Found problems: 1671
Assume $ a,b,c $ are arbitrary reals such that $ a+b+c = 0 $. Show that $$ \frac{33a^2-a}{33a^2+1}+\frac{33b^2-b}{33b^2+1}+\frac{33c^2-c}{33c^2+1} \ge 0 $$
The incircle of a triangle $ A_1A_2A_3$ is centered at $ O$ and meets the segment $ OA_j$ at $ B_j$ , $ j \equal{} 1, 2, 3$. A circle with center $ B_j$ is tangent to the two sides of the triangle having $ A_j$ as an endpoint and intersects the segment $ OB_j$ at $ C_j$. Prove that
\[ \frac{OC_1\plus{}OC_2\plus{}OC_3}{A_1A_2\plus{}A_2A_3\plus{}A_3A_1} \leq \frac{1}{4\sqrt{3}}\]
and find the conditions for equality.
Let real numbers a, b, c satisfy $a^2+2b^2+3c^2= \frac{3}{2}$, prove that $3^{-a}+9^{-b}+27^{-c}\ge1$.
Let, $\displaystyle{S =\sum_{i=1}^{k} {n_i}^2}$. Prove that for $n_i \in \mathbb{R}^+$
$$\sum_{i=1}^{k} \frac{n_i}{S-n_i^2} \geq \frac{4}{n_1+n_2+ \cdots+ n_k}$$
[i]Proposed by Kang Taeyoung, South Korea[/i]
Is the series
$$\sum_{n=0}^{\infty} \frac{n!}{(n+1)^{n}}\cdot \left(\frac{19}{7}\right)^{n}$$
convergent or divergent?
If the perimeter of rectangle $ABCD$ is $20$ inches, the least value of diagonal $\overline{AC}$, in inches, is:
$\textbf{(A)}\ 0\qquad
\textbf{(B)}\ \sqrt{50}\qquad
\textbf{(C)}\ 10\qquad
\textbf{(D)}\ \sqrt{200}\qquad
\textbf{(E)}\ \text{none of these}$
Solve the following equation for $x , y , z \in \mathbb{N}$ :
\[\left (1+ \frac{x}{y+z} \right )^2+\left (1+ \frac{y}{z+x} \right )^2+\left (1+ \frac{z}{x+y} \right )^2=\frac{27}{4}\]
Given positive integers $a,b,c$ which are pairwise coprime. Let $f(n)$ denotes the number of the non-negative integer solution $(x,y,z)$ to the equation
$$ax+by+cz=n.$$
Prove that there exists constants $\alpha, \beta, \gamma \in \mathbb{R}$ such that for any non-negative integer $n$,
$$|f(n)- \left( \alpha n^2+ \beta n + \gamma \right) | < \frac{1}{12} \left( a+b+c \right).$$
Let $a_1,a_2,\dots$ be real numbers. Suppose there is a constant $A$ such that for all $n,$
\[\int_{-\infty}^{\infty}\left(\sum_{i=1}^n\frac1{1+(x-a_i)^2}\right)^2\,dx\le An.\]
Prove there is a constant $B>0$ such that for all $n,$
\[\sum_{i,j=1}^n\left(1+(a_i-a_j)^2\right)\ge Bn^3.\]
$\boxed{A7}$Prove that for all $x,y,z>0$ with $\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=1$ and $0\leq a,b,c<1$ the following inequality holds
\[\frac{x^2+y^2}{1-a^z}+\frac{y^2+z^2}{1-b^x}+\frac{z^2+x^2}{1-c^y}\geq \frac{6(x+y+z)}{1-abc}\]
Find all triples of positive real numbers $(a, b, c)$ so that the expression
$M = \frac{(a + b)(b + c)(a + b + c)}{abc}$
gets its least value.
Let $f : \{ 1, 2, 3, \dots \} \to \{ 2, 3, \dots \}$ be a function such that $f(m + n) | f(m) + f(n) $ for all pairs $m,n$ of positive integers. Prove that there exists a positive integer $c > 1$ which divides all values of $f$.
