Find an integer pair of solutions $(x, y)$ to the following system of equations. $$\log_2 (y^x) = 16$$ $$\log_2 (x^y) = 8$$

Let $a$, $b$ and $c$ be positive real numbers such that $a+b+c=1$. Prove the inequality $$ \left ( \frac{1+a}{b}+2 \right ) \left ( \frac{1+b}{c}+2 \right ) \left ( \frac{1+c}{a}+2 \right )\geq 216.$$ When does equality hold? [i]Authored by Anastasija Trajanova[/i]

Find the largest constant $k$ such that the inequality $$a^2+b^2+c^2-ab-bc-ca \ge k \left|\frac{a^3-b^3}{a+b}+\frac{b^3-c^3}{b+c}+\frac{c^3-a^3}{c+a}\right|$$ holds for any for non negative real numbers $a,b,c$ with $(a+b)(b+c)(c+a)>0$.

Let $x_1,x_2,\ldots x_n$ be positive integers such that no one of them is an initial fragment of any other (for example, $12$ is an initial fragment of $\underline{12},\underline{12}5$ and $\underline{12}405$). Prove that \[\frac{1}{x_1}+\frac{1}{x_2}+\ldots+\frac{1}{x_n}<3. \]

Denote by $\mathbb{V}$ the real vector space of all real polynomials in one variable, and let $\gamma :\mathbb{V}\to \mathbb{R}$ be a linear map. Suppose that for all $f,g\in \mathbb{V}$ with $\gamma(fg)=0$ we have $\gamma(f)=0$ or $\gamma(g)=0$. Prove that there exist $c,x_0\in \mathbb{R}$ such that \[ \gamma(f)=cf(x_0)\quad \forall f\in \mathbb{V}\]

Show that there exists a constant $a > 1$ such that, for any positive integers $m$ and $n$, $\frac{m}{n} < \sqrt7$ implies that $$7-\frac{m^2}{n^2} \ge \frac{a}{n^2} .$$

Let $P(x)$ be a nonconstant polynomial of degree $n$ with rational coefficients which can not be presented as a product of two nonconstant polynomials with rational coefficients. Prove that the number of polynomials $Q(x)$ of degree less than $n$ with rational coefficients such that $P(x)$ divides $P(Q(x))$ a) is finite b) does not exceed $n$.

Let $a$ and $b$ be real numbers such that$$\frac{a}{a^2-5} = \frac{b}{5-b^2} = \frac{ab}{a^2b^2-5}$$where $a+b \neq 0$. $a^4 + b^4 =$ ?

Let $p$ be a certain prime number. Find all non-negative integers $n$ for which polynomial $P(x)=x^4-2(n+p)x^2+(n-p)^2$ may be rewritten as product of two quadratic polynomials $P_1, \ P_2 \in \mathbb{Z}[X]$.

Let be four real numbers. Find the polynom of least degree such that two of these numbers are some locally extreme values, and the other two are the respective points of local extrema.

Find all functions $f:\mathbb{R}\to\mathbb{R}$ for which \[f(x+y)=f(x-y)+f(f(1-xy))\] holds for all real numbers $x$ and $y$

Let $ABC$ be a triangle. Choose points $A'$, $B'$ and $C'$ independently on side segments $BC$, $CA$ and $AB$ respectively with a uniform distribution. For a point $Z$ in the plane, let $p(Z)$ denote the probability that $Z$ is contained in the triangle enclosed by lines $AA'$, $BB'$ and $CC'$. For which interior point $Z$ in triangle $ABC$ is $p(Z)$ maximised?

For every integer $d \geq 1$, let $M_d$ be the set of all positive integers that cannot be written as a sum of an arithmetic progression with difference $d$, having at least two terms and consisting of positive integers. Let $A = M_1$, $B = M_2 \setminus \{2 \}, C = M_3$. Prove that every $c \in C$ may be written in a unique way as $c = ab$ with $a \in A, b \in B.$

