There are 2026 integers greater than 1 written on a blackboard, not necessarily different. In a move, Confucius chooses two integers $m\gt 1$ and $n\gt 1$ from different places on the blackboard and replaces these two integers with
Let $ABC$ be a triangle and let points $M$ and $N$ be the midpoints of sides $AB$ and $AC$, respectively. Let points $K$ and $L$ be chosen strictly inside triangles $BMC$ and $BNC$, respectively, such that $K$ lies strictly inside triangle $ABL$ and $L$ lies strictly inside triangle $AKC$. Suppose that
Let $n$ be a positive integer. Liu Bang and Xiang Yu have a stick of length 1 and want to divide it between themselves. Liu marks at most $n$ points on the stick, and then Xiang marks at most $n$ points on the stick. The marked points are distinct. Then, the stick is cut at all marked points, creating a number of pieces. Afterwards, they take turns claiming any unclaimed piece of the stick, with Liu going first. Each player's goal is to maximise the total length of their own pieces. For each $n$, determine the largest value $c$ such that Liu may guarantee a total length of at least $c$, regardless of Xiang's play.
Shan-Yu and Mulan are playing a game. Let $\theta$ be an angle with $0^\circ\lt \theta\lt 180^\circ$ known to both players. Initially, Shan-Yu makes a paper triangle $T$ with measurements of his choice. Then, they repeatedly perform the following steps:
• If $T$ has at least one angle measuring exactly $\theta$, then the game stops and Mulan wins.
• Otherwise, Mulan chooses a point $P$ on the perimeter of $T$, different from its three vertices. She then makes a straight cut from $P$ to the opposite vertex of $T$, splitting it into two triangles.
• Shan-Yu discards one of the two triangles. The remaining triangle becomes the new $T$.
For which real values of $\theta$ can Mulan guarantee her victory in finitely many steps, no matter how Shan-Yu plays?
Let $a_1,a_2,a_3,\ldots$ be an infinite sequence of positive integers greater than 1. Suppose that for all positive integers $n$, the number $a_{n+1}$ is the smallest positive integer greater than $a_n$ such that $\gcd(a_{n+1},a_i)\gt 1$ for every $i=1,2,\ldots,n$. Prove that there exist positive integers $T$ and $L$ such that
$$a_{n+T}=a_n+L$$
for every positive integer $n$. Here $\gcd(x,y)$ denotes the greatest common divisor of positive integers $x,y$.
A line in the plane is called *sunny* if it is not parallel to any of the $x$-axis, the $y$-axis, or the line $x+y=0$.
Let $n \ge 3$ be a given integer. Determine all nonnegative integers $k$ such that there exist $n$ distinct lines in the plane satisfying both of the following:
• for all positive integers $a$ and $b$ with $a+b\le n+1$, the point $(a,b)$ lies on at least one of the lines; and
Let $\Omega$ and $\Gamma$ be circles with centers $M$ and $N$, respectively, such that the radius of $\Omega$ is less than the radius of $\Gamma$. Suppose $\Omega$ and $\Gamma$ intersect at two distinct points $A$ and $B$. Line $MN$ intersects $\Omega$ at $C$ and $\Gamma$ at $D$, so that $C$, $M$, $N$, $D$ lie on $MN$ in that order. Let $P$ be the circumcenter of triangle $ACD$. Line $AP$ meets $\Omega$ again at $E \neq A$ and meets $\Gamma$ again at $F \neq A$. Let $H$ be the orthocenter of triangle $PMN$.
Prove that the line through $H$ parallel to $AP$ is tangent to the circumcircle of triangle $BEF$. The orthocentre of a triangle is the point of intersection of its altitudes.
A function $f \colon \mathbb{N} \to \mathbb{N}$ is said to be *bonza* if $$
f(a)\quad\text{divides}\quad b^a-f(b)^{f(a)}
$$ for all positive integers $a$ and $b$.
Determine the smallest real constant $c$ such that $f(n) \leq cn$ for all bonza functions $f$ and all positive integers $n$.
An infinite sequence $a_1$, $a_2$, … consists of positive integers each of which has at least three proper divisors. Suppose that for each $n\geq 1$, $a_{n+1}$ is the sum of the three largest proper divisors of $a_n$. Determine all possible values of $a_1$. A proper divisor of a positive integer is a positive divisor other than the integer itself.
Alice and Bazza are playing the *inekoalaty game*, a two‑player game whose rules depend on a positive real number $\lambda$ which is known to both players. On the $n$th turn of the game (starting with $n=1$) the following happens:
• If $n$ is odd, Alice chooses a nonnegative real number $x_n$ such that $$
x_1 + x_2 + \cdots + x_n \le \lambda n.
$$
• If $n$ is even, Bazza chooses a nonnegative real number $x_n$ such that $$
x_1^2 + x_2^2 + \cdots + x_n^2 \le n.
$$
If a player cannot choose a suitable $x_n$, the game ends and the other player wins. If the game goes on forever, neither player wins. All chosen numbers are known to both players.
