that $-\frac{1}{3} < x,y,z < 1$.
For a quadrilateral $ABCD$, we call a square $amazing$ if all of its sides(extended if necessary) pass through distinct vertices of $ABCD$(no side passing through 2 vertices). Prove that for an arbitrary $ABCD$ such that its diagonals are not perpendicular, there exist at least 6 $amazing$ squares
Given a real number $a$ and a sequence $(x_n)_{n=1}^\infty$ defined by:
$$\left\{\begin{matrix} x_1=1 \\ x_2=0 \\ x_{n+2}=\frac{x_n^2+x_{n+1}^2}{4}+a\end{matrix}\right.$$
for all positive integers $n$.
1. For $a=0$, prove that $(x_n)$ converges.
2. Determine the largest possible value of $a$ such that $(x_n)$ converges.
Find all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying
\[xf(f(x)+y)=f(xy)+x^2\]
for all $x,y \in \mathbb{R}$.
Suppose that $u_0 , u_1 ,\ldots$ is a sequence of real numbers such that
$$u_n = \sum_{k=1}^{\infty} u_{n+k}^{2}\;\;\; \text{for} \; n=0,1,2,\ldots$$
Prove that if $\sum u_n$ converges, then $u_k=0$ for all $k$.
Given eleven real numbers, prove that there exist two of them such that their decimal representations agree infinitely often.
Find all pairs $(a,b)$ of real numbers such that
$$|\sqrt{1-x^2 }-ax-b| \leq \frac{\sqrt{2} -1}{2}$$
holds for all $x\in [0,1]$.
Let \( \mathbb{R} \) be the set of real numbers. Determine all functions \( f: \mathbb{R} \to \mathbb{R} \) such that, for any real numbers \( x \) and \( y \),
\[
f(x^2 y - y) = f(x)^2 f(y) + f(x)^2 - 1.
\]
The sequence $a_1, a_2, \ldots, a_n$ of positive real numbers satisfies the following conditions:
\begin{align*}
\sum_{i=1}^n \frac{1}{a_i} \le 1 \ \ \ \ \hbox{and} \ \ \ \ a_i \le a_{i-1}+1
\end{align*}
for all $i\in \lbrace 1, 2, \ldots, n \rbrace$, where $a_0$ is an integer. Prove that
\begin{align*}
n \le 4a_0 \cdot \sum_{i=1}^n \frac{1}{a_i}
\end{align*}
Find all real numbers $x$ satisfying the equation
\[\left\lfloor \frac{20}{x+18}\right\rfloor+\left\lfloor \frac{x+18}{20}\right\rfloor=1.\]
Let $\mathbb{R}^+$ the set of positive real numbers. Determine all the functions $f, g: \mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that, for all positive real numbers $x, y$ we have that
$$f(x + g(y)) = f(x + y) + g(y) \text{ and } g(x + f(y)) = g(x + y) + f(y)$$
if $a$, $b$ and $c$ are real numbers such that $(a-b)(b-c)(c-a) \neq 0$, prove the equality:
$\frac{b^2c^2}{(a-b)(a-c)}+\frac{c^2a^2}{(b-c)(b-a)}+\frac{a^2b^2}{(c-a)(c-b)}=ab+bc+ca$
Let $a$, $b$ and $c$ be real numbers such that $\mid a \mid >2$ and $a^2+b^2+c^2=abc+4$. Prove that numbers $x$ and $y$ exist such that $a=x+\frac{1}{x}$, $b=y+\frac{1}{y}$ and $c=xy+\frac{1}{xy}$.
Find all the functions $f:(0,+\infty) \to R $ that satisfy the equation
$$f(x)f(y)+f\big(\frac{\lambda}{x})f(\frac{\lambda}{y})=2f(xy)$$
for all pairs of $x,y$ real and positive numbers, where $\lambda$ is a positive real number such that $f(\lambda )=1$
Find all natural numbers $k$ for which there exists a set $M$ of ten real numbers such that there are exactly $k$ pairwise non-congruent triangles whose side lengths are three (not necessarily distinct) elements of $M$.
Find the least positive integer $n$, such that there is a polynomial \[ P(x) = a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_1x+a_0 \] with real coefficients that satisfies both of the following properties:
- For $i=0,1,\dots,2n$ it is $2014 \leq a_i \leq 2015$.
- There is a real number $\xi$ with $P(\xi)=0$.