There are $n \ge 2$ positive integers written on the whiteboard. A move consists of three steps: calculate the least common multiple $N$ of all numbers then choose any number $a$ and replace $a$ by $N/a$ . Prove that, using a finite number of moves, you can always make all the numbers on the whiteboard equal to $ 1$.

8. You have been kidnapped by a witch and are stuck in the [i]Terrifying Tower[/i], which has an infinite number of floors, starting with floor 1, each initially having 0 boxes. The witch allows you to do the following two things:[list] [*] For a floor $i$, put 2 boxes on floor $i+5$, 6 on floor $i+4$, 13 on floor $i+3$, 12 on floor $i+2$, 8 on floor $i+1$, and 1 on floor $i$, or remove the corresponding number of boxes from each floor if possible. [*] For a floor $i$, put 1 box on floor $i+4$, put 3 boxes on floor $i+3$, 6 on floor $i+2$, 5 on floor $i+1$, and 3 on floor $i$, or remove the corresponding number of boxes from each floor if possible. [/list] At the end, suppose the witch wishes to have exactly $n$ boxes in the tower. Specifically, she wants them to be on the first 10 floors. Let $T(n)$ be the number of distinct distributions of these $n$ boxes that you can make. Find $\displaystyle\sum_{n=1}^{15} T(n)$. [i]Proposed by Monkey_king1[/i]

Given an equilateral triangle $ABC$ and a straight line $\ell$, passing through its center. Intersection points of this line with sides $AB$ and $BC$ are reflected wrt to the midpoints of these sides respectively. Prove that the line passing through the resulting points, touches the inscribed circle triangle $ABC$.

The set $S=\{ \frac{1}{n} \; \vert \; n \in \mathbb{N} \}$ contains arithmetic progressions of various lengths. For instance, $\frac{1}{20}$, $\frac{1}{8}$, $\frac{1}{5}$ is such a progression of length $3$ and common difference $\frac{3}{40}$. Moreover, this is a maximal progression in $S$ since it cannot be extended to the left or the right within $S$ ($\frac{11}{40}$ and $\frac{-1}{40}$ not being members of $S$). Prove that for all $n \in \mathbb{N}$, there exists a maximal arithmetic progression of length $n$ in $S$.

A partition of a positive integer is even if all its elements are even numbers. Similarly, a partition is odd if all its elements are odd. Determine all positive integers $n$ such that the number of even partitions of $n$ is equal to the number of odd partitions of $n$. Remark: A partition of a positive integer $n$ is a non-decreasing sequence of positive integers whose sum of elements equals $n$. For example, $(2; 3; 4), (1; 2; 2; 2; 2)$ and $(9) $ are partitions of $9.$

Let $f: \mathbb{Z}^2\to \mathbb{R}$ be a function. It is known that for any integer $C$ the four functions of $x$ \[f(x,C), f(C,x), f(x,x+C), f(x, C-x)\] are polynomials of degree at most $100$. Prove that $f$ is equal to a polynomial in two variables and find its maximal possible degree. [i]Remark: The degree of a bivariate polynomial $P(x,y)$ is defined as the maximal value of $i+j$ over all monomials $x^iy^j$ appearing in $P$ with a non-zero coefficient.[/i]

Prove that every interior point of a parallelepiped with edges $a,b,c$ is on the distance at most $\frac12 \sqrt{a^2 +b^2 +c^2}$ from some vertex of the parallelepiped.

A cannon fires projectiles on a flat range at a fixed speed but with variable angle. The maximum range of the cannon is $L$. What is the range of the cannon when it fires at an angle $\frac{\pi}{6}$ above the horizontal? Ignore air resistance. (a) $\frac{\sqrt{3}}{2}L$ (b) $\frac{1}{\sqrt{2}}L$ (c) $\frac{1}{\sqrt{3}}L$ (d) $\frac{1}{2}L$ (e) $\frac{1}{3}L$

Find the number of all infinite sequences $a_1$, $a_2$, ... of positive integers such that $a_n+a_{n+1}=2a_{n+2}a_{n+3}+2005$ for all positive integers $n$.

A binoku is a $9 \times 9$ grid that is divided into nine $3 \times 3$ subgrids with the following properties: - each cell contains either a $0$ or a $1$, - each row contains at least one $0$ and at least one $1$, - each column contains at least one $0$ and at least one $1$, and - each of the nine subgrids contains at least one $0$ and at least one $1$. An incomplete binoku is obtained from a binoku by removing the numbers from some of the cells. What is the largest number of empty cells that an incomplete binoku can contain if it can be completed into a binoku in a unique way? [i]Proposed by Stijn Cambie, South Korea[/i]

What is the largest integer $n$ such that one can find $n$ points on the surface of a cube, not all lying on one face and being the vertices of a regular $n$-gon? (A Shapovalov)

Find all positive integers $m$ and $n$ with no common divisor greater than 1 such that $m^3 + n^3$ divides $m^2 + 20mn + n^2$. [i](Professor Yongjin Song)[/i]

Which is the largest four-digit number that has all four of its digits among its divisors and its digits are all different?

