Found problems: 1671
Let $a_1, b_1, a_2, b_2, \dots , a_n, b_n$ be nonnegative real numbers. Prove that
\[
\sum_{i, j = 1}^{n} \min\{a_ia_j, b_ib_j\} \le \sum_{i, j = 1}^{n} \min\{a_ib_j, a_jb_i\}.
\]
Let $n\ge 2$ be a natural number. Let $a_1\le a_2\le a_3\le \cdots \le a_n$ be real numbers such that $a_1+a_2+\cdots +a_n>0$ and $n(a_1^2+a_2^2+\cdots +a_n^2)=2(a_1+a_2+\cdots +a_n)^2.$ If $m=\lfloor n/2\rfloor+1$, the smallest integer larger than $n/2$, then show that $a_m>0.$
The numbers $$\frac{1}{i+j-1} \,\,\,\,\,\,\, (i = 1,2,...,n; j = 1,2,...,n)$$ are written in an $n$ by $n$ table: the number $1/(i + j - 1)$ stands at the intersection of the $i$-th row and $j$-th column. Chose any $n$ squares of the table so that no two of them stand in the same row and no two of them stand in the same column. Prove that the sum of the numbers in these $n$ squares is not less than $1$.
(Sergey Ivanov, St Petersburg)
Let $a,b,c \in \mathbb{R}^{+}$ such that $abc=1$. Prove that:
$$a^2b + b^2c + c^2a \ge \sqrt{(a+b+c)(ab + bc +ca)}$$
Prove that if $x$, $y$, $z$ are positive real numbers and $xyz = 1$ then
\[\frac{x^3}{x^2+y}+\frac{y^3}{y^2+z}+\frac{z^3}{z^2+x}\geq \dfrac {3} {2}.\]
[i]A. Golovanov[/i]
In the plane, consider a point $ X$ and a polygon $ \mathcal{F}$ (which is not necessarily convex). Let $ p$ denote the perimeter of $ \mathcal{F}$, let $ d$ be the sum of the distances from the point $ X$ to the vertices of $ \mathcal{F}$, and let $ h$ be the sum of the distances from the point $ X$ to the sidelines of $ \mathcal{F}$. Prove that $ d^2 \minus{} h^2\geq\frac {p^2}{4}.$
Let $a, b$, and $c$ be positive real numbers. Prove: $\frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\le \frac{a^2}{b^2} + \frac{b^2}{c^2} + \frac{c^2}{a^2}$ .
Let $a,b,c,d$ be the areas of the triangular faces of a tetrahedron, and let $h_a, h_b, h_c, h_d$ be the corresponding altitudes of the tetrahedron. If $V$ denotes the volume of tetrahedron, prove that
\[ (a+b+c+d)(h_a+h_b+h_c+h_d) \geq 48V \]
Let $a,\ b$ and $c$ be positive real numbers such that $a b+b c+c a=1$. Show that
$$
\frac{a^3}{a^2+3 b^2+3 a b+2 b c}+\frac{b^3}{b^2+3 c^2+3 b c+2 c a}+\frac{c^3}{c^2+3 a^2+3 c a+2 a b}>\frac{1}{6\left(a^2+b^2+c^2\right)^2} .
$$
Determine the minimum value of $a^{2} + b^{2}$ when $(a,b)$ traverses all the pairs of real numbers for which the equation \[ x^{4} + ax^{3} + bx^{2} + ax + 1 = 0 \] has at least one real root.
Find all functions $f:\mathbb{R}\to\mathbb{R}$ for which
\[ x(f(x+1)-f(x)) = f(x), \]
for all $x\in\mathbb{R}$ and
\[ | f(x) - f(y) | \leq |x-y| , \]
for all $x,y\in\mathbb{R}$.
[i]Mihai Piticari[/i]
Suppose $a_1, a_2, \ldots, a_n$ and $b_1, b_2, \ldots, b_n$ are real numbers such that \[ (a_1 ^ 2 + a_2 ^ 2 + \cdots + a_n ^ 2 -1)(b_1 ^ 2 + b_2 ^ 2 + \cdots + b_n ^ 2 - 1) > (a_1 b_1 + a_2 b_2 + \cdots + a_n b_n - 1)^2. \] Prove that $a_1 ^ 2 + a_2 ^ 2 + \cdots + a_n ^ 2 > 1$ and $b_1 ^ 2 + b_2 ^ 2 + \cdots + b_n ^ 2 > 1$.
Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$, such that $f(x+f(x)+f(y))=2f(x)+y$ for all positive reals $x,y$.
[i]Proposed by Athanasios Kontogeorgis, Greece[/i]
Let $f(x)$ be a continuous function defined on $0\leq x\leq \pi$ and satisfies $f(0)=1$ and
\[\left\{\int_0^{\pi} (\sin x+\cos x)f(x)dx\right\}^2=\pi \int_0^{\pi}\{f(x)\}^2dx.\]
Evaluate $\int_0^{\pi} \{f(x)\}^3dx.$
For $ t \equal{} 1, 2, 3, 4$, define $ \displaystyle S_t \equal{} \sum_{i \equal{} 1}^{350}a_i^t$, where $ a_i \in \{1,2,3,4\}$. If $ S_1 \equal{} 513$ and $ S_4 \equal{} 4745$, find the minimum possible value for $ S_2$.
Let $a,b,c,d \ge 0$ real numbers so that $a+b+c+d=1$.Prove that
$\sqrt{a+\frac{(b-c)^2}{6}+\frac{(c-d)^2}{6}+\frac{(d-b)^2}{6}} +\sqrt{b}+\sqrt{c}+\sqrt{d} \le 2.$
Let $a,b,c,d$ be positive real numbers with $abcd=1$. Prove that
$$\sqrt{\frac{a}{b+c+d^2+a^3}}+\sqrt{\frac{b}{c+d+a^2+b^3}}+\sqrt{\frac{c}{d+a+b^2+c^3}}+\sqrt{\frac{d}{a+b+c^2+d^3}} \leq 2$$
Triangle $ABC$ is inscribed in a semicircle of radius $r$ so that its base $AB$ coincides with diameter $AB$. Point $C$ does not coincide with either $A$ or $B$. Let $s=AC+BC$. Then, for all permissible positions of $C$:
$ \textbf{(A)}\ s^2\le8r^2$
$\qquad\textbf{(B)}\ s^2=8r^2$
$\qquad\textbf{(C)}\ s^2 \ge 8r^2$
${\qquad\textbf{(D)}\ s^2\le4r^2 }$
${\qquad\textbf{(E)}\ x^2=4r^2 } $
Find all the functions $f: \mathbb{Z}\to \mathbb{Z}$ satisfying the following property: if $a$, $b$ and $c$ are integers such that $a+b+c=0$, then
$$f(a)+f(b)+f(c)=a^2+b^2+c^2.$$
For any positive reals $a,b,c,d$ that satisfy $a^2 + b^2 + c^2 + d^2 = 4,$ show that
$$\frac{a^3}{a+b} + \frac{b^3}{b+c} + \frac{c^3}{c+d} + \frac{d^3}{d+a} + 4abcd \leq 6.$$
Let $a$, $b$, $c$ be positive real numbers such that $\min(ab,bc,ca) \ge 1$. Prove that $$\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.$$
[i]Proposed by Tigran Margaryan, Armenia[/i]
Let, $ a,b,c$ real positive numbers with $ abc \equal{} 1$
Prove:
$ (\frac {a}{1 \plus{} ab})^2 \plus{} (\frac {b}{1 \plus{} bc})^2 \plus{} (\frac {c}{1 \plus{} ca})^2\geq \frac {3}{4}$
Thanks!
Consider real numbers $x,y,z$ such that $x,y,z>0$. Prove that \[ (xy+yz+xz)\left(\frac{1}{x^2+y^2}+\frac{1}{x^2+z^2}+\frac{1}{y^2+z^2}\right) > \frac{5}{2}. \]
Let $p, q$ be real numbers with $0 < p < q$ and let $t_1, t_2, \cdots, t_n$ be real numbers in the interval $[p, q]$. Denote by $A$ and $B$ the arithmetic means of $t_1, t_2, \cdots, t_n$ and of $t_1^2, t_2^2,\cdots , t_n^2$, respectively. Prove that
\[\frac{A^2}{B}\ge\frac{4pq}{(p + q)^2}.\]
For arbitrary non-negative numbers $a$ and $b$ prove inequality
$\frac{a}{\sqrt{b^2+1}}+\frac{b}{\sqrt{a^2+1}}\ge\frac{a+b}{\sqrt{ab+1}}$,
and find, where equality occurs.
(Day 2, 6th problem
authors: Tomáš Jurík, Jaromír Šimša)