Metadieu, Tercieu and Quartieu are three bodybuilder warriors who fight against an $n$-headed monster. Each of them can attack the monster according to the following rules: (1) Metadieu's attack consists of cutting off half of the monster's heads, then cutting off one more head. If the monster's number of heads is odd, Metadieu cannot attack; (2) Tercieu's attack consists of cutting off a third of the monster's heads, then cutting off two more heads. If the monster's number of heads is not a multiple of 3, Tercieu cannot attack; (3) Quartieu's attack consists of cutting off a quarter of the monster's heads, then cutting off three more heads. If the monster's number of heads is not a multiple of 4, Quartieu cannot attack; If none of the three warriors can attack the monster at some point, then it will devour our three heroes. The objective of the three warriors is to defeat the monster, and for that they need to cut off all its heads, one warrior attacking at a time. For what positive integer values of $n$ is it possible for the three warriors to combine a sequence of attacks in order to defeat the monster?

Decide if the vertices of a regular $30$-gon can be numbered by numbers $1, 2,.., 30$ in such a way that the sum of the numbers of every two neighboring to be a square of a certain natural number.

Let $\varphi(n)$ denotes the number of positive integers less than or equal to $n$ which are relatively prime to $n$. Determine the number of positive integers $2\leq n\leq 50$ such that all coefficients of the polynomial \[ \left(x^{\varphi(n)} - 1\right) - \prod_{\substack{1\leq k\leq n\\\gcd(k,n) = 1}}(x-k) \] are divisible by $n$.

Starting from a positive integer $n$, we can replace the current number with a multiple of the current number or by deleting one or more zeroes from the decimal representation of the current number. Prove that for all values of $n$, it is possible to obtain a single-digit number by applying the above algorithm a finite number of times. There is a nice solution to this...

Two positive whole numbers differ by $3$. The sum of their squares is $117$. Find the larger of the two numbers.

Let $ABC$ be a triangle contained in one of the halfplanes determined by a line $r$. Points $A',B',C'$ are the reflections of $A,B,C$ in $r,$ respectively. Consider the line through $A'$ parallel to $BC$, the line through $B'$ parallel to $AC$ and the line through $C'$ parallel to $AB$. Show that these three lines have a common point.

Elmer the emu takes $44$ equal strides to walk between consecutive telephone poles on a rural road. Oscar the ostrich can cover the same distance in $12$ equal leaps. The telephone poles are evenly spaced, and the $41$st pole along this road is exactly one mile ($5280$ feet) from the first pole. How much longer, in feet, is Oscar's leap than Elmer's stride? $\textbf{(A) }6\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }11\qquad\textbf{(E) }15$

Given a positive integer $n$ and $x_1 \leq x_2 \leq \ldots \leq x_n, y_1 \geq y_2 \geq \ldots \geq y_n$, satisfying \[\displaystyle\sum_{i = 1}^{n} ix_i = \displaystyle\sum_{i = 1}^{n} iy_i\] Show that for any real number $\alpha$, we have \[\displaystyle\sum_{i =1}^{n} x_i[i\alpha] \geq \displaystyle\sum_{i =1}^{n} y_i[i\alpha]\] Here $[\beta]$ denotes the greastest integer not larger than $\beta$.

In an isosceles triangle \(ABC\) with \(AB = AC\), \(BK\) is the altitude and \(H\) is the orthocenter. On the side \(AB\), a point \(N\) is chosen such that \(AN = HN\). Prove that the circumcircles of triangles \(BCK\) and \(ABH\) have a common point on the line \(KN\). [i]Proposed by Fedir Yudin[/i]

Let $(x_n)_{n=1}^\infty$ be a sequence defined recursively with: $x_1=2$ and $x_{n+1}=\frac{x_n(x_n+n)}{n+1}$ for all $n \ge 1$. Prove that $$n(n+1) >\frac{(x_1+x_2+ \ldots +x_n)^2}{x_{n+1}}.$$ [i]Proposed by Nikola Velov[/i]

Let $ABCD$ be a tangential quadrilateral. Let $AB$ meet $CD$ at $E, AD$ intersect $BC$ at $F$. Two arbitrary lines through $E$ meet $AD,BC$ at $M,N, P,Q$ respectively ($M,N \in AD$, $P,Q \in BC$). Another arbitrary pair of lines through $F$ intersect $AB,CD$ at $X, Y,Z, T$ respectively ($X, Y \in AB$,$Z, T \in CD$). Suppose that $d_1, d_2$ are the second tangents from $E$ to the incircles of triangles $FXY, FZT,d_3, d_4$ are the second tangents from $F$ to the incircles of triangles $EMN,EPQ$. Prove that the four lines $d_1, d_2, d_3, d_4$ meet each other at four points and these intersections make a tangential quadrilateral. Nguyễn Văn Linh

Numbers from $1$ to $8$ are placed on the vertices of a cube, one on each of the eight vertices, so that the sum of the numbers on any three vertices of the same face is greater than $9$. Determines the minimum value that the sum of the numbers on one side can have.

$n$ volleyball teams have competed to each other (each $2$ teams have competed exactly $1$ time.). For every $2$ distinct teams like $A,B$, there exist exactly $t$ teams which have lost their match with $A,B$. Prove that $n=4t+3$. (Notabene that in volleyball, there doesn’t exist tie!)