Let $ABC$ be a right triangle with $\angle ACB = 90^o$, $BC = 16$, and $AC = 12$. Let the angle bisectors of $\angle BAC$ and $\angle ABC$ intersect $BC$ and $AC$ at $D$ and $E$ respectively. Let $AD$ and $BE$ intersect at $I$, and let the circle centered at $I$ passing through $C$ intersect $AB$ at $P$ and $Q$ such that $AQ < AP$. Compute the area of quadrilateral $DP QE$.

(a) Let $a,x,y$ be positive integers. Prove: if $x\ne y$, the also \[ax+\gcd(a,x)+\text{lcm}(a,x)\ne ay+\gcd(a,y)+\text{lcm}(a,y).\] (b) Show that there are no two positive integers $a$ and $b$ such that \[ab+\gcd(a,b)+\text{lcm}(a,b)=2014.\]

Ana has an iron material of mass $20.2$ kg. She asks Bilyana to make $n$ weights to be used in a classical weighning scale with two plates. Bilyana agrees under the condition that each of the $n$ weights is at least $10$ g. Determine the smallest possible value of $n$ for which Ana would always be able to determine the mass of any material (the mass can be any real number between $0$ and $20.2$ kg) with an error of at most $10$ g.

A ray of light reflects from the rays of a given angle. A ray that enters the vertex of the angle is absorbed. Prove that there is a natural number $n$ such that any ray can reflect at most $n$ times

Let $AA_1$, $BB_1$, $CC_1$ be the altitudes of an acute-angled, nonisosceles triangle $ABC$, and $A_2$, $B_2$, $C_2$ be the touching points of sides $BC$, $CA$, $AB$ with the correspondent excircles. It is known that line $B_1C_1$ touches the incircle of $ABC$. Prove that $A_1$ lies on the circumcircle of $A_2B_2C_2$.

Does there exist on the Cartesian plane a convex $2023$-gon with vertices at integer points, such that the lengths of all its sides are equal? [i]Proposed by Anton Trygub[/i]

A $9\times 7$ rectangle is tiled with tiles of the two types: L-shaped tiles composed by three unit squares (can be rotated repeatedly with $90^\circ$) and square tiles composed by four unit squares. Let $n\ge 0$ be the number of the $2 \times 2 $ tiles which can be used in such a tiling. Find all the values of $n$.

Find all pairs $(p, q)$ of prime numbers such that $$p(p^2 -p - 1) = q(2q + 3).$$

Odd numbers $x,y,z$ such that $gcd(x,y,z)=1$ are given. It turned out that $x^2+y^2+z^2 \vdots x+y+z$ Prove that $x+y+z-2$ is not divisible by $3$

Let $a_1,a_2,\cdots ,a_n$ be real numbers, not all zero. Prove that the equation: \[\sqrt{1+a_1x}+\sqrt{1+a_2x}+\cdots +\sqrt{1+a_nx}=n\] has at most one real nonzero root.

Let $S=\{ 1,2,3,\dots,26 \}.$ Find the number of ways in which $S$ can be partitioned into thirteen subsets such that the following is satisfied: [list] [*]each subset contains two elements of $S,$ and [*]the positive difference between the elements of a subset is $1$ or $13.$ [/list]

[b]4.[/b] We see, in Figures 1 and 2, examples of lock screens from a cellphone that only works with a password that is not typed but drawn with straight line segments. Those segments form a polygonal line with vertices in a lattice. When drawing the pattern that corresponds to a password, the finger can't lose contact with the screen. Every polygonal line corresponds to a sequence of digits and this sequence is, in fact, the password. The tracing of the polygonal obeys the following rules: [i]i.[/i] The tracing starts at some of the detached points which correspond to the digits from $1$ to $9$ (Figure 3). [i]ii.[/i] Each segment of the pattern must have as one of its extremes (on which we end the tracing of the segment) a point that has not been used yet. [i]iii.[/i] If a segment connects two points and contains a third one (its middle point), then the corresponding digit to this third point is included in the password. That does not happen if this point/digit has already been used. [i]iv.[/i] Every password has at least four digits. Thus, every polygonal line is associated to a sequence of four or more digits, which appear in the password in the same order that they are visited. In Figure 1, for instance, the password is 218369, if the first point visited was $2$. Notice how the segment connecting the points associated with $3$ and $9$ includes the points associated to digit $6$. If the first visited point were the $9$, then the password would be $963812$. If the first visited point were the $6$, then the password would be $693812$. In this case, the $6$ would be skipped, because it can't be repeated. On the other side, the polygonal line of Figure 2 is associated to a unique password. Determine the smallest $n (n \geq 4)$ such that, given any subset of $n$ digits from $1$ to $9$, it's possible to elaborate a password that involves exactly those digits in some order.