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-{"url": "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, or 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(i) For all positive integers $a$ and $b$ with $a+b\\leq n+1$, the point $(a,b)$ lies on at least one of the lines; and\n\n(ii) Exactly $k$ of the $n$ lines are sunny.", "answer": "N/A"}
-{"url": "2025-imo-p2", "problem": "Find all functions $f: \\mathbb{R} \\to \\mathbb{R}$ such that\n\n$$f(x^3 + y^3) = f(x + y)(x^2 - xy + y^2)$$\n\nfor all real numbers $x$ and $y$.", "answer": "N/A"}
-{"url": "2025-imo-p3", "problem": "Find all positive integers $n$ such that there exist positive integers $a_1, a_2, \\ldots, a_n$ with the property that for any positive integers $b_1, b_2, \\ldots, b_n$, there exists an integer $k$ such that\n\n$$a_1b_1 + a_2b_2 + \\cdots + a_nb_n + k$$\n\nis a perfect square.", "answer": "N/A"}
-{"url": "2025-imo-p4", "problem": "Penguins Ella, Distin, and Glara play a game on an infinite chessboard. Ella starts by placing a white stone at the origin $(0,0)$. On Distin's turn, Distin must move one stone from a cell $(a,b)$ to an adjacent cell sharing a side (possible moves: $(a\\pm 1, b)$ or $(a, b\\pm 1)$). After each of Distin's moves, if there is no stone at $(0,0)$, Glara immediately places a white stone at $(0,0)$. Otherwise, Glara places no stone. The game continues indefinitely: Distin moves on odd turns (turn $1, 3, 5, \\ldots$) and Glara places on even turns (turn $2, 4, 6, \\ldots$) if the conditions are met.\n\nShow that for each positive integer $d$, there is a moment after which there is always at least one white stone among the cells with Manhattan distance $d$ from the origin. (The Manhattan distance from $(0,0)$ to $(x,y)$ is $|x| + |y|$.)", "answer": "N/A"}
-{"url": "2025-imo-p5", "problem": "A finite set $S$ of positive integers is called *primitive* if for every $s \\in S$, none of the proper divisors of $s$ belong to $S$.\n\nDetermine all positive integers $n$ for which there exists a primitive set $S$ with $|S| = n$ such that\n\n$$\\sum_{s \\in S} \\frac{1}{s} = 1.$$", "answer": "N/A"}
-{"url": "2025-imo-p6", "problem": "Let $ABC$ be a triangle with incenter $I$ and incircle $\\omega$. Let $X$ be a point on the arc $BC$ of the circumcircle of triangle $ABC$ not containing $A$. Lines $XA$, $XB$, $XC$ meet $\\omega$ again at $D$, $E$, $F$ respectively. Prove that the circumcenter $O$ of triangle $DEF$ and the orthocenter $H$ of triangle $ABC$ are collinear with $I$.", "answer": "N/A"}