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As a UK resident for tax purposes, you are taxed on your worldwide income and gains.
You should declare income you get from your family business in Slovakia and also dividends you receive. Although both these dividends and income are taxed at source in that country, they should be reported on your tax return here by completing supplementary pages SA106 - Foreign to the main tax return.
There is a double tax agreement between the Slovak Republic and the UK whereby you are not taxed on the same income twice.
More information on this can be found here
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You claim foreign tax credit relief against any tax suffered on the income and investment income in Slovak Republic.
The money your family gave you to help you buy a car is not taxable and you don't report it on your tax return.
This is deemed a gift and gifts are not taxable in the UK.
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