Let the sequence $(a_{n})_{n\geqslant 1}$ be defined as: $$a_{n}=\sqrt{A_{n+2}^{1}\sqrt[3]{A_{n+3}^{2}\sqrt[4]{A_{n+4}^{3}\sqrt[5]{A_{n+5}^{4}}}}},$$where $A_{m}^{k}$ are defined by $$A_{m}^{k}=\binom{m}{k}\cdot k!.$$Prove that $$a_{n}<\frac{119}{120}\cdot n+\frac{7}{3}.$$

Let $$f(X)=a_{n}X^{n}+a_{n-1}X^{n-1}+\cdots +a_{1}X+a_{0}$$be a polynomial with real coefficients which satisfies $$a_{n}\geq a_{n-1}\geq \cdots \geq a_{1}\geq a_{0}>0.$$Prove that for every complex root $z$ of this polynomial, we have $|z|\leq 1$.

Let $\omega$ be the circumcircle of the acute nonisosceles triangle $\Delta ABC$. Point $P$ lies on the altitude from $A$. Let $E$ and $F$ be the feet of the altitudes from P to $CA$, $BA$ respectively. Circumcircle of triangle $\Delta AEF$ intersects the circle $\omega$ in $G$, different from $A$. Prove that the lines $GP$, $BE$ and $CF$ are concurrent.

Determine all natural numbers $n$ of the form $n=[a,b]+[b,c]+[c,a]$ where $a,b,c$ are positive integers and $[u,v]$ is the least common multiple of the integers $u$ and $v$.

Find all continuous functions $f : R \rightarrow R$ such, that $f(xy)= f\left(\frac{x^2+y^2}{2}\right)+(x-y)^2$ for any real numbers $x$ and $y$

Let $a,b,c$ be positive real numbers that satisfy $a+b+c=abc$. Prove that $$\sqrt{(1+a^2)(1+b^2)}+\sqrt{(1+b^2)(1+c^2)}+\sqrt{(1+a^2)(1+c^2)}-\sqrt{(1+a^2)(1+b^2)(1+c^2)} \ge 4.$$

Let $ABC$ be an acute triangle, and $H$ its orthocenter. The distance from $H$ to rays $BC$, $CA$, and $AB$ is denoted by $d_a$, $d_b$, and $d_c$, respectively. Let $R$ be the radius of circumcenter of $\triangle ABC$ and $r$ be the radius of incenter of $\triangle ABC$. Prove the following inequality: $$d_a+d_b+d_c \le \frac{3R^2}{4r}$$.

At a summer school there are $7$ courses. Each participant was a student in at least one course, and each course was taken by exactly $40$ students. It is known that for each $2$ courses there were at most $9$ students who took them both. Prove that at least $120$ students participated at this summer school.

Let $$P(X)=a_{0}X^{n}+a_{1}X^{n-1}+\cdots+a_{n}$$be a polynomial with real coefficients such that $a_{0}>0$ and $$a_{n}\geq a_{i}\geq 0,$$for all $i=0,1,2,\ldots,n-1.$ Prove that if $$P^{2}(X)=b_{0}X^{2n}+b_{1}X^{2n-1}+\cdots+b_{n-1}X^{n+1}+\cdots+b_{2n},$$then $P^2(1)\geq 2b_{n-1}.$

Let $p$ be an odd prime. Prove that the number $$\left\lfloor \left(\sqrt{5}+2\right)^{p}-2^{p+1}\right\rfloor$$is divisible by $20p$.

Find all ordered pairs of nonnegative integers $(x,y)$ such that \[x^4-x^2y^2+y^4+2x^3y-2xy^3=1.\]

There are $75$ points in the plane, no three collinear. Prove that the number of acute triangles is no more than $70\%$ from the total number of triangles with vertices in these points.