Determine the smallest and the greatest possible values of the expression $$\left( \frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}\right)\left( \frac{a^2}{a^2+1}+\frac{b^2}{b^2+1}+\frac{c^2}{c^2+1}\right)$$ provided $a,b$ and $c$ are non-negative real numbers satisfying $ab+bc+ca=1$. [i]Proposed by Walther Janous, Austria [/i]

In terms of $n\ge2$, find the largest constant $c$ such that for all nonnegative $a_1,a_2,\ldots,a_n$ satisfying $a_1+a_2+\cdots+a_n=n$, the following inequality holds: \[\frac1{n+ca_1^2}+\frac1{n+ca_2^2}+\cdots+\frac1{n+ca_n^2}\le \frac{n}{n+c}.\] [i]Calvin Deng.[/i]

Let $n \ge 1$ and $x_1, \ldots, x_n \ge 0$. Prove that $$ (x_1 + \frac{x_2}{2} + \ldots + \frac{x_n}{n}) (x_1 + 2x_2 + \ldots + nx_n) \le \frac{(n+1)^2}{4n} (x_1 + x_2 + \ldots + x_n)^2 .$$

In a triangle $ABC$, the bisector of the angle at $A$ of a triangle $ABC$ intersects the segment $BC$ and the circumcircle of $ABC$ at points $A_1$ and $A_2$, respectively. Points $B_1,B_2,C_1,C_2$ are analogously defined. Prove that $$\frac{A_1A_2}{BA_2+CA_2}+\frac{B_1B_2}{CB_2+AB_2}+\frac{C_1C_2}{AC_2+BC_2}\ge\frac34.$$

Consider $n \geq 2 $ positive numbers $0<x_1 \leq x_2 \leq ... \leq x_n$, such that $x_1 + x_2 + ... + x_n = 1$. Prove that if $x_n \leq \dfrac{2}{3}$, then there exists a positive integer $1 \leq k \leq n$ such that $\dfrac{1}{3} \leq x_1+x_2+...+x_k < \dfrac{2}{3}$.

Let $G=G(V,E)$ be a simple graph with vertex set $V$ and edge set $E$. Suppose $|V|=n$. A map $f:\,V\rightarrow\mathbb{Z}$ is called good, if $f$ satisfies the followings: (1) $\sum_{v\in V} f(v)=|E|$; (2) color arbitarily some vertices into red, one can always find a red vertex $v$ such that $f(v)$ is no more than the number of uncolored vertices adjacent to $v$. Let $m(G)$ be the number of good maps. Prove that if every vertex in $G$ is adjacent to at least one another vertex, then $n\leq m(G)\leq n!$.

Find all real $ a$, such that there exist a function $ f: \mathbb{R}\rightarrow\mathbb{R}$ satisfying the following inequality: \[ x\plus{}af(y)\leq y\plus{}f(f(x)) \] for all $ x,y\in\mathbb{R}$

Let $k$ and $l$ be two given positive integers and $a_{ij}(1 \leq i \leq k, 1 \leq j \leq l)$ be $kl$ positive integers. Show that if $q \geq p > 0$, then \[(\sum_{j=1}^{l}(\sum_{i=1}^{k}a_{ij}^{p})^{q/p})^{1/q}\leq (\sum_{i=1}^{k}(\sum_{j=1}^{l}a_{ij}^{q})^{p/q})^{1/p}.\]

In $\triangle ABC$, consider points $D$ and $E$ on $AC$ and $AB$, respectively, and assume that they do not coincide with any of the vertices $A$, $B$, $C$. If the segments $BD$ and $CE$ intersect at $F$, consider areas $w$, $x$, $y$, $z$ of the quadrilateral $AEFD$ and the triangles $BEF$, $BFC$, $CDF$, respectively. [list=a] [*] Prove that $y^2 > xz$. [*] Determine $w$ in terms of $x$, $y$, $z$. [/list] [asy] import graph; size(10cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(12); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -2.8465032978885407, xmax = 9.445649196374966, ymin = -1.7618066305534972, ymax = 4.389732795464592; /* image dimensions */ draw((3.8295013012181283,2.816337276198864)--(-0.7368327629589799,-0.5920813291311117)--(5.672613975760373,-0.636902634996282)--cycle, linewidth(0.5)); /* draw figures */ draw((3.8295013012181283,2.816337276198864)--(-0.7368327629589799,-0.5920813291311117), linewidth(0.5)); draw((-0.7368327629589799,-0.5920813291311117)--(5.672613975760373,-0.636902634996282), linewidth(0.5)); draw((5.672613975760373,-0.636902634996282)--(3.8295013012181283,2.816337276198864), linewidth(0.5)); draw((-0.7368327629589799,-0.5920813291311117)--(4.569287648059735,1.430279997142299), linewidth(0.5)); draw((5.672613975760373,-0.636902634996282)--(1.8844000180622977,1.3644681598392678), linewidth(0.5)); label("$y$",(2.74779188172294,0.23771684184669772),SE*labelscalefactor); label("$w$",(3.2941097703568736,1.8657441499758196),SE*labelscalefactor); label("$x$",(1.6660824622277512,1.0025618859342047),SE*labelscalefactor); label("$z$",(4.288408327670633,0.8168138037986672),SE*labelscalefactor); /* dots and labels */ dot((3.8295013012181283,2.816337276198864),dotstyle); label("$A$", (3.8732067323088435,2.925600853925651), NE * labelscalefactor); dot((-0.7368327629589799,-0.5920813291311117),dotstyle); label("$B$", (-1.1,-0.7565817154670613), NE * labelscalefactor); dot((5.672613975760373,-0.636902634996282),dotstyle); label("$C$", (5.763466626982254,-0.7784344310124186), NE * labelscalefactor); dot((4.569287648059735,1.430279997142299),dotstyle); label("$D$", (4.692683565259744,1.5051743434774234), NE * labelscalefactor); dot((1.8844000180622977,1.3644681598392678),dotstyle); label("$E$", (1.775346039954538,1.4942479857047448), NE * labelscalefactor); dot((2.937230516274804,0.8082418657164665),linewidth(4.pt) + dotstyle); label("$F$", (2.889834532767763,0.954), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]

