Let $n$ be a positive integer, $n\geq 2$. Let $M=\{0,1,2,\ldots n-1\}$. For an integer nonzero number $a$ we define the function $f_{a}: M\longrightarrow M$, such that $f_{a}(x)$ is the remainder when dividing $ax$ at $n$. Find a necessary and sufficient condition such that $f_{a}$ is bijective. And if $f_{a}$ is bijective and $n$ is a prime number, prove that $a^{n(n-1)}-1$ is divisible by $n^{2}$.

Can a tetrahedron scan turn out to be a triangle with sides $3, 4$ and $5$ (a tetrahedron can be cut only along the edges)?

Is rational or irrational,the number $$\left(\dfrac{2}{\sqrt[3]{25}+\sqrt[3]{15}+\sqrt[3]{9}}+\dfrac{1}{\sqrt[3]{9}+\sqrt[3]{6}+\sqrt[3]{4}}\right) \times \left(\sqrt[3]{25}+\sqrt[3]{10}+\sqrt[3]{4}\right)?$$

Prove that for any polynomial $P$ with real coefficients, and for any positive integer $n$, there exists a polynomial $Q$ with real coefficients such that $P(x)^2 +Q(x)^2$ is divisible by $(1+x^2)^n$.

Let $ABC$ be a non-isosceles and acute triangle. $X$ is a point on arc $BC$ not containing $A$ such that $BA-CA = CX-BX$. The incircle of $\triangle ABC$ touches $AC$ and $AB$ at $E$ and $F$ respectively. The $X$-excircle of $\triangle XBC$ touches $XC$ and $XB$ at $Y$ and $Z$ respectively. Let $T$ be such that $TA$ and $TX$ bisects $\angle BAC$ and $\angle BXC$ respectively. Prove that $T$ lies on the radical axis of circles $(BFZ)$ and $(CEY)$. [i](Proposed by Chuah Jia Herng)[/i]

Given a triangle $ABC$, let $D$ be the midpoint of the side $AC$ and let $M$ be the point that divides the segment $BD$ in the ratio $1/2$; that is, $MB/MD=1/2$. The rays $AM$ and $CM$ meet the sides $BC$ and $AB$ at points $E$ and $F$, respectively. Assume the two rays perpendicular: $AM\perp CM$. Show that the quadrangle $AFED$ is cyclic if and only if the median from $A$ in triangle $ABC$ meets the line $EF$ at a point situated on the circle $ABC$.

Let $ABC$ be a triangle inscribed in the ellipse $\frac{x^2}{4} +\frac{y^2}{9}= 1$. If its centroid is the origin $(0,0)$, find its area.

Determine all positive integers $n$ for which there exists a polynomial $P_n(x)$ of degree $n$ with integer coefficients that is equal to $n$ at $n$ different integer points and that equals zero at zero.

Find the largest integer $n$, where $2009^n$ divides $2008^{2009^{2010}} + 2010^{2009^{2008}}$ .

Right $ \triangle ABC$ has $ AB \equal{} 3$, $ BC \equal{} 4$, and $ AC \equal{} 5$. Square $ XYZW$ is inscribed in $ \triangle ABC$ with $ X$ and $ Y$ on $ \overline{AC}$, $ W$ on $ \overline{AB}$, and $ Z$ on $ \overline{BC}$. What is the side length of the square? [asy]size(200);defaultpen(fontsize(10pt)+linewidth(.8pt)); real s = (60/37); pair A = (0,0); pair C = (5,0); pair B = dir(60)*3; pair W = waypoint(B--A,(1/3)); pair X = foot(W,A,C); pair Y = (X.x + s, X.y); pair Z = (W.x + s, W.y); label("$A$",A,SW); label("$B$",B,NW); label("$C$",C,SE); label("$W$",W,NW); label("$X$",X,S); label("$Y$",Y,S); label("$Z$",Z,NE); draw(A--B--C--cycle); draw(X--W--Z--Y);[/asy] $ \textbf{(A)}\ \frac {3}{2}\qquad \textbf{(B)}\ \frac {60}{37}\qquad \textbf{(C)}\ \frac {12}{7}\qquad \textbf{(D)}\ \frac {23}{13}\qquad \textbf{(E)}\ 2$

Fine all positive integers $m,n\geq 2$, such that (1) $m+1$ is a prime number of type $4k-1$; (2) there is a (positive) prime number $p$ and nonnegative integer $a$, such that \[\frac{m^{2^n-1}-1}{m-1}=m^n+p^a.\]

