Σελίδα:   Αρχείο Ασκήσεων Μήνα

\boldsymbol{\color{black} To \,\,\, \pi\alpha\rho\alpha\kappa\acute{\alpha}\tau\omega \,\,\, \kappa\epsilon\acute{\iota}\mu\epsilon\nu o \,\,\, \epsilon\acute{\iota}\nu\alpha\iota \,\,\, \gamma\rho\alpha\mu\acute{\epsilon}\nu o \,\,\, \sigma\epsilon \,\,\, \color{red} LATEX.}

\boldsymbol{\color{Black} 3/2017. \,\,\,\, \color{Blue} \,\,\,\Delta\acute{\iota}\delta\epsilon\tau\alpha\iota\,\,\,\iota\sigma o\sigma\kappa\epsilon\lambda\acute{\epsilon}\varsigma\,\,\,\tau\rho\acute{\iota}\gamma\omega\nu o \,\,\,AB\Gamma\,\,\,\mu\epsilon\,\,\,AB=A\Gamma \,\,\, \kappa\alpha\iota \,\,\, \widehat{BA\Gamma}=20^{\circ}.}

\boldsymbol{\color{Blue}\Theta\epsilon\omega\rho o \acute{\upsilon}\mu\epsilon,\,\,\sigma\eta\mu\epsilon\acute{\iota} o\,\,\, P \,\,\, \tau\eta\varsigma \,\,\, \pi\lambda\epsilon\upsilon\rho\acute{\alpha}\varsigma \,\,\, \tau o \upsilon \,\,\, AB \,\,\, \acute{\epsilon}\tau\sigma\iota \,\,\,\acute{\omega}\sigma\tau\epsilon:\,\,\,\widehat{A\Gamma P}=30^{\circ}\,\,\, \kappa\alpha\iota}

\boldsymbol{\color{Blue}\sigma\eta\mu\epsilon\acute{\iota} o \,\,\,\Sigma\,\,\, \tau\eta\varsigma \,\,\,A\Gamma\,\,\, \acute{\epsilon}\tau\sigma\iota \,\,\,\acute{\omega}\sigma\tau\epsilon:\,\,\,\widehat{AB\Sigma}=20^{\circ}\,\,\, .}

\boldsymbol{\color{Blue} N\alpha \,\,\, \upsilon\pi o \lambda o \gamma\acute{\iota}\sigma\epsilon\tau\epsilon \,\,\, \tau\eta\nu \,\,\, \gamma\omega\nu\acute{\iota}\alpha \,\,\,\widehat{P\Sigma B}}.

\boldsymbol{\color{Black} 4/2016.}

\boldsymbol{\color{Blue} \bullet \,\,\, N\alpha \,\, \delta\epsilon\acute{\iota}\xi\epsilon\tau\epsilon\,\, \acute{o}\tau\iota :\,\,\, K\acute{\alpha}\theta\epsilon\,\, \sigma\upsilon\nu\acute{\alpha}\rho\tau\eta\sigma\eta \quad f\,:\,IR\,\,\rightarrow \,\,IR \quad \gamma\rho\acute{\alpha}\phi\epsilon\tau\alpha\iota \,\, \omega\varsigma}

\boldsymbol{\color{Blue} \acute{\alpha}\theta\rho o \iota\sigma\mu\alpha \,\, \mu\iota\alpha\varsigma\,\,\, \acute{\alpha}\rho\tau\iota\alpha\varsigma \,\, \kappa\alpha\iota \,\, \mu\iota\alpha\varsigma \,\, \pi\epsilon\rho\iota\tau\tau\acute{\eta}\varsigma \,\, \sigma\upsilon\nu\acute{\alpha}\rho\tau\eta\sigma\eta\varsigma.}

\boldsymbol{\color{Blue} \bullet \,\,\,N\alpha \,\, \beta\rho\epsilon\acute{\iota}\tau\epsilon \,\, \sigma\upsilon\nu\acute{\alpha}\rho\tau\eta\sigma\eta \quad f\,: \: IR\,\,\rightarrow \,\,IR \quad \mu\epsilon : \: f(x)f(-x)=4 \quad \gamma\iota\alpha \: \kappa\acute{\alpha}\theta\epsilon \: x \in IR.}

