Prove that $0\le yz+zx+xy-2xyz\le{7\over27}$, where $x,y$ and $z$ are non-negative real numbers satisfying $x+y+z=1$.

In right triangle $ABC$ the hypotenuse $\overline{AB}=5$ and leg $\overline{AC}=3$. The bisector of angle $A$ meets the opposite side in $A_1$. A second right triangle $PQR$ is then constructed with hypotenuse $\overline{PQ}=A_1B$ and leg $\overline{PR}=A_1C$. If the bisector of angle $P$ meets the opposite side in $P_1$, the length of $PP_1$ is: $\textbf{(A)}\ \frac{3\sqrt{6}}{4}\qquad \textbf{(B)}\ \frac{3\sqrt{5}}{4}\qquad \textbf{(C)}\ \frac{3\sqrt{3}}{4}\qquad \textbf{(D)}\ \frac{3\sqrt{2}}{2}\qquad \textbf{(E)}\ \frac{15\sqrt{2}}{16}$

Let $g:[0,1]\rightarrow \mathbb{R}$ be a continuous function and let $f_{n}:[0,1]\rightarrow \mathbb{R}$ be a sequence of functions defined by $f_{0}(x)=g(x)$ and $$f_{n+1}(x)=\frac{1}{x}\int_{0}^{x}f_{n}(t)dt.$$ Determine $\lim_{n\to \infty}f_{n}(x)$ for every $x\in (0,1]$.

The value of $x$ that satisfies $\log_{2^x} 3^{20} = \log_{2^{x+3}} 3^{2020}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Let $n$ be a positive integer and let $P$ be the set of monic polynomials of degree $n$ with complex coefficients. Find the value of \[ \min_{p \in P} \left \{ \max_{|z| = 1} |p(z)| \right \} \]

Let $P_1 , P_2 , ...$ be arbitrary points and A be a connected compact set in the plane with a diameter greater than 4. Show that for some point P in A , $\overline {PP_1} \cdot \overline {PP_2} \cdots \overline {PP_n}>1$. Furthermore, prove that this is no longer necessarily true for compact sets of diameter 4.

Some books are placed on each other. Someone first, reverses the upper book. Then he reverses the $2$ upper books. Then he reverses the $3$ upper books and continues like this. After he reversed all the books, he starts this operation from the first. Prove that after finite number of movements, the books become exactly like their initial configuration.

Let $S$ be the set of circles in the coordinate plane that are tangent to each of the three circles with equations $x^{2}+y^{2}=4$, $x^{2}+y^{2}=64$, and $(x-5)^{2}+y^{2}=3$. What is the sum of the areas of all circles in $S$? $\textbf{(A)}~48\pi\qquad\textbf{(B)}~68\pi\qquad\textbf{(C)}~96\pi\qquad\textbf{(D)}~102\pi\qquad\textbf{(E)}~136\pi\qquad$

Two circles with centres $A$ and $B$ lie inside an angle. They touch each other and both sides of the angle. Prove that the circle with the diameter $AB$ touches both sides of the angle. (V. Prasolov)

There exists an infinite set of triangles with the following properties: (a) the lengths of the sides are integers with no common factors, and (b) one and only one angle is $60^\circ$. One such triangle has side lengths $5,7,8$. Find two more.

Prove that the number \[\sum_{k=0}^{n}\binom{2n+1}{2k+1}2^{3k}\] is not divisible by $5$ for any integer $n\geq 0$.

Mr. Ambulando is at the intersection of $5^{\text{th}}$ and $\text{A St}$, and needs to walk to the intersection of $1^{\text{st}}$ and $\text{F St}$. There's an accident at the intersection of $4^{\text{th}}$ and $\text{B St}$, which he'd like to avoid. [center]<see attached>[/center] Given that Mr. Ambulando wants to walk the shortest distance possible, how many different routes through downtown can he take?

If $p,q,$ and $r$ are primes with $pqr=7(p+q+r)$, find $p+q+r$.

