Found problems: 2529
Evaluate the limit:
\[ \lim_{n \to \infty}{n \cdot \sin{1} \cdot \sin{2} \cdot \dots \cdot \sin{n}}.\]
Proposed to "Unirea" Intercounty contest, grade 11, Romania
Let $A = (a_1, a_2, \ldots, a_{2001})$ be a sequence of positive integers. Let $m$ be the number of 3-element subsequences $(a_i,a_j,a_k)$ with $1 \leq i < j < k \leq 2001$, such that $a_j = a_i + 1$ and $a_k = a_j + 1$. Considering all such sequences $A$, find the greatest value of $m$.
Let $a_1,a_2,\dots,a_m$ be a finite sequence of positive integers. Prove that there exist nonnegative integers $b,c,$ and $N$ such that $$\left\lfloor \sum_{i=1}^m \sqrt{n+a_i} \right\rfloor =\left\lfloor \sqrt{bn+c} \right\rfloor$$ holds for all integers $n>N.$
[i]Proposed by Carl Schildkraut[/i]
Show that for a triangle with radii of circumcircle and incircle equal to $ R$, $ r$ respectively, the inequality $ R \geq 2r$ holds.
Given $P$ a point m inside a triangle acute-angled $ABC$ and $DEF$ intersections of lines with that $AP$, $BP$, $CP$ with$\left[ BC \right],\left[ CA \right],$respective $\left[ AB \right]$
a) Show that the area of the triangle $DEF$ is at most a quarter of the area of the triangle $ABC$
b) Show that the radius of the circle inscribed in the triangle $DEF$ is at most a quarter of the radius of the circle circumscribed on triangle $4ABC.$
Let $a, b, c$ be positive real numbers such that $abc=8$. Prove that
\[
\frac{a^2}{\sqrt{(1+a^3)(1+b^3)}} +\frac{b^2}{\sqrt{(1+b^3)(1+c^3)}} +\frac{c^2}{\sqrt{(1+c^3)(1+a^3)}} \geq \frac{4}{3}
\]
There are $110$ guinea pigs for each of the $110$ species, arranging as a $110\times 110$ array. Find the maximum integer $n$ such that, no matter how the guinea pigs align, we can always find a column or a row of $110$ guinea pigs containing at least $n$ different species.
If $x, y, z$ are positive real numbers with sum $12$, prove that $\frac{x}{y}+\frac{y}{z}+\frac{z}{x}+ 3 \ge \sqrt{x} +\sqrt{y }+\sqrt{z}$.
When equality is valid?
Suppose real numbers $a, b, c, d$ satisfy $a + b + c + d = 17$ and $ab + bc + cd + da = 46$. If the minimum possible value of $a^2 + b^2 + c^2 + d^2$ can be expressed as a rational number $\frac{p}{q}$ in simplest form, find $p + q$.
Consider a matrix of size $n\times n$ whose entries are real numbers of absolute value not exceeding $1$. The sum of all entries of the matrix is $0$. Let $n$ be an even positive integer. Determine the least number $C$ such that every such matrix necessarily has a row or a column with the sum of its entries not exceeding $C$ in absolute value.
[i]Proposed by Marcin Kuczma, Poland[/i]
For some integer $m$, the polynomial $x^3-2011x+m$ has the three integer roots $a$, $b$, and $c$. Find $|a|+|b|+|c|$.
