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Київська міська олімпіада 2025, 2 тур, 9.4

Нехай 𝐻 – точка перетину висот, а точка 𝑂 – центр описаного кола трикутника 𝐴𝐵𝐶. Пряма 𝐴𝐻 вдруге перетинає описане коло ∆𝐴𝐵𝐶 в точці 𝑁. Описане коло ∆𝐵𝑂𝐶 з центром в точці 𝑄 вдруге перетинає пряму 𝑂𝐻 в точці 𝑋. Доведіть, що точки 𝑂,𝑄, 𝑁, 𝑋 лежать на одному колі.

Матвій Курський



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Київська міська олімпіада 2025, 2 тур, 9.4

Нехай 𝐻 – точка перетину висот, а точка 𝑂 – центр описаного кола трикутника 𝐴𝐵𝐶. Пряма 𝐴𝐻 вдруге перетинає описане коло ∆𝐴𝐵𝐶 в точці 𝑁. Описане коло ∆𝐵𝑂𝐶 з центром в точці 𝑄 вдруге перетинає пряму 𝑂𝐻 в точці 𝑋. Доведіть, що точки 𝑂,𝑄, 𝑁, 𝑋 лежать на одному колі.

Матвій Курський

BY Geometry Ukraine




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The channel appears to be part of the broader information war that has developed following Russia's invasion of Ukraine. The Kremlin has paid Russian TikTok influencers to push propaganda, according to a Vice News investigation, while ProPublica found that fake Russian fact check videos had been viewed over a million times on Telegram. Since its launch in 2013, Telegram has grown from a simple messaging app to a broadcast network. Its user base isn’t as vast as WhatsApp’s, and its broadcast platform is a fraction the size of Twitter, but it’s nonetheless showing its use. While Telegram has been embroiled in controversy for much of its life, it has become a vital source of communication during the invasion of Ukraine. But, if all of this is new to you, let us explain, dear friends, what on Earth a Telegram is meant to be, and why you should, or should not, need to care. But Kliuchnikov, the Ukranian now in France, said he will use Signal or WhatsApp for sensitive conversations, but questions around privacy on Telegram do not give him pause when it comes to sharing information about the war. Artem Kliuchnikov and his family fled Ukraine just days before the Russian invasion. What distinguishes the app from competitors is its use of what's known as channels: Public or private feeds of photos and videos that can be set up by one person or an organization. The channels have become popular with on-the-ground journalists, aid workers and Ukrainian President Volodymyr Zelenskyy, who broadcasts on a Telegram channel. The channels can be followed by an unlimited number of people. Unlike Facebook, Twitter and other popular social networks, there is no advertising on Telegram and the flow of information is not driven by an algorithm.
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