Δείξε τη Λύση

\boldsymbol{\color{Black} 3/2016. \,\,\, \color{blue} \,\,\, \Theta\epsilon\omega\rho o\acute{\upsilon}\mu\epsilon \,\,\, \tau\epsilon\tau\rho\acute{\alpha}\pi\lambda\epsilon\upsilon\rho o \,\,\, AB\Gamma\Delta \,\,\, \mu\epsilon \,\,\, \pi\epsilon\rho\acute{\iota}\mu\epsilon\tau\rho o \,\,\, 2S.}

\boldsymbol{\color{blue} N\alpha \,\,\, \delta\epsilon\acute{\iota}\xi\epsilon\tau\epsilon \,\,\, \acute{o}\tau\iota:}

\boldsymbol{\color{Blue} \bullet \:\:\: (A\Gamma)\: \textless \: S \,\,\,\,\, \kappa\alpha\iota}

\boldsymbol{\color{Blue} \bullet \:\:\: (A\Gamma)^2+(B\Delta)^2 \:\textless\: S[(A\Gamma)+(B\Delta)].}

Δείξε τη Λύση

\boldsymbol{\color{Black} 2/2016.\:\:\:}\boldsymbol{\color{Blue} \Delta\acute{i}\delta o\nu\tau\alpha\iota \:\: o\iota \:\: \pi\rho\alpha\gamma\mu\alpha\tau\iota\kappa o\acute{i} \:\: \alpha\rho\iota\theta\mu o\acute{i} \:\:\: a, x, y \in IR \:\:\: \gamma\iota\alpha \: \tau o \upsilon s \: o\pi o\acute{i} o \upsilon s \: \epsilon\acute{i}\nu\alpha\iota: }

\boldsymbol{\color{Blue}\ 2a-2x+y\geq 0\:\:\:\:\:\:\:\:(1)\:\:\:\:\:\:\:\: \kappa\alpha\iota\:\:\:\:\:\:\:\: 4a-x\leq 0 \,\,\,\,\,\,(2).}

\boldsymbol{\color{Blue} \:\:\: N\alpha \,\,\, \delta\epsilon\acute{i}\xi\epsilon\tau\epsilon \:\: \acute{o}\tau\iota :\:\:\:\:}

\boldsymbol{\color{Blue}\ 6a^2+2x^2-10ax+4ay-xy+3x-2y\leq 0 \:\:\:\:\:\:\:\:\:\:\:\:(3)}.

Δείξε τη Λύση

\boldsymbol{\color{Black} 1/2016 \,\,\, \color{blue} \Sigma\tau o \,\,\, \epsilon\sigma\omega\tau\epsilon\rho\iota\kappa\acute{o} \,\,\, \kappa\upsilon\rho\tau\acute{\eta}\varsigma \,\,\, \gamma\omega\nu\acute{\iota}\alpha\varsigma \,\,\, \widehat{xOy} \,\,\, \theta\epsilon\omega\rho o \acute{\upsilon}\mu\epsilon \,\,\, \sigma\eta\mu\epsilon\acute{\iota}o \,\,\, M. }

\boldsymbol{\color{Blue} A\nu \,\,\, \left( MA \right) \,\,\, \kappa\alpha\iota \,\,\, \left( MB \right) \,\,\, \epsilon\acute{\iota}\nu\alpha\iota \,\,\, \alpha\nu\tau\acute{\iota}\sigma\tau o \iota\chi\alpha \,\,\, o\iota \,\,\, \alpha\pi o\sigma\tau\acute{\alpha}\sigma\epsilon\iota\varsigma \,\,\, \tau o\upsilon \,\,\,\sigma\eta\mu\epsilon\acute{\iota}o\upsilon \,\,\, M }

