VTMO 2026 - Grade 8 - Final - SET 7 - Self-study

Question 1

Real numbers $a,b$ satisfy $a^3+b^3+6ab=8$. What values can $a+b$ take?

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The units digit of $n^3+3n^2+3n$ is $4$, where $n$ is a positive integer. Find the units digit of $31n^2+n+2021$.

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Real numbers $x,y,z,t$ satisfy $5x+4y+3z+6t=20$ and $x\ge y\ge z\ge t\ge0$. Find the maximum and minimum of $x+y+z+t$.

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Given $a+b+c=3$, $a^2+b^2+c^2=7$ and $a^3+b^3+c^3=15$, calculate $a^4+b^4+c^4$.

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Acute triangle $ABC$ has orthocenter $H$ and $AH=BC$. Find $\angle BAC$.

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