Let $n\geq3$ be a given integer, and let $a_1,a_2,\cdots,a_{2n},b_1,b_2,\cdots,b_{2n}$ be $4n$ nonnegative reals, such that $$a_1+a_2+\cdots+a_{2n}=b_1+b_2+\cdots+b_{2n}>0,$$ and for any $i=1,2,\cdots,2n,$ $a_ia_{i+2}\geq b_i+b_{i+1},$ where $a_{2n+1}=a_1,$ $a_{2n+2}=a_2,$ $b_{2n+1}=b_1.$ Detemine the minimum of $a_1+a_2+\cdots+a_{2n}.$

Let $x,y,z$ be positive real numbers. Prove the following inequality: \[ \frac 1{x^2+yz} + \frac 1{y^2+zx } + \frac 1{z^2+xy} \leq \frac 12 \left( \frac 1{xy} + \frac 1{yz} + \frac 1{zx} \right). \]

Prove that for every integer $n$ greater than $1:$ $$\frac{3n+1}{2n+2} < \left( \frac{1}{n} \right)^{n} + \left( \frac{2}{n} \right)^{n}+ \ldots+\left( \frac{n}{n} \right)^{n} <2.$$

Let $ a$, $ b$, and $ c$ be positive real numbers such that $ ab + bc + ca = 3$. Prove the inequality \[ 3 + \sum_{\mathrm{\cyc}} (a - b)^2 \ge \frac {a + b^2c^2}{b + c} + \frac {b + c^2a^2}{c + a} + \frac {c + a^2b^2}{a + b} \ge 3. \]

Let $ a_1$, $ a_2$, ..., $ a_n$ and $ b_1$, $ b_2$, ..., $ b_n$ be $ 2 \cdot n$ real numbers. Prove that the following two statements are equivalent: [b]i)[/b] For any $ n$ real numbers $ x_1$, $ x_2$, ..., $ x_n$ satisfying $ x_1 \leq x_2 \leq \ldots \leq x_ n$, we have $ \sum^{n}_{k \equal{} 1} a_k \cdot x_k \leq \sum^{n}_{k \equal{} 1} b_k \cdot x_k,$ [b]ii)[/b] We have $ \sum^{s}_{k \equal{} 1} a_k \leq \sum^{s}_{k \equal{} 1} b_k$ for every $ s\in\left\{1,2,...,n\minus{}1\right\}$ and $ \sum^{n}_{k \equal{} 1} a_k \equal{} \sum^{n}_{k \equal{} 1} b_k$.

For positive reals $a \geq b \geq c \geq 0$ prove the following inequality: $$\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \geq \frac{a+b}{a+c}+\frac{b+c}{b+a}+\frac{c+a}{c+b}$$

The real numbers $a, b, c$ are positive, and the real numbers $p, q, r \in [0,1/2]$ satisfy equality $p + q + r = 1$. Prove the inequality $$pab + qbc + rca \le \frac18 (a + b + c)^2.$$

Let $a,b,c,d$ be any real numbers such that $a+b+c+d=0$, prove that \[1296(a^7+b^7+c^7+d^7)^2\le637(a^2+b^2+c^2+d^2)^7\]

Let $x, y$ and $z$ be positive real numbers such that $x \geq y+z$. Proof that $$\frac{x+y}{z} + \frac{y+z}{x} +\frac{z+x}{y} \geq 7$$ When does equality occur? (Walther Janous)

If $a,b,c$ are the sides of a triangle and $r$ the inradius of the triangle, prove that \[\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\le \frac{1}{4r^2} \]

Positive integer numbers $k$ and $n$ satisfy the inequality $k > n!$. Prove that there exist pairwisely different prime numbers $p_1, p_2, \ldots, p_n$ which are divisors of the numbers $k+1, k+2, \ldots, k+n$ respectively (i.e. $p_i|k+i$).

Consider two infinite sequences $a_0,a_1,a_2,\dots$ and $b_0,b_1,b_2,\dots$ of real numbers such that $a_0=0$, $b_0=0$ and \[a_{k+1}=b_k, \quad b_{k+1}=\frac{a_kb_k+a_k+1}{b_k+1}\] for each integer $k \ge 0$. Prove that $a_{2024}+b_{2024} \ge 88$.

Show that $ \frac{1}{2^2} +\frac{1}{3^2} +...+\frac{1}{n^2} <\frac{2}{3}$ for all $n \ge 2 $.

Let $a,b,c$ be positive real numbers such that $a^2+b^2+c^2\geqslant 3$. Prove that \[\frac{a^2}{a+b^2}+\frac{b^2}{b+c^2}+\frac{c^2}{c+a^2}\geqslant\frac{3}{2}.\]

Find all continuous functions $f : \mathbb R \to \mathbb R$ such that: (a) $\lim_{x \to \infty}f(x)$ exists; (b) $f(x) = \int_{x+1}^{x+2}f(t) \, dt$, for all $x \in \mathbb R$.

Let $a,b,c,d,e,f$ be positive real numbers. Given that $def+de+ef+fd=4$, show that \[ ((a+b)de+(b+c)ef+(c+a)fd)^2 \geq\ 12(abde+bcef+cafd). \][i]Proposed by Allen Liu[/i]