Let $ a,b,c$ be positive real numbers such that $ ab\plus{}bc\plus{}ca\equal{}3$. Prove that \[ \frac 1{1\plus{}a^2(b\plus{}c)} \plus{} \frac 1{1\plus{}b^2(c\plus{}a)} \plus{} \frac 1 {1\plus{}c^2(a\plus{}b) } \leq \frac 3 {1\plus{}2abc} .\]

Prove that if \( a \), \( b \), and \( c \) are positive real numbers, then the following inequality holds: \[ \frac{a + 3c}{a + b} + \frac{c + 3a}{b + c} + \frac{4b}{c + a} \geq 6. \]

Let $x_1,x_2,\ldots, x_{10}$ be non-negative real numbers such that $\frac{x_1}{1}+ \frac{x_2}{2} +\cdots+ \frac{x_{10}}{10}$ $\leq9$. Find the maximum possible value of $\frac{{x_1}^2}{1}+\frac{{x_2}^2}{2}+\cdots+\frac{{x_{10}}^2}{10}$.

$\boxed{\text{A1}}$Let $a,b,c$ be positive reals numbers such that $a+b+c=1$.Prove that $2(a^2+b^2+c^2)\ge \frac{1}{9}+15abc$

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f\left(x^3+f(y)\right)=x^2f(x)+y,$$for all $x,y\in\mathbb{R}.$ (Here $\mathbb{R}$ denotes the set of all real numbers.)

Find all real numbers $a, b, c$ that satisfy $$ 2a - b =a^2b, \qquad 2b-c = b^2 c, \qquad 2c-a= c^2 a.$$

[b]a)[/b] For $ a,b\ge 0 $ and $ x,y>0, $ show that: $$ \frac{a^3}{x^2} +\frac{b^3}{y^2}\ge \frac{(a+b)^3}{(x+y)^2} . $$ [b]b)[/b] For $ a,b,c\ge 0 $ and $ x,y,z>0 $ under the condition $ a+b+c=x+y+z, $ prove that: $$ \frac{a^3}{x^2} +\frac{b^3}{y^2} +\frac{c^3}{z^2} \ge a+b+c. $$

p1. Given $N = 9 + 99 + 999 + ... +\underbrace{\hbox{9999...9}}_{\hbox{121\,\,numbers}}$. Determine the value of N. p2. The triangle $ABC$ in the following picture is isosceles, with $AB = AC =90$ cm and $BC = 108$ cm. The points $P$ and $Q$ are located on $BC$, respectively such that $BP: PQ: QC = 1: 2: 1$. Points $S$ and $R$ are the midpoints of $AB$ and $AC$ respectively. From these two points draw a line perpendicular to $PR$ so that it intersects at $PR$ at points $M$ and $N$ respectively. Determine the length of $MN$. [img]https://cdn.artofproblemsolving.com/attachments/7/1/e1d1c4e6f067df7efb69af264f5c8de5061a56.png[/img] p3. If eight equilateral triangles with side $ 12$ cm are arranged as shown in the picture on the side, we get a octahedral net. Define the volume of the octahedron. [img]https://cdn.artofproblemsolving.com/attachments/4/8/18cdb8b15aaf4d92f9732880784facf9348a84.png[/img] p4. It is known that $a^2 + b^2 = 1$ and $x^2 + y^2 = 1$. Continue with the following algebraic process. $(a^2 + b^2)(x^2 + y^2) – (ax + by)^2 = ...$ a. What relationship can be concluded between $ax + by$ and $1$? b. Why? p5. A set of questions consists of $3$ questions with a choice of answers True ($T$) or False ($F$), as well as $3$ multiple choice questions with answers $A, B, C$, or $D$. Someone answer all questions randomly. What is the probability that he is correct in only $2$ questions?

Let $P$ be a point chosen on the interior of side $\overline{BC}$ of triangle $\triangle ABC$ with side lengths $\overline{AB} = 10, \overline{BC} = 10, \overline{AC} = 12$. If $X$ and $Y$ are the feet of the perpendiculars from $P$ to the sides $AB$ and $AC$, then the minimum possible value of $PX^2 + PY^2$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by Andrew Wen[/i]

Let $ P(x)$ be a polynomial with a non-negative coefficients. Prove that if the inequality $ P\left(\frac {1}{x}\right)P(x)\geq 1$ holds for $ x \equal{} 1$, then this inequality holds for each positive $ x$.

Suppose that $a_1,a_2,\dots a_n$ are real $(n>1)$ and $$A+ \sum_{i=1}^{n} a^2_i< \frac{1}{n-1} (\sum_{i=1}^{n} a_i)^2.$$ Prove that $A<2a_ia_j$ for $1\leq i<j\leq n.$

Let $a$, $b$ and $c$ be positive real numbers such that $a^2+b^2+c^2=3$. Prove the following inequality: $\frac{a}{3c(a^2-ab+b^2)} + \frac{b}{3a(b^2-bc+c^2)} + \frac{c}{3b(c^2-ca+a^2)} \leq \frac{1}{abc}$

Find the largest real constant $a$ such that for all $n \geq 1$ and for all real numbers $x_0, x_1, ... , x_n$ satisfying $0 = x_0 < x_1 < x_2 < \cdots < x_n$ we have \[\frac{1}{x_1-x_0} + \frac{1}{x_2-x_1} + \dots + \frac{1}{x_n-x_{n-1}} \geq a \left( \frac{2}{x_1} + \frac{3}{x_2} + \dots + \frac{n+1}{x_n} \right)\]

Show that for positive reals $a, b, c, d$ we have \[ \left(\dfrac{ab + ac + ad + bc + bd + cd}{6} \right)^3 \geq \left(\dfrac{abc + abd + acd + bcd}{4}\right)^2 \]

Let $a,b,c$ be positive real numbers such that $a+b+c=1$. Prove that \[\frac {a}{b} + \frac {a}{c} + \frac {c}{b} + \frac {c}{a} + \frac {b}{c} + \frac {b}{a} + 6 \geq 2\sqrt{2}\left (\sqrt{\frac{1-a}{a}} + \sqrt{\frac{1-b}{b}} + \sqrt{\frac{1-c}{c}}\right ).\] When does equality hold?