For the positive real numbers $ a,b,c$ prove that \[ \frac c{a \plus{} 2b} \plus{} \frac d{b \plus{} 2c} \plus{} \frac a{c \plus{} 2d} \plus{} \frac b{d \plus{} 2a} \geq \frac 43.\]
Let $ a,b,c $ be positive real numbers, prove the inequality:
$ (a+b+c)^2+ab+bc+ac\geq 6\sqrt{abc(a+b+c)} $
If $a,b,c$ are positive numbers with $a^2 +b^2 +c^2 = 1$, prove that $a+b+c+\frac{1}{abc} \ge 4\sqrt3$
Let $a,b,c>0$ satisfy $a\geq b\geq c$. Prove that
$$\frac{4}{a^2(b+c)}+\frac{4}{b^2(c+a)}+\frac{4}{c^2(a+b)} \leq \left(\sum_{cyc} \frac{a^2+1} {b^2} \right)\left(\sum_{cyc} \frac{b^3}{a^2(a^3+2b^3)}\right).$$
Given positive reals $a,b,c;$ show that we have
\[\left(a+\frac 1b\right)\left(b+\frac 1c\right)\left(c+\frac 1a\right)\geq 8.\]
Let $n$ be an even positive integer, and let $G$ be an $n$-vertex graph with exactly $\tfrac{n^2}{4}$ edges, where there are no loops or multiple edges (each unordered pair of distinct vertices is joined by either 0 or 1 edge). An unordered pair of distinct vertices $\{x,y\}$ is said to be [i]amicable[/i] if they have a common neighbor (there is a vertex $z$ such that $xz$ and $yz$ are both edges). Prove that $G$ has at least $2\textstyle\binom{n/2}{2}$ pairs of vertices which are amicable.
[i]Zoltán Füredi (suggested by Po-Shen Loh)[/i]
Let $n$ be a positive integer. Find the largest real number $\lambda$ such that for all positive real numbers $x_1,x_2,\cdots,x_{2n}$ satisfying the inequality
\[\frac{1}{2n}\sum_{i=1}^{2n}(x_i+2)^n\geq \prod_{i=1}^{2n} x_i,\]
the following inequality also holds
\[\frac{1}{2n}\sum_{i=1}^{2n}(x_i+1)^n\geq \lambda\prod_{i=1}^{2n} x_i.\]
$ u,v,w > 0$,such that $ u \plus{} v \plus{} w \plus{} \sqrt {uvw} \equal{} 4$
prove that $ \sqrt {\frac {uv}{w}} \plus{} \sqrt {\frac {vw}{u}} \plus{} \sqrt {\frac {wu}{v}}\geq u \plus{} v \plus{} w$
Let $A_{1}A_{2}A_{3}$ be an acute triangle, and let $O$ and $H$ be its circumcenter and orthocenter, respectively. For $1\leq i \leq 3$, points $P_{i}$ and $Q_{i}$ lie on lines $OA_{i}$ and $A_{i+1}A_{i+2}$ (where $A_{i+3}=A_{i}$), respectively, such that $OP_{i}HQ_{i}$ is a parallelogram. Prove that
\[\frac{OQ_{1}}{OP_{1}}+\frac{OQ_{2}}{OP_{2}}+\frac{OQ_{3}}{OP_{3}}\geq 3.\]
For non-negative real numbers $x_1, x_2, \ldots, x_n$ which satisfy $x_1 + x_2 + \cdots + x_n = 1$, find the largest possible value of $\sum_{j = 1}^{n} (x_j^{4} - x_j^{5})$.
Given real numbers $x,y,z$ such that $x+y+z=0$, show that
\[\dfrac{x(x+2)}{2x^2+1}+\dfrac{y(y+2)}{2y^2+1}+\dfrac{z(z+2)}{2z^2+1}\ge 0\]
When does equality hold?
If $ x$, $ y$, and $ z$ are positive numbers satisfying \[x \plus{} 1/y \equal{} 4,\quad y \plus{} 1/z \equal{} 1,\quad\text{and}\quad z \plus{} 1/x \equal{} 7/3,\] then $ xyz \equal{}$
$ \textbf{(A)}\ 2/3 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 4/3 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 7/3$
Given an integer $n \geq 3$, determine the maximum value of product of $n$ non-negative real numbers $x_1,x_2, \ldots , x_n$ when subjected to the condition
\begin{align*} \sum_{k=1}^n \frac{x_k}{1+x_k} =1 \end{align*}
Given $a_i\ge 0,x_i\in\mathbb{R},(i=1,2,\ldots,n)$. Prove that
$$((1-\sum_{i=1}^n a_i\cos x_i)^2+(1-\sum_{i=1}^n a_i\sin x_i)^2)^2\ge 4(1-\sum_{i=1}^n a_i)^3$$