Found problems: 1671
Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ satisfying the following conditions :
1) $f(x+y)-f(x)-f(y) \in \{0,1\} $ for all $x,y \in \mathbb{R}$
2) $\lfloor f(x) \rfloor = \lfloor x \rfloor $ for all real $x$.
Let $a$, $b$, $c$ be positive real numbers such that $ab+bc+ca\leq 3abc$. Prove that
\[\sqrt{\frac{a^2+b^2}{a+b}}+\sqrt{\frac{b^2+c^2}{b+c}}+\sqrt{\frac{c^2+a^2}{c+a}}+3\leq \sqrt{2}\left(\sqrt{a+b}+\sqrt{b+c}+\sqrt{c+a}\right)\]
[i]Proposed by Dzianis Pirshtuk, Belarus[/i]
Find all continuous and nondecreasing functions $ f:[0,\infty)\longrightarrow\mathbb{R} $ that satisfy the inequality:
$$ \int_0^{x+y} f(t) dt\le \int_0^x f(t) dt +\int_0^y f(t) dt,\quad\forall x,y\in [0,\infty) . $$
a) Prove that for all positive reals $u,v,x,y$ the following inequality takes place:
\[ \frac ux + \frac vy \geq \frac {4(uy+vx)}{(x+y)^2} . \]
b) Let $a,b,c,d>0$. Prove that
\[ \frac a{b+2c+d} + \frac b{c+2d+a} + \frac c{d+2a+b} + \frac d{a+2b+c} \geq 1.\]
[i]Traian Tămâian[/i]
Let $a,b,c$ be positive real numbers so that $a+b+c = \frac{1}{\sqrt{3}}$. Find the maximum value of
$$27abc+a\sqrt{a^2+2bc}+b\sqrt{b^2+2ca}+c\sqrt{c^2+2ab}.$$
Let $ a$, $ b$, $ c$ be positive real numbers. Prove the inequality
\[ \frac{1}{a\left(b+1\right)}+\frac{1}{b\left(c+1\right)}+\frac{1}{c\left(a+1\right)}\geq \frac{3}{1+abc}. \]
Let $ a_{1},a_{2},\cdots,a_{n}$ be positive real numbers satisfying $ a_{1} \plus{} a_{2} \plus{} \cdots \plus{} a_{n} \equal{} 1$. Prove that
\[\left(a_{1}a_{2} \plus{} a_{2}a_{3} \plus{} \cdots \plus{} a_{n}a_{1}\right)\left(\frac {a_{1}}{a_{2}^2 \plus{} a_{2}} \plus{} \frac {a_{2}}{a_{3}^2 \plus{} a_{3}} \plus{} \cdots \plus{} \frac {a_{n}}{a_{1}^2 \plus{} a_{1}}\right)\ge\frac {n}{n \plus{} 1}\]
$a,b,c$ are positive real numbers. prove the following inequality:
$\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}+\frac{1}{(a+b+c)^2}\ge \frac{7}{25}(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a+b+c})^2$
(20 points)
Prove that
\[ (x^4+y)(y^4+z)(z^4+x) \geq (x+y^2)(y+z^2)(z+x^2) \]
for all positive real numbers $x, y, z$ satisfying $xyz \geq 1$.
250 numbers are chosen among positive integers not exceeding 501. Prove that for every integer $ t$ there are four chosen numbers $ a_1$, $ a_2$, $ a_3$, $ a_4$, such that $ a_1 \plus{} a_2 \plus{} a_3 \plus{} a_4 \minus{} t$ is divisible by 23.
[i]Author: K. Kokhas[/i]
In $\triangle ABC$ , then \[\frac{1}{1+\cos^2 A+\cos^2 B}+\frac{1}{1+\cos^2 B+\cos^2 C}+\frac{1}{1+\cos^2 C+\cos^2 A}\le 2\]
Prove for all positive real numbers $a,b,c$, such that $a^2+b^2+c^2=1$:
\[\frac{a^3}{b^2+c}+\frac{b^3}{c^2+a}+\frac{c^3}{a^2+b}\ge \frac{\sqrt{3}}{1+\sqrt{3}}.\]
Consider infinite sequences $\{x_n\}$ of positive reals such that $x_0=1$ and $x_0\ge x_1\ge x_2\ge\ldots$.
