Let $f: \mathbb{Z} \rightarrow \mathbb{Z}$ be a function such that: For all $a$ and $b$ in $\mathbb{Z} - \{0\}$, $f(ab) \geq f(a) + f(b)$. Show that for all $a \in \mathbb{Z} - \{0\}$ we have $f(a^n) = nf(a)$ for all $n \in \mathbb{N}$ if and only if $f(a^2) = 2f(a)$

Given is an acute-angled triangle $ABC$ with angles $\alpha$, $\beta$ and $\gamma$. On side $AB$ lies a point $P$ such that the line connecting the feet of the perpendiculars from $P$ on $AC$ and $BC$ is parallel to $AB$. Express the ratio $\frac{AP}{BP}$ in terms of $\alpha$ and $\beta$.

In a regular hexagon $ABCDEF$ with center $O$, points $M$ and $N$ are the midpoints of the sides $CD$ and $DE$, and $L$ the intersection point of $AM$ and $BN$. Prove that: (a) $ABL$ and $DMLN$ have equal areas; (b) $\angle ALD=\angle OLN=60^\circ$; (c) $\angle OLD=90^\circ$.

Let an integer $ n > 3$. Denote the set $ T\equal{}\{1,2, \ldots,n\}.$ A subset S of T is called [i]wanting set[/i] if S has the property: There exists a positive integer $ c$ which is not greater than $ \frac {n}{2}$ such that $ |s_1 \minus{} s_2|\ne c$ for every pairs of arbitrary elements $ s_1,s_2\in S$. How many does a [i]wanting set[/i] have at most are there ?

Prove that the number $\lfloor (2+\sqrt5)^{2019} \rfloor$ is not prime.

Find all polynomials $P(x)$ with real coefficients satisfying: $P(2017) = 2016$ and $$(P(x)+1)^2=P(x^2+1).$$

A quadrilateral is inscribed in a circle. If angles are inscribed in the four arcs cut off by the sides of the quadrilateral, their sum will be: $ \textbf{(A)}\ 180^\circ \qquad \textbf{(B)}\ 540^\circ \qquad \textbf{(C)}\ 360^\circ \qquad \textbf{(D)}\ 450^\circ \qquad \textbf{(E)}\ 1080^\circ$

5. In tetrahedron $ABCD$ following equalities hold: $\angle BAC+\angle BDC=\angle ABD+\angle ACD$ $\angle BAD+\angle BCD=\angle ABC+\angle ADC$ Prove that center of sphere circumscribed about ABCD lies on a line through midpoints of $AB$ and $CD$.

Let $p$ be an odd prime number, and $k$ be an odd number not divisible by $p$. Consider a field $K$ be a field with $kp+1$ elements, and $A = \{x_1,x_2, \dots, x_t\}$ be the set of elements of $K^*$, whose order is not $k$ in the multiplicative group $(K^*,\cdot)$. Prove that the polynomial $P(X)=(X+x_1)(X+x_2)\dots(X+x_t)$ has at least $p$ coefficients equal to $1$.

If $ g(x)\equal{}1\minus{}x^2$ and $ f(g(x)) \equal{} \frac{1\minus{}x^2}{x^2}$ when $ x\neq0$, then $ f(1/2)$ equals $ \textbf{(A)}\ 3/4 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ \sqrt2/2 \qquad \textbf{(E)}\ \sqrt2$

1. Solve in real numbers x,y,z : $\{\begin{array}{ccc} x^2=yz+1 \\ y^2=zx+2 \\ z^2=xy+4 \\ \end{array}$

Let $P$ be a point inside a triangle $ABC$ such that $$ \frac{AP + BP}{AB} = \frac{BP + CP}{BC} = \frac{CP + AP}{CA} .$$ Lines $AP$, $BP$, $CP$ intersect the circumcircle of triangle $ABC$ again in $A'$, $B'$, $C'$. Prove that the triangles $ABC$ and $A'B'C'$ have a common incircle.

Is $\cos 1^\circ$ rational? Prove.

