| [{"id": "2025-imo-p1", "problem": "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 $x + y = 0$.\n\nLet $n \\geq 3$ be a given integer. Determine all nonnegative integers $k$ such that there exist $n$ distinct lines in the plane satisfying both of the following:\n\n* 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\n* exactly $k$ of the $n$ lines are sunny.\n", "solution": null}, {"id": "2025-imo-p2", "problem": "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 $PMN$.\n\nProve that the line through $H$ parallel to $AP$ is tangent to the circumcircle of triangle $BEF$.\n\n(The *orthocentre* of a triangle is the point of intersection of its altitudes.)\n", "solution": null}, {"id": "2025-imo-p3", "problem": "Let $\\mathbb{N}$ denote the set of positive integers. A function $f: \\mathbb{N} \\to \\mathbb{N}$ is said to be *bonza* if\n\n$$f(a) \\text{ divides } b^a - f(b)^{f(a)}$$\n\nfor all positive integers $a$ and $b$.\n\nDetermine the smallest real constant $c$ such that $f(n) \\leq cn$ for all bonza functions $f$ and all positive integers $n$.\n", "solution": null}, {"id": "2025-imo-p4", "problem": "A *proper divisor* of a positive integer $N$ is a positive divisor of $N$ other than $N$ itself.\n\nThe infinite sequence $a_1, a_2, \\ldots$ consists of positive integers, each of which has at least three proper divisors.\n\nFor each $n \\geq 1$, the integer $a_{n+1}$ is the sum of the three largest proper divisors of $a_n$.\n\nDetermine all possible values of $a_1$.\n", "solution": null}, {"id": "2025-imo-p5", "problem": "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:\n\n- If $n$ is odd, Alice chooses a nonnegative real number $x_n$ such that\n\n$$x_1 + x_2 + \\cdots + x_n \\leq \\lambda n.$$\n\n- If $n$ is even, Bazza chooses a nonnegative real number $x_n$ such that\n\n$$x_1^2 + x_2^2 + \\cdots + x_n^2 \\leq n.$$\n\nIf 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.\n\nDetermine all values of $\\lambda$ for which Alice has a winning strategy and all those for which Bazza has a winning strategy.\n", "solution": null}, {"id": "2025-imo-p6", "problem": "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.\n\nDetermine the minimum number of tiles Matilda needs to place to satisfy these conditions.", "solution": null}] |