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"value": "\nProblem 2. Let $n$ three-digit numbers satisfy the following properties:\n\n(1) No number contains the digit 0 .\n\n(2) The sum of the digits of each number is 9 .\n\n(3) The units digits of any two numbers are different.\n\n(4) The tens digits of any two numbers are different.... |
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"value": "## Problem 2\n\nLet the circles $k_{1}$ and $k_{2}$ intersect at two distinct points $A$ and $B$, and let $t$ be a common tangent of $k_{1}$ and $k_{2}$, that touches $k_{1}$ and $k_{2}$ at $M$ and $N$, respectively. If $t \\perp A M$ and $M N=2 A M$, evaluate $\\angle N M B$.\n\... |
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"value": "\nProblem 3. Let $A L$ and $B K$ be angle bisectors in the non-isosceles triangle $A B C$ ( $L$ lies on the side $B C, K$ lies on the side $A C$ ). The perpendicular bisector of $B K$ intersects the line $A L$ at point $M$. Point $N$ lies on the line $B K$ such that $L N$ is para... |
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"value": "\nProblem 4. Let $M$ be a subset of the set of 2021 integers $\\{1,2,3, \\ldots, 2021\\}$ such that for any three elements (not necessarily distinct) $a, b, c$ of $M$ we have $|a+b-c|>10$. Determine the largest possible number of elements of $M$.\n"
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"value": "\nProblem 1. Find all distinct prime numbers $p, q$ and $r$ such that\n\n$$\n3 p^{4}-5 q^{4}-4 r^{2}=26\n$$\n"
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"value": "<think>\nOkay, let's see. I need to find distinct prime numbers p, q, and r such that 3p⁴ - 5q⁴ - 4r² = 26. Hmm, primes a... |
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"value": "\nProblem 2. Consider an acute triangle $A B C$ with area S. Let $C D \\perp A B \\quad(D \\in A B)$, $D M \\perp A C \\quad(M \\in A C)$ and $\\quad D N \\perp B C \\quad(N \\in B C)$. Denote by $H_{1}$ and $H_{2}$ the orthocentres of the triangles $M N C$ and $M N D$ respective... |
[{"from":"user","value":"\nProblem 4. For a positive integer $n$, two players A and B play the follo(...TRUNCATED) |
[{"from":"user","value":"\nProblem 1. Find all pairs $(a, b)$ of positive integers such that\n\n$$\n(...TRUNCATED) |
[{"from":"user","value":"## Problem 1.\n\nFind all pairs $(a, b)$ of positive integers such that $a!(...TRUNCATED) |
[{"from":"user","value":"\nProblem 4. Find all pairs $(p, q)$ of prime numbers such that\n\n$$\n1+\\(...TRUNCATED) |
End of preview. Expand in Data Studio
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