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

The product of three consecutive positive integers is $ 8$ times their sum. What is the sum of their squares? $ \textbf{(A)}\ 50 \qquad \textbf{(B)}\ 77 \qquad \textbf{(C)}\ 110 \qquad \textbf{(D)}\ 149 \qquad \textbf{(E)}\ 194$

Let $a_n = 2^{n-1}$ for $n > 0$. Let \[ b_n = \sum\limits_{r+s \leq n} a_ra_s \] Find $b_n-b_{n-1}$, $b_n-2b_{n-1}$ and $b_n$.

On a circle $100$ points are chosen and for each point we wrote the multiple of its distances to the rest. Could the written numbers be $1,2,\dots, 100$ in some order?

Prove the following theorem: If the intersection of any plane that has more than one point in common with the surface $F$ is a circle, then $F$ is a sphere (surface).

Barry has a standard die containing the numbers 1-6 on its faces. He rolls the die continuously, keeping track of the sum of the numbers he has rolled so far, starting from 0. Let $E_n$ be the expected number of time he needs to until his recorded sum is at least $n$. It turns out that there exist positive reals $a, b$ such that $$\lim_{n \rightarrow \infty} E_n - (an + b) = 0$$ Find $(a,b)$. [i]Proposed by Dilhan Salgado[/i]

Does there exist a functional equation[sup]1[/sup] that has a solution and the range of any of its solutions is the set of integers? [sup]1[/sup][size=75]A [i]functional equation[/i] has the form $\mbox{\footnotesize \(E = 0\)}$, where $\mbox{\footnotesize \(E\)}$ is a function form. The set of function forms is the smallest set $\mbox{\footnotesize \(\mathcal{F}\)}$ which contains the variables $\mbox{\footnotesize \(x_1, x_2, \dots\)}$, the real numbers $\mbox{\footnotesize \(r \in \mathbb{R}\)}$, and for which $\mbox{\footnotesize \(E, E_1, E_2 \in \mathcal{F}\)}$ implies $\mbox{\footnotesize \(E_1+E_2 \in \mathcal{F}\)}$, $\mbox{\footnotesize \(E_1 \cdot E_2 \in \mathcal{F}\)}$, and $\mbox{\footnotesize \(f(E) \in \mathcal{F}\)}$, where $\mbox{\footnotesize \(f\)}$ is a fixed function symbol. The solution of the functional equation $\mbox{\footnotesize \(E = 0\)}$ is a function $\mbox{\footnotesize \(f: \mathbb{R} \to \mathbb{R}\)}$ such that $\mbox{\footnotesize \(E = 0\)}$ holds for all values of the variables. E.g. $\mbox{\footnotesize \(f\big(x_1 + f(\sqrt{2} \cdot x_2 \cdot x_2)\big) + (-\pi) + (-1) \cdot x_1 \cdot x_1 \cdot x_2 = 0\)}$ is a functional equation.[/size]

Solve into $N$: $$a^2 = 2^b +c^4$$

Find all pairs of real numbers $(x, y)$ that satisfy the equation $$\frac{(x+y)(2-\sin(x+y))}{4\sin^2(x+y)}=\frac{xy}{x+y}$$

Consider a sequence $P_{1}, P_{2}, \ldots$ of points in the plane such that $P_{1}, P_{2}, P_{3}$ are non-collinear and for every $n \geq 4, P_{n}$ is the midpoint of the line segment joining $P_{n-2}$ and $P_{n-3}$. Let $L$ denote the line segment joining $P_{1}$ and $P_{5}$. Prove the following: [list=a] [*] The area of the triangle formed by the points $P_{n}, P_{n-1}, P_{n-2}$ converges to zero as $n$ goes to infinity. [*] The point $P_{9}$ lies on $L$. [/list]

Let $ABCD$ be a trapezoid with $AB\parallel DC$. Let $M$ be the midpoint of $CD$. If $AD\perp CD, AC\perp BM,$ and $BC\perp BD$, find $\frac{AB}{CD}$. [i]Proposed by Nathan Ramesh

Let $ABC$ be a triangle inscribed in a circle $(O)$, and $M$ be some point on the perpendicular bisector of $BC$. Let $I_1, I_2$ be the incenters of triangles $MAB,MAC$. Prove that the incenters of triangles $A_II_1I_2$ are on a fixed line when $M$ varies on the perpendicular bisector. Trần Quang Hùng