Let $a, b, c$ be positive real numbers satisfying the condition $\frac{1}{ab}+\frac{1}{ac}+\frac{1}{bc}=1$ . Prove the inequality $$\frac{a}{\sqrt{1+a^2}}+\frac{b}{\sqrt{1+b^2}}+\frac{c}{\sqrt{1+c^2}} \le \frac{3\sqrt3}{2}$$ When does the equality hold? (L. Parts)

Let $D$ be an interior point on the side $BC$ of an acute-angled triangle $ABC$. Let the circumcircle of triangle $ADB$ intersect $AC$ again at $E(\ne A)$ and the circumcircle of triangle $ADC$ intersect $AB$ again at $F(\ne A)$. Let $AD$, $BE$, and $CF$ intersect the circumcircle of triangle $ABC$ again at $D_1(\ne A)$, $E_1(\ne B)$ and $F_1(\ne C)$, respectively. Let $I$ and $I_1$ be the incentres of triangles $DEF$ and $D_1E_1F_1$, respectively. Prove that $E,F, I, I_1$ are concyclic.

Prove that the equation $(\sqrt5 +1)^{2x}+ (\sqrt5 -1)^{2x}=2^x(y^2+2)$ has an infinite number of solutions in natural numbers.

Let $ABC$ be an acute triagnle with its circumcircle $\omega$. Let point $D$ be the foot of triangle $ABC$ from point $A$. Let points $E,F$ be midpoints of sides $AB,AC$, respectively. Let points $P$ and $Q$ be the second intersections of of circle $\omega$ with circumcircle of triangles $BDE$ and $CDF$, respectively. Suppose that $A,P,B,Q$ and $C$ be on a circle in this order. Show that the lines $EF,BQ$ and $CP$ are concurrent.

Let $p(x)$ and $q(x)$ be two polynomials, both of which have their sum of coefficients equal to $s.$ Let $p,q$ satisfy $p(x)^3-q(x)^3=p(x^3)-q(x^3).$ Show that (i) There exists an integer $a\geq1$ and a polynomial $r(x)$ with $r(1)\neq0$ such that \[p(x)-q(x)=(x-1)^ar(x).\] (ii) Show that $s^2=3^{a-1},$ where $a$ is described as above.

Angle $A$ in triangle $ABC$ is equal to $120^o$. Prove that the distance from the center of the circumscribed circle to the orthocenter is equal to $AB + AC$. (V. Protasov)

An infinite sequence $(x_{n})$ is defined by its first term $x_{1}>1$, which is a rational number, and the relation $x_{n+1}=x_{n}+\frac{1}{\lfloor x_{n}\rfloor}$ for all positive integers $n$. Prove that this sequence contains an integer. [i]A. Golovanov[/i]

Let $n$ be an odd positive integer. Prove that $\sigma(n)^3 <n^4$.

Find all positive integers $n$ such that $n$ is equal to $100$ times the number of positive divisors of $n$.

When Pablo turns $15$, he throws a party inviting $43$ friends. He presents them with a cake n the form of a regular $15$-sided polygon and on it $15$ candles. The candles are arranged so that between candles and vertices there are never three aligned (any three candles are not aligned, nor are any two candles with a vertex of the polygon, nor are any two vertices of the polygon with a candle). Then Pablo divides the cake into triangular pieces, by means of cuts that join candles to each other or candles and vertices, but also do not intersect with others already made. Why, by doing this, Paul was able to distribute a piece to each of his guests but he was left without eating?

Find all integers $x,y$ that satisfy the equation $x+y=x^2-xy+y^2$

Let $ X $ be the power set of set of $ \{ 0\}\cup\mathbb{N} , $ and let be a function $ d:X^2\longrightarrow\mathbb{R} $ defined as $$ d(U,V)=\sum_{n\in\mathbb{N}}\frac{\chi_U (n) +\chi_V (n) -2\chi_{U\cap V} (n)}{2} , $$ where $ \chi_W (n)=\left\{ \begin{matrix} 1,& n\in W\\ 0,& n\not\in W \end{matrix} \right. ,\quad\forall W\in X,\forall n\in\mathbb{N} . $ [b]a)[/b] Prove that there exists an unique $ V' $ such that $ \lim_{k\to\infty} d\left( \{ k+i|i\in\mathbb{N}\} , V'\right) =0. $ [b]b)[/b] Demonstrate that for all $ V\in X $ there exists a $ v\in\mathbb{N} $ with $ d\left( \left\{ \frac{3}{2} -\frac{1}{2}(-1)^{v} \right\} , V \right) >\frac{1}{k} . $ [b]c)[/b] Let $ f: X\longrightarrow X,\quad f(X)=\left\{ 1+x|x\in X\right\} . $ Calculate $ d\left( f(A),f(B) \right) $ in terms of $ d(A,B) $ and prove that $ f $ admits an unique fixed point.