Let b > c > a be positive real number. What is the minimum integral value of $b+\frac{1}{(b-c)a(c-a)}?$

<p>Assuming that</p><p>b – c = p b = c + p</p><p>c – a = q c = a + q</p><p>Hence b = c + p</p><p>= a + q + p</p><p><math><mrow><mo>? </mo></mrow></math>&nbsp;<math><mrow><mi>y</mi><mo>+</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mo stretchy="false"> (</mo><mi>b</mi><mo>−</mo><mi>c</mi><mo stretchy="false">)</mo><mi>a</mi><mo stretchy="false"> (</mo><mi>c</mi><mo>−</mo><mi>a</mi><mo stretchy="false">)</mo></mrow></mfrac></mrow></math>&nbsp;can be written as a + q + p +&nbsp;<math><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mi>a</mi><mo>.</mo><mi>p</mi><mo>.</mo><mi>q</mi></mrow></mfrac></mrow></math></p><p>Apply AM GM in equally concept in above equation</p><p><math><mrow><mfrac><mrow><mi>a</mi><mo>+</mo><mi>q</mi><mo>+</mo><mi>p</mi><mfrac><mrow><mn>1</mn></mrow><mrow><mi>a</mi><mo>.</mo><mi>p</mi><mo>.</mo><mi>q</mi></mrow></mfrac></mrow><mrow><mn>4</mn></mrow></mfrac><mo>≥</mo><mn>4</mn><mroot><mrow><mi>a</mi><mo>×</mo><mi>p</mi><mo>×</mo><mi>q</mi><mo>×</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>a</mi><mo>.</mo><mi>p</mi><mo>.</mo><mi>q</mi></mrow></mfrac></mrow><mrow></mrow></mroot></mrow></math></p><p><math><mrow><mfrac><mrow><mi>a</mi><mo>+</mo><mi>q</mi><mo>+</mo><mi>p</mi><mfrac><mrow><mn>1</mn></mrow><mrow><mi>a</mi><mo>.</mo><mi>p</mi><mo>.</mo><mi>q</mi></mrow></mfrac></mrow><mrow><mn>4</mn></mrow></mfrac><mo>≥</mo><mn>1</mn></mrow></math></p><p><math><mrow><mi>a</mi><mo>+</mo><mi>q</mi><mo>+</mo><mi>p</mi><mfrac><mrow><mn>1</mn></mrow><mrow><mi>a</mi><mo>.</mo><mi>p</mi><mo>.</mo><mi>q</mi></mrow></mfrac><mo>≥</mo><mn>4</mn></mrow></math></p><p>Hence, minimum value is 4</p>

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