id
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stringclasses 6
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|---|---|---|
2025-imo-p1
|
A line in the plane is called *sunny* if it is **not** parallel to any of the $x$-axis, the $y$-axis, and the line $x + y = 0$.
Let $n \geq 3$ 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 $a + b \leq n + 1$, the point $(a, b)$ is on at least one of the lines; and
* exactly $k$ of the $n$ lines are sunny.
| null |
2025-imo-p2
|
Let $\Omega$ and $\Gamma$ be circles with centres $M$ and $N$, respectively, such that the radius of $\Omega$ is less than the radius of $\Gamma$. Suppose circles $\Omega$ and $\Gamma$ intersect at two distinct points $A$ and $B$. Line $MN$ intersects $\Omega$ at $C$ and $\Gamma$ at $D$, such that points $C, M, N$ and $D$ lie on the line in that order. Let $P$ be the circumcentre of triangle $ACD$. Line $AP$ intersects $\Omega$ again at $E \neq A$. Line $AP$ intersects $\Gamma$ again at $F \neq A$. Let $H$ be the orthocentre 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.)
| null |
2025-imo-p3
|
Let $\mathbb{N}$ denote the set of positive integers. A function $f: \mathbb{N} \to \mathbb{N}$ is said to be *bonza* if
$$f(a) \text{ divides } b^a - f(b)^{f(a)}$$
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 $n$.
| null |
2025-imo-p4
|
A *proper divisor* of a positive integer $N$ is a positive divisor of $N$ other than $N$ itself.
The infinite sequence $a_1, a_2, \ldots$ consists of positive integers, each of which has at least three proper divisors.
For each $n \geq 1$, the integer $a_{n+1}$ is the sum of the three largest proper divisors of $a_n$.
Determine all possible values of $a_1$.
| null |
2025-imo-p5
|
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 $n^{\text{th}}$ 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
$$x_1 + x_2 + \cdots + x_n \leq \lambda n.$$
- If $n$ is even, Bazza chooses a nonnegative real number $x_n$ such that
$$x_1^2 + x_2^2 + \cdots + x_n^2 \leq n.$$
If a player cannot choose a suitable number $x_n$, 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.
| null |
2025-imo-p6
|
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 to satisfy these conditions.
| null |
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