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{"problem_id": "apex_2025_3-part11", "original_problem_id": "apex_2025_3", "problem": "Let $\\triangle A B C$ be an acute triangle with circumcircle $\\omega_{1}$, and let $D$ be a point on segment $B C$. Circle $\\omega_{2}$ is tangent to segment $A D$, segment $B D$, and $\\omega_{1}$. Circle $\\omega_{3}$ is tangent... |
{"problem_id": "apex_2025_4-part3", "original_problem_id": "apex_2025_4", "problem": "We call a $n \\times n$ table filled with positive integers \\emph{divisoral} if it holds that:\n\\begin{itemize}\n\\item numbers in $i$-th row are exactly all divisors of some positive integer $r_i$,\n\\item numbers in $j$-th column ... |
{"problem_id": "apex_2025_4-part4", "original_problem_id": "apex_2025_4", "problem": "We call a $n \\times n$ table filled with positive integers \\emph{divisoral} if it holds that:\n\\begin{itemize}\n\\item numbers in $i$-th row are exactly all divisors of some positive integer $r_i$,\n\\item numbers in $j$-th column ... |
{"problem_id": "apex_2025_4-part5", "original_problem_id": "apex_2025_4", "problem": "We call a $n \\times n$ table filled with positive integers \\emph{divisoral} if it holds that:\n\\begin{itemize}\n\\item numbers in $i$-th row are exactly all divisors of some positive integer $r_i$,\n\\item numbers in $j$-th column ... |
{"problem_id": "apex_2025_4-part6", "original_problem_id": "apex_2025_4", "problem": "We call a $n \\times n$ table filled with positive integers \\emph{divisoral} if it holds that:\n\\begin{itemize}\n\\item numbers in $i$-th row are exactly all divisors of some positive integer $r_i$,\n\\item numbers in $j$-th column ... |
{"problem_id": "apex_2025_4-part7", "original_problem_id": "apex_2025_4", "problem": "We call a $n \\times n$ table filled with positive integers \\emph{divisoral} if it holds that:\n\\begin{itemize}\n\\item numbers in $i$-th row are exactly all divisors of some positive integer $r_i$,\n\\item numbers in $j$-th column ... |
{"problem_id": "apex_2025_4-part8", "original_problem_id": "apex_2025_4", "problem": "We call a $n \\times n$ table filled with positive integers \\emph{divisoral} if it holds that:\n\\begin{itemize}\n\\item numbers in $i$-th row are exactly all divisors of some positive integer $r_i$,\n\\item numbers in $j$-th column ... |
{"problem_id": "apex_2025_4-part9", "original_problem_id": "apex_2025_4", "problem": "We call a $n \\times n$ table filled with positive integers \\emph{divisoral} if it holds that:\n\\begin{itemize}\n\\item numbers in $i$-th row are exactly all divisors of some positive integer $r_i$,\n\\item numbers in $j$-th column ... |
{"problem_id": "apex_2025_4-part10", "original_problem_id": "apex_2025_4", "problem": "We call a $n \\times n$ table filled with positive integers \\emph{divisoral} if it holds that:\n\\begin{itemize}\n\\item numbers in $i$-th row are exactly all divisors of some positive integer $r_i$,\n\\item numbers in $j$-th column... |
{"problem_id": "apex_2025_4-part11", "original_problem_id": "apex_2025_4", "problem": "We call a $n \\times n$ table filled with positive integers \\emph{divisoral} if it holds that:\n\\begin{itemize}\n\\item numbers in $i$-th row are exactly all divisors of some positive integer $r_i$,\n\\item numbers in $j$-th column... |
{"problem_id": "apex_2025_4-part12", "original_problem_id": "apex_2025_4", "problem": "We call a $n \\times n$ table filled with positive integers \\emph{divisoral} if it holds that:\n\\begin{itemize}\n\\item numbers in $i$-th row are exactly all divisors of some positive integer $r_i$,\n\\item numbers in $j$-th column... |
{"problem_id": "apex_2025_4-part13", "original_problem_id": "apex_2025_4", "problem": "We call a $n \\times n$ table filled with positive integers \\emph{divisoral} if it holds that:\n\\begin{itemize}\n\\item numbers in $i$-th row are exactly all divisors of some positive integer $r_i$,\n\\item numbers in $j$-th column... |
{"problem_id": "apex_2025_4-part14", "original_problem_id": "apex_2025_4", "problem": "We call a $n \\times n$ table filled with positive integers \\emph{divisoral} if it holds that:\n\\begin{itemize}\n\\item numbers in $i$-th row are exactly all divisors of some positive integer $r_i$,\n\\item numbers in $j$-th column... |
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