Found problems: 2529
Let $a,b,c$ be positive real numbers. Prove that
\[ \frac{2a}{a^{2}+bc}+\frac{2b}{b^{2}+ca}+\frac{2c}{c^{2}+ab}\leq\frac{a}{bc}+\frac{b}{ca}+\frac{c}{ab} \]
For all positive real numbers $a, b,c.$ Prove the folllowing inequality$$\frac{a}{c+5b}+\frac{b}{a+5c}+\frac{c}{b+5a}\geq\frac{1}{2}.$$
Let $ n$ be a positive integer and let $ a_1,a_2,\ldots,a_n$ be positive real numbers such that:
\[ \sum^n_{i \equal{} 1} a_i \equal{} \sum^n_{i \equal{} 1} \frac {1}{a_i^2}.
\]
Prove that for every $ i \equal{} 1,2,\ldots,n$ we can find $ i$ numbers with sum at least $ i$.
Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
Let $ a,b,c,d$ be nonnegative real numbers. Prove that
\[ a^4\plus{}b^4\plus{}c^4\plus{}d^4\plus{}2abcd \ge a^2b^2\plus{}a^2c^2\plus{}a^2d^2\plus{}b^2c^2\plus{}b^2d^2\plus{}c^2d^2.\]
A [i]special number[/i] is a positive integer $n$ for which there exists positive integers $a$, $b$, $c$, and $d$ with \[ n = \frac {a^3 + 2b^3} {c^3 + 2d^3}. \] Prove that
i) there are infinitely many special numbers;
ii) $2014$ is not a special number.
[i]Romania[/i]
Find all pairs $(x,y)$ of real numbers such that
\[16^{x^{2}+y} + 16^{x+y^{2}} = 1\]
For each $P$ inside the triangle $ABC$, let $A(P), B(P)$, and $C(P)$ be the points of intersection of the lines $AP, BP$, and $CP$ with the sides opposite to $A, B$, and $C$, respectively. Determine $P$ in such a way that the area of the triangle $A(P)B(P)C(P)$ is as large as possible.
We have $2^m$ sheets of paper, with the number $1$ written on each of them. We perform the following operation. In every step we choose two distinct sheets; if the numbers on the two sheets are $a$ and $b$, then we erase these numbers and write the number $a + b$ on both sheets. Prove that after $m2^{m -1}$ steps, the sum of the numbers on all the sheets is at least $4^m$ .
[i]Proposed by Abbas Mehrabian, Iran[/i]
Let $k$ be one of the integers $2, 3,4$ and let $n = 2^k -1$. Prove the inequality
\[1+ b^k + b^{2k} + \cdots+ b^{nk} \geq (1 + b^n)^k\]
for all real $b \geq 0.$
Prove that, for all positive integers $m$ and $n$, we have $$\left\lfloor m\sqrt{2} \right\rfloor\cdot\left\lfloor n\sqrt{7} \right\rfloor<\left\lfloor mn\sqrt{14} \right\rfloor.$$
Let $n\geqslant 2$ be a positive integer and $a_1,a_2, \ldots ,a_n$ be real numbers such that \[a_1+a_2+\dots+a_n=0.\]
Define the set $A$ by
\[A=\left\{(i, j)\,|\,1 \leqslant i<j \leqslant n,\left|a_{i}-a_{j}\right| \geqslant 1\right\}\]
Prove that, if $A$ is not empty, then
\[\sum_{(i, j) \in A} a_{i} a_{j}<0.\]
Prove the inequality \[ \sin^n (2x) + \left( \sin^n x - \cos^n x \right)^2 \le 1. \]
Let $n$ be a fixed positive integer. Initially, $n$ 1's are written on a blackboard. Every minute, David picks two numbers $x$ and $y$ written on the blackboard, erases them, and writes the number $(x+y)^4$ on the blackboard. Show that after $n-1$ minutes, the number written on the blackboard is at least $2^{\frac{4n^2-4}{3}}$.
[i]Proposed by Calvin Deng[/i]
Let real $a$, $b$, and $c$ satisfy $$abc+a+b+c=ab+bc+ca+5.$$ Find the least possible value of $a^2+b^2+c^2$.
Let $n(n\geq2)$ be a natural number and $a_1,a_2,...,a_n$ natural positive real numbers. Determine the least possible value of the expression $$E_n=\frac{(1+a_1)\cdot(a_1+a_2)\cdot(a_2+a_3)\cdot...\cdot(a_{n-1}+a_n)\cdot(a_n+3^{n+1})} {a_1\cdot a_2\cdot a_3\cdot...\cdot a_n}$$
Let $a$, $b$, $c$ be real numbers. Prove that \begin{align*}&\quad\,\,\sqrt{2(a^2+b^2)}+\sqrt{2(b^2+c^2)}+\sqrt{2(c^2+a^2)}\\&\ge \sqrt{3[(a+b)^2+(b+c)^2+(c+a)^2]}.\end{align*}
Ali and Naqi are playing a game. At first, they have Polynomial $P(x) = 1+x^{1398}$.
Naqi starts. In each turn one can choice natural number $k \in [0,1398]$ in his trun, and add $x^k$ to the polynomial. For example after 2 moves $P$ can be : $P(x) = x^{1398} + x^{300} + x^{100} +1$. If after Ali's turn, there exist $t \in R$ such that $P(t)<0$ then Ali loses the game. Prove that Ali can play forever somehow he never loses the game!
Let $a,b,c$ be positive real numbers such that $\frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}=3$. Prove that
$$\frac{a+b}{a^2+ab+b^2}+ \frac{b+c}{b^2+bc+c^2}+ \frac{c+a}{c^2+ca+a^2}\le 2$$
When is the equality valid?
For every $x,y\ge0$ prove that $(x+1)^2+(y-1)^2\ge2\sqrt{2xy}.$
Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers.
[i]Proposed by Dorlir Ahmeti, Albania[/i]
What is the largest possible area of an isosceles trapezoid in which the largest side is $13$ and the perimeter is $28$?
$
\textbf{(A)}\ 13
\qquad\textbf{(B)}\ 24
\qquad\textbf{(C)}\ 27
\qquad\textbf{(D)}\ 28
\qquad\textbf{(E)}\ 30
$
A square of side $N=n^2+1$, $n\in \mathbb{N}^*$, is partitioned in unit squares (of side $1$), along $N$ rows and $N$ columns. The $N^2$ unit squares are colored using $N$ colors, $N$ squares with each color. Prove that for any coloring there exists a row or a column containing unit squares of at least $n+1$ colors.
Let $a_1,a_2,a_3, \cdots ,a_n$ be positive real numbers. For the integers $n\ge 2$, prove that\[ \left (\frac{\sum_{j=1}^{n} \left (\prod_{k=1}^{j}a_k \right )^{\frac{1}{j}}}{\sum_{j=1}^{n}a_j} \right )^{\frac{1}{n}}+\frac{\left (\prod_{i=1}^{n}a_i \right )^{\frac{1}{n}}}{\sum_{j=1}^{n} \left (\prod_{k=1}^{j}a_k \right )^{\frac{1}{j}}}\le \frac{n+1}{n}\]
Show that
\[1 \leq \frac{(x+y)(x^3+y^3)}{(x^2+y^2)^2} \leq \frac98\]
holds for all positive real numbers $x,y$.