A sphere is inscribed in a truncated right circular cone as shown. The volume of the truncated cone is twice that of the sphere. What is the ratio of the radius of the bottom base of the truncated cone to the radius of the top base of the truncated cone? [asy] real r=(3+sqrt(5))/2; real s=sqrt(r); real Brad=r; real brad=1; real Fht = 2*s; import graph3; import solids; currentprojection=orthographic(1,0,.2); currentlight=(10,10,5); revolution sph=sphere((0,0,Fht/2),Fht/2); //draw(surface(sph),green+white+opacity(0.5)); //triple f(pair t) {return (t.x*cos(t.y),t.x*sin(t.y),t.x^(1/n)*sin(t.y/n));} triple f(pair t) { triple v0 = Brad*(cos(t.x),sin(t.x),0); triple v1 = brad*(cos(t.x),sin(t.x),0)+(0,0,Fht); return (v0 + t.y*(v1-v0)); } triple g(pair t) { return (t.y*cos(t.x),t.y*sin(t.x),0); } surface sback=surface(f,(3pi/4,0),(7pi/4,1),80,2); surface sfront=surface(f,(7pi/4,0),(11pi/4,1),80,2); surface base = surface(g,(0,0),(2pi,Brad),80,2); draw(sback,rgb(0,1,0)); draw(sfront,rgb(.3,1,.3)); draw(base,rgb(.4,1,.4)); draw(surface(sph),rgb(.3,1,.3)); [/asy] $ \textbf {(A) } \dfrac {3}{2} \qquad \textbf {(B) } \dfrac {1+\sqrt{5}}{2} \qquad \textbf {(C) } \sqrt{3} \qquad \textbf {(D) } 2 \qquad \textbf {(E) } \dfrac {3+\sqrt{5}}{2} $

Prove that $\frac{5^{125}-1}{5^{25}-1}$ is a composite number.

Let $a, b$ and $c$ be pairwise different natural numbers. Prove $\frac{a^3 + b^3 + c^3}{3} \ge abc + a + b + c$. When does equality holds? (Karl Czakler)

Let $f: \mathbb R \to\mathbb R$ be a function such that $|f(x)| \leq 1$ for all $x \in \mathbb R$ and \[f \left( x + \frac{13}{42} \right) + f(x) = f \left( x + \frac 17 \right) + f \left( x + \frac 16 \right), \quad \forall x \in \mathbb R.\] Show that $f$ is a periodic function.

Let $n \ge 1$ be a fixed positive integer. We consider all the sets $S$ which consist of sub-sequences of the sequence $0, 1,2, ..., n$ satisfying the following conditions: i) If $(a_i)_{i=0}^k$ belongs to $S$, then $a_0 = 0$, $a_k = n$ and $a_{i+1} - a_i \le 2$ for all $0 \le i \le k - 1$. ii) If $(a_i)_{i=0}^k$ and $(b_j)^h_{j=0}$ both belong to $S$, then there exist $0 \le i_0 \le k - 1$ and $0 \le j_0 \le h - 1$ such that $a_{i_0} = b_{j_0}$ and $a_{i_0+1} = b_{j_0+1}$. Find the maximum value of $|S|$ (among all the above-mentioned sets $S$).

Prove that all positive real $x, y, z$ satisfy the inequality $x^y + y^z + z^x > 1$. (D. Bazylev)

The sequence $a_1, a_2, a_3, ...$ of positive reals is such that $\sum a_i$ diverges. Show that there is a sequence $b_1, b_2, b_3, ...$ of positive reals such that $\lim b_n = 0$ and $\sum a_ib_i$ diverges.

The numbers $x, y, z, t, a$ and $b$ are positive integers, so that $xt-yz = 1$ and $$\frac{x}{y} \ge \frac{a}{b} \ge \frac{z}{t} .$$Prove that $$ab \le (x + z) (y +t)$$

Find all functions $f :\mathbb R \to \mathbb R$ such that for all real $x, y$ \[f(f(x)^2 + f(y)) = xf(x) + y.\]

Let $p(x) = (x- x_1)(x- x_2)(x- x_3)$, where $x_1, x_2$ and $x_3$ are real. Show that $p(x) p''(x) \le p'(x)^2$ for all $x$.

Prove that there exists a positive integer $N$ such that for every polynomial $P(x)$ of degree $2019$, there exist $N$ linear polynomials $p_1,p_2, \ldots p_N$ such that $P(x)=p_1(x)^{2019}+p_2(x)^{2019}+ \ldots + p_N(x)^{2019}$. (Assume all polynomials in this problem have real coefficients, and leading coefficients cannot be zero.)

Determine all real numbers $a, b, c, d$ such that the polynomial $f(x) = ax^3 +bx^2 + cx + d$ satisfies simultaneously the folloving conditions $\begin {cases} |f(x)| \le 1 \,for \, |x| \le 1 \\ f(2) = 26 \end {cases}$