A sphere $S$ with center $O$ and radius $R$ is given. Let $P$ be a fixed point on this sphere. Points $A,B,C$ move on the sphere $S$ such that we have $\angle APB = \angle BPC = \angle CPA = 90^\circ.$ Prove that the plane of triangle $ABC$ passes through a fixed point.

Point $X$ is chosen inside the non trapezoid quadrilateral $ABCD$ such that $\angle AXD +\angle BXC=180$. Suppose the angle bisector of $\angle ABX$ meets the $D$-altitude of triangle $ADX$ in $K$, and the angle bisector of $\angle DCX$ meets the $A$-altitude of triangle $ADX$ in $L$.We know $BK \perp CX$ and $CL \perp BX$. If the circumcenter of $ADX$ is on the line $KL$ prove that $KL \perp AD$. Proposed by [i]Alireza Dadgarnia[/i]

Let $ABC$ be a triangle, and let $D$ be the point where the incircle meets side $BC$. Let $J_b$ and $J_c$ be the incentres of the triangles $ABD$ and $ACD$, respectively. Prove that the circumcentre of the triangle $AJ_bJ_c$ lies on the angle bisector of $\angle BAC$.

Let $(a_i)_{i\in \mathbb{N}}$ be a sequence with $a_1=\frac{3}2$ such that $$a_{n+1}=1+\frac{n}{a_n}$$ Find $n$ such that $2020\le a_n <2021$

Determine all positive integers whose squares end in $196$.

If $f$ is a permutation of $S = \{0, 1,..., 14\}$, then for integers $k \ge 1$, define $$f^k(x) =\underbrace{f(f...(f(x))... ))}_{k\,\,\, applications \,\,\, of \,\,\, f}$$ Compute the number of permutations $f$ of $S$ such that, for some $k \ge 1$, $f^k(x) = (x + 5) \mod \,\,\, 15$ for all $x \in S$.

Determine all positive integers $ n\geq 2$ that satisfy the following condition: for all $ a$ and $ b$ relatively prime to $ n$ we have \[a \equiv b \pmod n\qquad\text{if and only if}\qquad ab\equiv 1 \pmod n.\]

A thin piece of wood of uniform density in the shape of an equilateral triangle with side length $3$ inches weighs $12$ ounces. A second piece of the same type of wood, with the same thickness, also in the shape of an equilateral triangle, has side length of $5$ inches. Which of the following is closest to the weight, in ounces, of the second piece? $\textbf{(A)}\ 14.0\qquad\textbf{(B)}\ 16.0\qquad\textbf{(C)}\ 20.0\qquad\textbf{(D)}\ 33.3\qquad\textbf{(E)}\ 55.6$

Suppose we have some proteins that each protein is a sequence of 7 "AMINO-ACIDS" $A,\ B,\ C,\ H,\ F,\ N$. For example $AFHNNNHAFFC$ is a protein. There are some steps that in each step an amino-acid will change to another one. For example with the step $NA\rightarrow N$ the protein $BANANA$ will cahnge to $BANNA$("in Persian means workman"). We have a set of allowed steps that each protein can change with these steps. For example with the set of steps: $\\ 1)\ AA\longrightarrow A\\ 2)\ AB\longrightarrow BA\\ 3)\ A\longrightarrow \mbox{null}$ Protein $ABBAABA$ will change like this: $\\ ABB\underline{AA}BA\\ \underline{AB}BABA\\ B\underline{AB}ABA\\ BB\underline{AA}BA\\ BB\underline{AB}A\\ BBB\underline{AA}\\ BBB\underline{A}\\ BBB$ You see after finite steps this protein will finish it steps. Set of allowed steps that for them there exist a protein that may have infinitely many steps is dangerous. Which of the following allowed sets are dangerous? a) $NO\longrightarrow OONN$ b) $\left\{\begin{array}{c}HHCC\longrightarrow HCCH\\ CC\longrightarrow CH\end{array}\right.$ c) Design a set of allowed steps that change $\underbrace{AA\dots A}_{n}\longrightarrow\underbrace{BB\dots B}_{2^{n}}$ d) Design a set of allowed steps that change $\underbrace{A\dots A}_{n}\underbrace{B\dots B}_{m}\longrightarrow\underbrace{CC\dots C}_{mn}$ You see from $c$ and $d$ that we acn calculate the functions $F(n)=2^{n}$ and $G(M,N)=mn$ with these steps. Find some other calculatable functions with these steps. (It has some extra mark.)

For any function $f: \mathbb{R}^2\to \mathbb{R}$ consider the function $\Phi_f:\mathbb{R}^2\to [-\infty,\infty]$ for which $\Phi_f(x,y)=\limsup_{ z \to y} f(x,z)$ for any $(x,y) \in \mathbb{R}^2$. [list=a] [*]Is it true that if $f$ is Lebesgue measurable then $\Phi_f$ is also Lebesgue measurable?[/*] [*]Is it true that if $f$ is Borel measurable then $\Phi_f$ is also Borel measurable?[/*] [/list]

The Fibonacci sequence $f_1,f_2,f_3,\dots$ is defined by $f_1=f_2=1$, $f_{n+2}=f_{n+1}+f_n$. Find all $n$ such that $f_n = n^2$.

Let $a$, $b$ be positive integers such that $b^n+n$ is a multiple of $a^n+n$ for all positive integers $n$. Prove that $a=b$. [i]Proposed by Mohsen Jamali, Iran[/i]