We define the function $Z(A)$ where we write the digits of $A$ in base $10$ form in reverse. (For example: $Z(521)=125$). Call a number $A$ $good$ if the first and last digits of $A$ are different, none of it's digits are $0$ and the equality: $$Z(A^2)=(Z(A))^2$$ happens. Find all such good numbers greater than $10^6$.\\

The entries in a $ 3\times3$ array include all the digits from 1 through 9, arranged so that the entries in every row and column are in increasing order. How many such arrays are there? $ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 24\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 42\qquad\textbf{(E)}\ 60$

The points $D,E,F$ belong to the sides $(AB), (BC)$ and $(CA)$ of the triangle $ABC$ respectively (but they are not vertices). Let us denote with $d_0, d_1, d_2$, and $d_3$ the maximal side length of the triangles $DEF$, $DEA$, $DBF$, $CEF$, respectively. Prove that $$d_0 \ge \frac{\sqrt3}{2} min\{d_1, d_2, d_3\}$$ When the equality takes place?

Given two sequences $x_n$ and $y_n$ defined by $x_0 = y_0 = 7$, \[x_n = 4x_{n-1}+3y_{n-1}, \text{ and}\]\[y_n = 3y_{n-1}+2x_{n-1},\] find $\lim_{n \to \infty} \frac{x_n}{y_n}$.

In a triangle $ABC$, the angle bisector of $A$ and the cevians $BD$ and $CE$ concur at a point $P$ inside the triangle. Show that the quadrilateral $ADPE$ has an incircle if and only if $AB=AC$.

What is the smallest positive integer $n$ such that $2016n$ is a perfect cube?

Let $\alpha, \beta, \gamma$ be the angles of some triangle. Prove that there is a triangle whose sides are equal to $\sin \alpha$, $\sin \beta$, $\sin \gamma$.

Let $a_1,a_2,a_3,\dots$ be a sequence of numbers such that $a_{n+2} = 2a_n$ for all integers $n.$ Suppose $a_1 = 1,$ $a_2 = 3,$ \[ \sum_{n=1}^{2021} a_{2n} = c, \quad \text{and} \quad \sum\limits_{n=1}^{2021} a_{2n-1} = b. \] Then $c - b + \tfrac{c-b}{b}$ can be written in the form $x^y,$ where $x$ and $y$ are integers such that $x$ is as small as possible. Find $x + y.$

a) Prove that if $x_1,x_2,\ldots,x_n,y_1,y_2,\ldots,y_n$ are positive real numbers satisfying the conditions [list=i] [*] $x_1y_1<x_2y_2<\ldots<x_ny_n$; [*] $x_1+x_2+\ldots+x_k \ge y_1+y_2+ \ldots +y_k,$ for $k=\overline{1,n},$[/list] then $$\frac{1}{x_1}+\frac{1}{x_2}+\ldots+\frac{1}{x_n} \le \frac{1}{y_1}+\frac{1}{y_2}+\ldots+\frac{1}{y_n}.$$ b) Let $A=\{a_1,a_2,\ldots,a_n\}$ be a set of positive integers with the property that for any distinct subsets $B$ and $C$ of $A$ we have $\sum_{x \in B} x \neq \sum_{x \in C} x.$ Prove that $$\frac{1}{a_1}+\frac{1}{a_2}+\ldots+\frac{1}{a_n}<2.$$

The product of positive numbers $x, y$ and $z$ is equal to $1$. Prove that if it holds that $$\frac1x +\frac1y + \frac1z \ge x + y + z,$$ then for any natural $k$, holds the inequality $$\frac{1}{x^k} +\frac{1}{y^k} + \frac{1}{z^k} \ge x^k + y^k + z^k.$$

Answer both (i) and (ii). Test for convergence the series (i) \[ \frac 1{\log (2!)} + \frac 1{\log (3!)} + \frac 1{\log (4!)} + \cdots + \frac 1{\log (n!)} +\cdots\] (ii) \[ \frac 13 + \frac 1{3\sqrt3} + \frac 1{3\sqrt3 \sqrt[3]3} + \cdots + \frac 1{3\sqrt3 \sqrt[3]3 \cdots \sqrt[n]3} + \cdots\]

Let $\mathbb{Z}$ be the set of integers. Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that, for all integers $a$ and $b$, $$f(2a)+2f(b)=f(f(a+b)).$$ [i]Proposed by Liam Baker, South Africa[/i]