Determine all values of $\lambda$ for which Alice has a winning strategy and all those for which Bazza has a winning strategy.
Consider a $2025 \times 2025$ grid of unit squares. Matilda wishes to place on the grid some rectangular tiles, possibly of different sizes, such that each side of every tile lies on a grid line and every unit square is covered by at most one tile.
Determine the minimum number of tiles Matilda needs to place so that each row and each column of the grid has exactly one unit square that is not covered by any tile.
Determine all real numbers $\alpha$ such that the number
$$
\lfloor\alpha\rfloor+\lfloor 2 \alpha\rfloor+\cdots+\lfloor n \alpha\rfloor
$$
is a multiple of $n$ for every positive integer $n$. (Here $\lfloor z\rfloor$ denotes the greatest integer less than or equal to $z$.) For example, $\lfloor-\pi\rfloor=-4$ and $\lfloor2\rfloor=\lfloor2.9\rfloor=2$.
Determine all positive integers $a$ and $b$ such that there exists a positive integer $g$ such that $\operatorname{gcd}\left(a^{n}+b, b^{n}+a\right)=g$ for all sufficiently large $n$.
Let $N$ be a positive integer and let $a_{1}, a_{2}, \ldots$ be an infinite sequence of positive integers. Suppose that, for each $n \gt N$, $a_{n}$ is equal to the number of times $a_{n-1}$ appears in the list $a_{1}, a_{2}, \ldots, a_{n-1}$.
Prove that at least one of the sequences $a_{1}, a_{3}, a_{5}, \ldots$ and $a_{2}, a_{4}, a_{6}, \ldots$ is eventually periodic.
Let $ABC$ be a triangle with $AB\lt AC\lt BC$. Let the incentre and incircle of triangle $ABC$ be $I$ and $\omega$, respectively. Let $X$ be the point on line $BC$ different from $C$ such that the line through $X$ parallel to $AC$ is tangent to $\omega$. Similarly, let $Y$ be the point on line $BC$ different from $B$ such that the line through $Y$ parallel to $AB$ is tangent to $\omega$. Let $AI$ intersect the circumcircle of triangle $ABC$ again at $P\ne A$. Let $K$ and $L$ be the midpoints of $AC$ and $AB$, respectively.
Prove that $\angle KIL+\angle YPX=180^\circ$.
Turbo the snail plays a game on a board with 2024 rows and 2023 columns. There are hidden monsters in 2022 of the cells. Initially, Turbo does not know where any of the monsters are, but he knows that there is exactly one monster in each row except the first row and the last row, and that each column contains at most one monster.
Turbo makes a series of attempts to go from the first row to the last row. On each attempt, he chooses to start on any cell in the first row, then repeatedly moves to an adjacent cell sharing a common side. (He is allowed to return to a previously visited cell.) If he reaches a cell with a monster, his attempt ends and he is transported back to the first row to start a new attempt. The monsters do not move, and Turbo remembers whether or not each cell he has visited contains a monster. If he reaches any cell in the last row, his attempt ends and the game is over.
Determine the minimum value of $n$ for which Turbo has a strategy that guarantees reaching the last row on the $n$th attempt or earlier, regardless of the locations of the monsters.
Let $\mathbb{Q}$ be the set of rational numbers. A function $f:\mathbb{Q}\to\mathbb{Q}$ is called aquaesulian if the following property holds: for every $x,y\in\mathbb{Q}$,
$$f(x+f(y))=f(x)+y\quad\text{or}\quad f(f(x)+y)=x+f(y).$$
Show that there exists an integer $c$ such that for any aquaesulian function $f$ there are at most $c$ different rational numbers of the form $f(r)+f(-r)$ for some rational number $r$, and find the smallest possible value of $c$.
Determine all positive, composite integers $n$ that satisfy the following property: if the positive divisors of $n$ are $1=d_{1}\lt d_{2}\lt \cdots\lt d_{k}=n$, then $d_{i}$ divides $d_{i+1}+d_{i+2}$ for every $1 \leqslant i \leqslant k-2$.
Let $ABC$ be an acute-angled triangle with $AB \lt AC$. Denote its circumcircle by $\Omega$ and denote the midpoint of $\operatorname{arc} CAB$ by $S$. Let the perpendicular from $A$ to $BC$ meet $BS$ and $\Omega$ at $D$ and $E \neq A$ respectively. Let the line through $D$ parallel to $BC$ meet line $BE$ at $L$ and denote the circumcircle of triangle $BDL$ by $\omega$. Let $\omega$ meet $\Omega$ again at $P \neq B$.
Prove that the line tangent to $\omega$ at $P$, and line $BS$ intersect on the internal bisector of $\angle BAC$.