\boldsymbol{\color{Blue} \,\,\, \alpha\pi o \,\,\, \tau\iota\varsigma \,\,\, \pi\lambda\epsilon\upsilon\rho\acute{\epsilon}\varsigma \,\,\, Ox \,\,\, \kappa\alpha\iota \,\,\, Oy \,\,\, \tau\eta\varsigma \,\,\, \gamma\omega\nu\acute{\iota}\alpha\varsigma \,\,\, \widehat{xOy}, \,\,\, \nu\alpha \,\,\, \beta\rho\epsilon\acute{\iota}\tau\epsilon \,\,\, \tau o\nu \,\,\, \gamma .\,\, \tau\acute{o}\pi o \,\,\, \tau\omega\nu}

\boldsymbol{\color{Blue} \sigma\eta\mu\epsilon\acute{\iota}\omega\nu \,\,\, M \,\,\, \gamma\iota\alpha \,\,\, \tau\alpha \,\,\, o\pi o\acute{\iota}\alpha \,\,\, \epsilon\acute{\iota}\nu\alpha\iota: \,\,\, \frac{(MB)}{(MA)}=\frac{k}{l}.}

\boldsymbol{\color{Blue}}.Δείξε τη Λύση

\boldsymbol{\color{Black} 12/2015.\,\,\,\, \color{blue} \Delta\acute{\iota}\delta\epsilon\tau\alpha\iota \,\,\, \tau\rho\acute{\iota}\gamma\omega\nu o \,\,\, AB\Gamma \,\,\, \kappa\alpha\iota \,\,\, \tau o \,\,\, \acute{\upsilon}\psi o \varsigma \,\,\, \tau o \upsilon \,\,\, A\Delta.}

\boldsymbol{\color{Blue} H \,\, \pi\alpha\rho\acute{\alpha}\lambda\lambda\eta\lambda\eta \,\,\, \alpha\pi o \,\,\, \tau o \,\,\, H \,\,\, \sigma\tau\eta\nu \,\,\, AB \,\,\, \tau\acute{\epsilon}\mu\nu\epsilon\iota \,\,\, \tau o \,\,\, \tau\mu\acute{\eta}\mu\alpha \,\,\, BZ \,\,\, \sigma\tau o \,\,\, \sigma\eta\mu\epsilon\acute{\iota}o \,\,\, \Theta.}

\boldsymbol{\color{Blue} H \,\, \pi\alpha\rho\acute{\alpha}\lambda\lambda\eta\lambda\eta \,\,\, \alpha\pi o \,\,\, \tau o \,\,\, H \,\,\, \sigma\tau\eta\nu \,\,\, A\Gamma \,\,\, \tau\acute{\epsilon}\mu\nu\epsilon\iota \,\,\, \tau o \,\,\, \tau\mu\acute{\eta}\mu\alpha \,\,\, \Gamma E \,\,\, \sigma\tau o \,\,\, \sigma\eta\mu\epsilon\acute{\iota}o \,\,\, \tau o\upsilon \,\,\, I.}

\boldsymbol{\color{Black} 1. \,\,\,\, \color{blue} N\alpha \,\,\, \delta\epsilon\acute{\iota}\xi\epsilon\tau\epsilon \,\,\, \acute{o}\tau\iota \,\,\, \tau o \,\,\, \tau\epsilon\tau\rho\acute{\alpha}\pi\lambda\epsilon\upsilon\rho o \,\,\, H\Theta\Delta I \,\,\, \epsilon\acute{\iota}\nu\alpha\iota \,\,\, \pi\alpha\rho\alpha\lambda\lambda\eta\lambda\acute{o}\gamma\rho\alpha\mu\mu o.}

\boldsymbol{\color{Black} 2. \,\,\,\, \color{blue} \Pi\acute{o}\tau\epsilon \,\,\, \tau o \,\,\, H\Theta\Delta I \,\,\, \epsilon\acute{\iota}\nu\alpha\iota \,\,\, \rho\acute{o}\mu\beta o\varsigma;}

Δείξε τη Λύση