(a) Find all strictly monotone functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
\[f(x+f(y))=f(x)+y\quad\text{for all real}\ x,y. \]
(b) If $n>1$ is an integer, prove that there is no strictly monotone function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
\[ f(x+f(y))=f(x)+y^n\quad \text{for all real}\ x, y.\]
Determine the real numbers $ x,y,z $ from the interval $ (0,1) $ that satisfies $ x+y+z=1, $ and
$$ \sqrt{\frac{x(1-y^2)}{2}} +\sqrt{\frac{y(1-z^2)}{2}} +\sqrt{\frac{z(1-x^2)}{2}} =\sqrt{1+xy+yz+zx} . $$
[i]Gabriela Constantinescu[/i]
Let $\mathbb{R}^{*}$ be the set of non-zero real numbers. Find all functions $f : \mathbb{R}^{*} \to \mathbb{R}^{*}$ such that $f(x^{2}+y) = (f(x))^{2} + \frac{f(xy)}{f(x)}$, for all $x,y \in \mathbb{R}^{*}$ and $-x^{2} \not= y$.
Find all functions $f:\mathbb{Q}\to\mathbb{Q}$ such that the equation
\[f(xf(x)+y) = f(y) + x^2\]holds for all rational numbers $x$ and $y$.
Here, $\mathbb{Q}$ denotes the set of rational numbers.
Show: For all real numbers $ a,b,c$ with $ 0<a,b,c<1$ is:
\[ \sqrt{a^2bc\plus{}ab^2c\plus{}abc^2}\plus{}\sqrt{(1\minus{}a)^2(1\minus{}b)(1\minus{}c)\plus{}(1\minus{}a)(1\minus{}b)^2(1\minus{}c)\plus{}(1\minus{}a)(1\minus{}b)(1\minus{}c)^2}<\sqrt{3}.\]
The sum of non-negative numbers $ x $, $ y $ and $ z $ is $3$. Prove the inequality
$$ {1 \over x ^ 2 + y + z} + {1 \over x + y ^ 2 + z} + {1 \over x + y + z ^ 2} \leq 1. $$
Let $S$ be the set of all triangles $ABC$ for which \[ 5 \left( \dfrac{1}{AP} + \dfrac{1}{BQ} + \dfrac{1}{CR} \right) - \dfrac{3}{\min\{ AP, BQ, CR \}} = \dfrac{6}{r}, \] where $r$ is the inradius and $P, Q, R$ are the points of tangency of the incircle with sides $AB, BC, CA,$ respectively. Prove that all triangles in $S$ are isosceles and similar to one another.
We define a sequence $ \left(a_{1},a_{2},a_{3},\ldots \right)$ by
\[ a_{n} \equal{} \frac {1}{n}\left(\left\lfloor\frac {n}{1}\right\rfloor \plus{} \left\lfloor\frac {n}{2}\right\rfloor \plus{} \cdots \plus{} \left\lfloor\frac {n}{n}\right\rfloor\right),
\] where $\lfloor x\rfloor$ denotes the integer part of $x$.
[b]a)[/b] Prove that $a_{n+1}>a_n$ infinitely often.
[b]b)[/b] Prove that $a_{n+1}<a_n$ infinitely often.
[i]Proposed by Johan Meyer, South Africa[/i]
Find all function $f:\mathbb{N}\rightarrow\mathbb{N}$ such that for all $a,b\in\mathbb{N}$ ,
$(f(a)+b) f(a+f(b))=(a+f(b))^2$
For each positive integer $n$, find the largest real number $C_n$ with the following property. Given any $n$ real-valued functions $f_1(x), f_2(x), \cdots, f_n(x)$ defined on the closed interval $0 \le x \le 1$, one can find numbers $x_1, x_2, \cdots x_n$, such that $0 \le x_i \le 1$ satisfying
\[|f_1(x_1)+f_2(x_2)+\cdots f_n(x_n)-x_1x_2\cdots x_n| \ge C_n\]
[i]Marko Radovanović, Serbia[/i]
Let $a,b,c$ be distinct positive real numbers, and let $k$ be a positive integer greater than $3$. Show that
\[\left\lvert\frac{a^{k+1}(b-c)+b^{k+1}(c-a)+c^{k+1}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{k+1}{3(k-1)}(a+b+c)\]
and
\[\left\lvert\frac{a^{k+2}(b-c)+b^{k+2}(c-a)+c^{k+2}(a-b)}{a^k(b-c)+b^k(c-a)+c^k(a-b)}\right\rvert\ge \frac{(k+1)(k+2)}{3k(k-1)}(a^2+b^2+c^2).\]
[i]Calvin Deng.[/i]
Let $a$, $b$, $c$ and $d$ be real numbers such that no two of them are equal,
\[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=4\] and $ac=bd$. Find the maximum possible value of
\[\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}.\]
Find all functions $f,g: \mathbb{R} \to \mathbb{R}$ such that satisfies
$$f(x^2-g(y))=g(x)^2-y$$
for all $x,y \in \mathbb{R}$