Подошел к концу третий день турнира Колмогорова, и мы, как обычно, радуем вас задачами с него 💥
Задача. На сторонах 𝐵𝐶, 𝐶𝐴 и 𝐴𝐵 треугольника 𝐴𝐵𝐶 выбраны точки 𝐴₁, 𝐵₁ и 𝐶₁ соответственно. Четырёхугольники 𝐴𝐵₁𝐴₁𝐶₁, 𝐵𝐶₁𝐵₁𝐴₁ и 𝐶𝐴₁𝐶₁𝐵₁ описаны около окружностей с центрами 𝐼𝑎, 𝐼𝑏 и 𝐼𝑐 соответственно. Докажите, что площади треугольников 𝐴₁𝐵₁𝐶₁ и 𝐼𝑎𝐼𝑏𝐼𝑐 отличаются в четыре раза.
Подошел к концу третий день турнира Колмогорова, и мы, как обычно, радуем вас задачами с него 💥
Задача. На сторонах 𝐵𝐶, 𝐶𝐴 и 𝐴𝐵 треугольника 𝐴𝐵𝐶 выбраны точки 𝐴₁, 𝐵₁ и 𝐶₁ соответственно. Четырёхугольники 𝐴𝐵₁𝐴₁𝐶₁, 𝐵𝐶₁𝐵₁𝐴₁ и 𝐶𝐴₁𝐶₁𝐵₁ описаны около окружностей с центрами 𝐼𝑎, 𝐼𝑏 и 𝐼𝑐 соответственно. Докажите, что площади треугольников 𝐴₁𝐵₁𝐶₁ и 𝐼𝑎𝐼𝑏𝐼𝑐 отличаются в четыре раза.
In February 2014, the Ukrainian people ousted pro-Russian president Viktor Yanukovych, prompting Russia to invade and annex the Crimean peninsula. By the start of April, Pavel Durov had given his notice, with TechCrunch saying at the time that the CEO had resisted pressure to suppress pages criticizing the Russian government. The last couple days have exemplified that uncertainty. On Thursday, news emerged that talks in Turkey between the Russia and Ukraine yielded no positive result. But on Friday, Reuters reported that Russian President Vladimir Putin said there had been some “positive shifts” in talks between the two sides. 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. Just days after Russia invaded Ukraine, Durov wrote that Telegram was "increasingly becoming a source of unverified information," and he worried about the app being used to "incite ethnic hatred." Some people used the platform to organize ahead of the storming of the U.S. Capitol in January 2021, and last month Senator Mark Warner sent a letter to Durov urging him to curb Russian information operations on Telegram.
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