A circle $\omega$ is inscribed in a quadrilateral $ABCD$. Let $I$ be the center of $\omega$. Suppose that \[ (AI + DI)^2 + (BI + CI)^2 = (AB + CD)^2. \] Prove that $ABCD$ is an isosceles trapezoid.

The positive numbers $a,b,c,A,B,C$ satisfy a condition $$a + A = b + B = c + C = k$$ Prove that $$aB + bC + cA \le k^2$$

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]

[b]p16.[/b] When not writing power rounds, Eric likes to climb trees. The strength in his arms as a function of time is $s(t) = t^3 - 3t^2$. His climbing velocity as a function of the strength in his arms is $v(s) = s^5 + 9s^4 + 19s^3 - 9s^2 - 20s$. At how many (possibly negative) points in time is Eric stationary? [b]p17[/b]. Consider a triangle $\vartriangle ABC$ with angles $\angle ACB = 60^o$, $\angle ABC = 45^o$. The circumcircle around $\vartriangle ABH$, where $H$ is the orthocenter of $\vartriangle ABC$, intersects $BC$ for a second time in point $P$, and the center of that circumcircle is $O_c$. The line $PH$ intersects $AC$ in point $Q$, and $N$ is center of the circumcircle around $\vartriangle AQP$. Find $\angle NO_cP$. [b]p18.[/b] If $x, y$ are positive real numbers and $xy^3 = \frac{16}{9}$ , what is the minimum possible value of $3x + y$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a,b,c,d,x,y$ denote the lengths of the sides $AB, BC,CD,DA$ and the diagonals $AC,BD$ of a cyclic quadrilateral $ABCD$ respectively. Prove that $$(\frac{1}{a}+\frac{1}{c})^2+(\frac{1}{b}+\frac{1}{d})^2 \geq 8 ( \frac{1}{x^2}+\frac{1}{y^2})$$

Consider a prism with triangular base. The total area of the three faces containing a particular vertex $A$ is $K$. Show that the maximum possible volume of the prism is $\sqrt{\frac{K^3}{54}}$ and find the height of this largest prism.

$a,b,c\in\mathbb{R^+}$ and $a^2+b^2+c^2=48$. Prove that \[a^2\sqrt{2b^3+16}+b^2\sqrt{2c^3+16}+c^2\sqrt{2a^3+16}\le24^2\]

Given positive real numbers $a,b,c$ such that $abc=1$, find the maximum possible value of $$\frac{1}{(4a+4b+c)^3}+\frac{1}{(4b+4c+a)^3}+\frac{1}{(4c+4a+b)^3}.$$

Let $a, b, c, d$ three strictly positive real numbers such that \[a^{2}+b^{2}+c^{2}=d^{2}+e^{2},\] \[a^{4}+b^{4}+c^{4}=d^{4}+e^{4}.\] Compare \[a^{3}+b^{3}+c^{3}\] with \[d^{3}+e^{3},\]

Let $a_{1}, a_{2}, \ldots, a_{n}\in [0;1]$. If $S=a_{1}^{3}+a_{2}^{3}+\ldots+a_{n}^{3}$ then prove that \[\frac{a_{1}}{2n+1+S-a_{1}^{3}}+\frac{a_{2}}{2n+1+S-a_{2}^{3}}+\ldots+\frac{a_{n}}{2n+1+S-a_{n}^{3}}\leq \frac{1}{3}\]

Let $x_1, x_2, \dots ,x_n$ ($n\ge 2$) be positive real numbers satisfying $\sum^{n}_{i=1}x_i=1$. Prove that:\[\sum^{n}_{i=1}\dfrac{x_i}{\sqrt{1-x_i}}\ge \dfrac{\sum_{i=1}^{n}\sqrt{x_i}}{\sqrt{n-1}}.\]

Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.

Let $p \geq 5$ be a prime. (a) Show that exists a prime $q \neq p$ such that $q| (p-1)^{p}+1$ (b) Factoring in prime numbers $(p-1)^{p}+1 = \prod_{i=1}^{n}p_{i}^{a_{i}}$ show that: \[\sum_{i=1}^{n}p_{i}a_{i}\geq \frac{p^{2}}2 \]

Let $AB$ be a segment of unit length and let $C, D$ be variable points of this segment. Find the maximum value of the product of the lengths of the six distinct segments with endpoints in the set $\{A,B,C,D\}.$

a) Prove that, whatever the real number x would be, the following inequality takes place ${{x}^{4}}-{{x}^{3}}-x+1\ge 0.$ b) Solve the following system in the set of real numbers: ${{x}_{1}}+{{x}_{2}}+{{x}_{3}}=3,x_{1}^{3}+x_{2}^{3}+x_{3}^{3}=x_{1}^{4}+x_{2}^{4}+x_{3}^{4}$. The Mathematical Gazette