id stringclasses 6
values | problem stringclasses 6
values | solution stringclasses 0
values |
|---|---|---|
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 ... | 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 ... | 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 in... | 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 $a_n$... | 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
$$... | 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 satisf... | null |
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