Problema (nível 4): Quantas circunferências distintas passam por pelo menos três vértices do quadriculado seguinte? (image1)


Conteúdo referenciado:
[image1]
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Resolução: Solução 1: Cada três pontos não colineares determinam uma circunferência. Há $\dfrac{9\times 8\times 7}{3\times 2\times 1}=84$ conjuntos de três pontos, dos quais $8$ são constituídos por pontos colineares ($3$ verticais, $3$ horizontais e $2$ diagonais). Se uma circunferência passa por quatro pontos, então os quatro subconjuntos de três pontos determinam a mesma circunferência. Há $14$ circunferências que passam por quatro pontos: Assim, às $84-8=76$ circunferências têm que ser retiradas $3\times 14=42$ circunferências repetidas, pelo que há $76-42=34$ circunferências distintas. Solução 2: O centro da circunferência pode estar no ponto central, num dos eixos de simetria ou fora dos eixos de simetria do quadriculado. Há $2$ circunferências cujo centro está no ponto central do quadriculado: (image2) (image3) Há $24$ circunferências cujo centro está num dos eixos de simetria do quadriculado, mas não no ponto central (as $6$ indicadas a seguir e as $6\times 3$ que se obtêm por rotação): (image4) (image5) (image6) (image7) (image8) (image9) Há $8$ circunferências cujo centro está fora dos eixos de simetria do quadriculado (a indicada a seguir e as $7$ que se obtêm por rotações e reflexões): (image10) Logo, ao todo há $2+24+8=34$ circunferências distintas. Solução 1: Cada três pontos não colineares determinam uma circunferência. Há $\dfrac{9\times 8\times 7}{3\times 2\times 1}=84$ conjuntos de três pontos, dos quais $8$ são constituídos por pontos colineares ($3$ verticais, $3$ horizontais e $2$ diagonais). Se uma circunferência passa por quatro pontos, então os quatro subconjuntos de três pontos determinam a mesma circunferência. Há $14$ circunferências que passam por quatro pontos: Assim, às $84-8=76$ circunferências têm que ser retiradas $3\times 14=42$ circunferências repetidas, pelo que há $76-42=34$ circunferências distintas. Solução 2: O centro da circunferência pode estar no ponto central, num dos eixos de simetria ou fora dos eixos de simetria do quadriculado. Há $2$ circunferências cujo centro está no ponto central do quadriculado: (image11) (image12) Há $24$ circunferências cujo centro está num dos eixos de simetria do quadriculado, mas não no ponto central (as $6$ indicadas a seguir e as $6\times 3$ que se obtêm por rotação): (image13) (image14) (image15) (image16) (image17) (image18) Há $8$ circunferências cujo centro está fora dos eixos de simetria do quadriculado (a indicada a seguir e as $7$ que se obtêm por rotações e reflexões): (image19) Logo, ao todo há $2+24+8=34$ circunferências distintas. $9$ que passam por dois pontos numa linha horizontal e por dois pontos noutra linha horizontal (image20) (image21) (image22) Solução 1: Cada três pontos não colineares determinam uma circunferência. Há $\dfrac{9\times 8\times 7}{3\times 2\times 1}=84$ conjuntos de três pontos, dos quais $8$ são constituídos por pontos colineares ($3$ verticais, $3$ horizontais e $2$ diagonais). Se uma circunferência passa por quatro pontos, então os quatro subconjuntos de três pontos determinam a mesma circunferência. Há $14$ circunferências que passam por quatro pontos: Assim, às $84-8=76$ circunferências têm que ser retiradas $3\times 14=42$ circunferências repetidas, pelo que há $76-42=34$ circunferências distintas. Solução 2: O centro da circunferência pode estar no ponto central, num dos eixos de simetria ou fora dos eixos de simetria do quadriculado. Há $2$ circunferências cujo centro está no ponto central do quadriculado: (image23) (image24) Há $24$ circunferências cujo centro está num dos eixos de simetria do quadriculado, mas não no ponto central (as $6$ indicadas a seguir e as $6\times 3$ que se obtêm por rotação): (image25) (image26) (image27) (image28) (image29) (image30) Há $8$ circunferências cujo centro está fora dos eixos de simetria do quadriculado (a indicada a seguir e as $7$ que se obtêm por rotações e reflexões): (image31) Logo, ao todo há $2+24+8=34$ circunferências distintas. $5$ que passam por dois pontos numa linha horizontal e por um ponto em cada uma das outras linhas (image32) (image33)


Conteúdo referenciado:
[image2]
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[image32]
\begin{pspicture}(0,-1.5)(2,2)  \pspolygon(0,0)(2,0)(2,2)(0,2) \psline(1,0)(1,2) \psline(0,1)(2,1) \pscircle(0.5,0.5){1.581} \rput(0,0){\tiny $\bullet$} \rput(1,0){\tiny $\bullet$} \rput(2,0){\tiny $\bullet$} \rput(0,1){\tiny $\bullet$} \rput(1,1){\tiny $\bullet$} \rput(2,1){\tiny $\bullet$} \rput(0,2){\tiny $\bullet$} \rput(1,2){\tiny $\bullet$} \rput(2,2){\tiny $\bullet$} \rput(1,-2){\footnotesize e as três obtidas por rotação} \end{pspicture}
[image33]
\begin{pspicture}(0,-1.5)(2,2)  \pspolygon(0,0)(2,0)(2,2)(0,2) \psline(1,0)(1,2) \psline(0,1)(2,1) \pscircle(1,1){1} \rput(0,0){\tiny $\bullet$} \rput(1,0){\tiny $\bullet$} \rput(2,0){\tiny $\bullet$} \rput(0,1){\tiny $\bullet$} \rput(1,1){\tiny $\bullet$} \rput(2,1){\tiny $\bullet$} \rput(0,2){\tiny $\bullet$} \rput(1,2){\tiny $\bullet$} \rput(2,2){\tiny $\bullet$} \end{pspicture}