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1

Part of 1989 Greece National Olympiad

Problems(3)

periodic wanted, g(x)=f(x+a), f even, g odd - 1989 Greece MO Grade XI p1

Source:

9/7/2024
Consider two functions f, g :R→Rf , \,g \,:\mathbb{R} \to \mathbb{R} such that from some a>0a>0 holds g(x)=f(x+a)g(x)=f(x+a) for any x∈Rx \in \mathbb{R}. If ff is even and gg is odd, prove that both functions are periodic.
algebraperiodic functionperiodic
4x8 system of equations 1989 Greece MO Grade X p1

Source:

9/7/2024
Let a,b,c,dx,y,z,wa,b,c,d x,y,z, w be real numbers such that ax−by−cz−dw=0bx+ay−dz+cw=0cx+dy+az−bw=0dx−cy+bz+aw=0\begin{matrix} ax -by-c z-dw =0\\ b x +a y -d z +cw=0\\ c x+ d y +a z -b w=0\\ dx-c y+bz+aw=0 \end{matrix} prove that a=b=c=d=0,  or  x=y=z=w=0a=b=c=d=0, \ \ or \ \ x=y=z=w=0
algebrasystem of equations
\sqrt{9+x_1}+ \sqrt{9+x_2}+...+ \sqrt{9+x_{100}}=100\sqrt{10}

Source: 1989 Greece MO Grade XII p1

9/6/2024
Find all real solutions of 9+x1+9+x2+...+9+x100=1001016−x1+16−x2+...+16−x100=10015 \begin{matrix} \sqrt{9+x_1}+ \sqrt{9+x_2}+...+ \sqrt{9+x_{100}}=100\sqrt{10}\\ \sqrt{16-x_1}+ \sqrt{16-x_2}+...+ \sqrt{16-x_{100}}=100\sqrt{15} \end{matrix}
algebrasystem of equations