A block of wood has the shape of a right circular cylinder with radius $6$ and height $8$, and its entire surface has been painted blue. Points $A$ and $B$ are chosen on the edge on one of the circular faces of the cylinder so that $\overarc{AB}$ on that face measures $120^\circ$. The block is then sliced in half along the plane that passes through point $A$, point $B$, and the center of the cylinder, revealing a flat, unpainted face on each half. The area of one of those unpainted faces is $a\cdot\pi + b\sqrt{c}$, where $a$, $b$, and $c$ are integers and $c$ is not divisible by the square of any prime. Find $a+b+c$. [asy]import three; import solids; size(8cm); currentprojection=orthographic(-1,-5,3); picture lpic, rpic; size(lpic,5cm); draw(lpic,surface(revolution((0,0,0),(-3,3*sqrt(3),0)..(0,6,4)..(3,3*sqrt(3),8),Z,0,120)),gray(0.7),nolight); draw(lpic,surface(revolution((0,0,0),(-3*sqrt(3),-3,8)..(-6,0,4)..(-3*sqrt(3),3,0),Z,0,90)),gray(0.7),nolight); draw(lpic,surface((3,3*sqrt(3),8)..(-6,0,8)..(3,-3*sqrt(3),8)--cycle),gray(0.7),nolight); draw(lpic,(3,-3*sqrt(3),8)..(-6,0,8)..(3,3*sqrt(3),8)); draw(lpic,(-3,3*sqrt(3),0)--(-3,-3*sqrt(3),0),dashed); draw(lpic,(3,3*sqrt(3),8)..(0,6,4)..(-3,3*sqrt(3),0)--(-3,3*sqrt(3),0)..(-3*sqrt(3),3,0)..(-6,0,0),dashed); draw(lpic,(3,3*sqrt(3),8)--(3,-3*sqrt(3),8)..(0,-6,4)..(-3,-3*sqrt(3),0)--(-3,-3*sqrt(3),0)..(-3*sqrt(3),-3,0)..(-6,0,0)); draw(lpic,(6*cos(atan(-1/5)+3.14159),6*sin(atan(-1/5)+3.14159),0)--(6*cos(atan(-1/5)+3.14159),6*sin(atan(-1/5)+3.14159),8)); size(rpic,5cm); draw(rpic,surface(revolution((0,0,0),(3,3*sqrt(3),8)..(0,6,4)..(-3,3*sqrt(3),0),Z,230,360)),gray(0.7),nolight); draw(rpic,surface((-3,3*sqrt(3),0)..(6,0,0)..(-3,-3*sqrt(3),0)--cycle),gray(0.7),nolight); draw(rpic,surface((-3,3*sqrt(3),0)..(0,6,4)..(3,3*sqrt(3),8)--(3,3*sqrt(3),8)--(3,-3*sqrt(3),8)--(3,-3*sqrt(3),8)..(0,-6,4)..(-3,-3*sqrt(3),0)--cycle),white,nolight); draw(rpic,(-3,-3*sqrt(3),0)..(-6*cos(atan(-1/5)+3.14159),-6*sin(atan(-1/5)+3.14159),0)..(6,0,0)); draw(rpic,(-6*cos(atan(-1/5)+3.14159),-6*sin(atan(-1/5)+3.14159),0)..(6,0,0)..(-3,3*sqrt(3),0),dashed); draw(rpic,(3,3*sqrt(3),8)--(3,-3*sqrt(3),8)); draw(rpic,(-3,3*sqrt(3),0)..(0,6,4)..(3,3*sqrt(3),8)--(3,3*sqrt(3),8)..(3*sqrt(3),3,8)..(6,0,8)); draw(rpic,(-3,3*sqrt(3),0)--(-3,-3*sqrt(3),0)..(0,-6,4)..(3,-3*sqrt(3),8)--(3,-3*sqrt(3),8)..(3*sqrt(3),-3,8)..(6,0,8)); draw(rpic,(-6*cos(atan(-1/5)+3.14159),-6*sin(atan(-1/5)+3.14159),0)--(-6*cos(atan(-1/5)+3.14159),-6*sin(atan(-1/5)+3.14159),8)); label(rpic,"$A$",(-3,3*sqrt(3),0),W); label(rpic,"$B$",(-3,-3*sqrt(3),0),W); add(lpic.fit(),(0,0)); add(rpic.fit(),(1,0));[/asy]

Given an integer $a_0$, we define a sequence of real numbers $a_0, a_1, . . .$ using the relation $$a^2_i = 1 + ia^2_{i-1},$$ for $i \ge 1$. An index $j$ is called [i]good [/i] if $a_j$ can be an integer for some $a_0$. Determine the sum of the indices $j$ which lie in the interval $[0, 99]$ and which are not good.