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Found problems: 27

2013 Today's Calculation Of Integral, 867
Tags: calculus , derivative , integration , function , quadratic , linear equation , algebra
Express $\int_0^2 f(x)dx$ for any quadratic functions $f(x)$ in terms of $f(0),\ f(1)$ and $f(2).$

1976 Czech and Slovak Olympiad III A, 4
Tags: algebra , system of equations , linear algebra , linear equation
Determine all solutions of the linear system of equations \begin{align*} &x_1& &-x_2& &-x_3& &-\cdots& &-x_n& &= 2a, \\ -&x_1& &+3x_2& &-x_3& &-\cdots& &-x_n& &= 4a, \\ -&x_1& &-x_2& &+7x_3& &-\cdots& &-x_n& &= 8a, \\ &&&&&&&&&&&\vdots \\ -&x_1& &-x_2& &-x_3& &-\cdots& &+\left(2^n-1\right)x_n& &= 2^na, \end{align*} with unknowns $x_1,\ldots,x_n$ and a real parameter $a.$