[b]a)[/b] Prove that for every such sequence there is an $n\ge1$ such that: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}\ge3.999. \]
[b]b)[/b] Find such a sequence such that for all $n$: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}<4. \]
Let $x,y,z$ be positive integers such that $x\neq y\neq z \neq x$ .Prove that $$(x+y+z)(xy+yz+zx-2)\geq 9xyz.$$
When does the equality hold?
[i]Proposed by Dorlir Ahmeti, Albania[/i]
Four real numbers $ p$, $ q$, $ r$, $ s$ satisfy $ p+q+r+s = 9$ and $ p^{2}+q^{2}+r^{2}+s^{2}= 21$. Prove that there exists a permutation $ \left(a,b,c,d\right)$ of $ \left(p,q,r,s\right)$ such that $ ab-cd \geq 2$.
Given $n$ non-negative real numbers, $n\geq 3$, such that the sum of the $n$ numbers is less than or equal to $3$ and the sum of the squares of the $n$ numbers is greater than or equal to $1$, prove that among the $n$ numbers three can be chosen whose sum is greater than or equal to $1$.
Suppose the points $D, E, F$ lie on sides $BC, CA, AB$, respectively, so that $AD, BE, CF$ are angle bisectors. Define $P_1$, $P_2$, $P_3$ respectively as the intersection point of $AD$ with $EF$, $BE$ with $DF$, $CF$ with $DE$ respectively. Prove that
$$\frac{AD}{AP_1}+\frac{BE}{BP_2}+\frac{CF}{CP_3} \ge 6$$
Let $x_1 \le x_2 \le \ldots \le x_{2n-1}$ be real numbers whose arithmetic mean equals $A$. Prove that
$$2\sum_{i=1}^{2n-1}\left( x_{i}-A\right)^2 \ge \sum_{i=1}^{2n-1}\left( x_{i}-x_{n}\right)^2.$$
Let $f$ be a function such that $f (x + y) = f (x) + f (y)$ for all $x,y \in R$ and $f (1) = 100$. Calculate $\sum_{k = 1}^{10}f (k!)$.
Prove that for all positive real numbers $a,b,c$ the following inequality takes place
\[ \frac{1}{b(a+b)}+ \frac{1}{c(b+c)}+ \frac{1}{a(c+a)} \geq \frac{27}{2(a+b+c)^2} . \]
[i]Laurentiu Panaitopol, Romania[/i]
Let $x_1,..., x_n$ be non-negative real numbers, not all of which are zeros.
(i) Prove that
$$1 \le \frac{\left(x_1+\frac{x_2}{2}+\frac{x_3}{3}+...+\frac{x_n}{n}\right)(x_1+2x_2+3x_3+...+nx_n)}{(x_1+x_2+x_3+...+x_n)^2} \le \frac{(n+1)^2}{4n}$$
(ii) Show that, for each $n > 1$, both inequalities can hold as equalities.
Find all functions $f:\mathbb R\to\mathbb R$ so that the following relation holds for all $x, y\in\mathbb R$.
$$f(f(x)f(y)-1) = xy - 1$$
Let $a, b, c$ be non-negative numbers with $a+b+c = 3$. Prove the inequality
\[\frac{a}{b^2+1}+\frac{b}{c^2+1}+\frac{c}{a^2+1} \geq \frac 32.\]
Prove that for all reas $a,b,c,d\in(0,1)$ we have $$\left(ab-cd\right)\left(ac+bd\right)\left(ad-bc\right)+\min{\left(a,b,c,d\right)} < 1.$$
If $p$ and $q$ are positive numbers with $p+q = 1$,
knowing that any real numbers $x,y$ satisfy $(x-y)^2 \ge 0$, show that
$\frac{x+y}{2} \ge \sqrt{xy}$,
$\frac{x^2+y^2}{2} \ge \big(\frac{x+y}{2}\big)^2$,
$\big(p+\frac{1}{p}\big)^2+\big(q+\frac{1}{q}\big)^2 \ge \frac{25}{2}$