document.write( "Question 1196190: a+b+c=3 and a^2 + b^2 +c^2 =3 solve for a^2022 +b^2022 +c^2022 show proof \n" ); document.write( "

Algebra.Com's Answer #828929 by ikleyn(52781) $\"\"$ $\"About$

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\n" ); document.write( "a+b+c=3 and a^2 + b^2 +c^2 =3 solve for a^2022 +b^2022 +c^2022 show proof
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document.write( "I will start the solution from proving another statement\r\n" );
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document.write( "    |   If 3*(a^2 + b^2 + c^2) = (a+b+c)^2,        |\r\n" );
document.write( "    |   where \"a\", \"b\" and \"c\" are real numbers,   |\r\n" );
document.write( "    |            then a = b = c.                   |\r\n" );
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document.write( "Proof\r\n" );
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document.write( "From 3*(a^2+b^2+c^2) = (a+b+c)^2,  you can easily deduce making FOIL, that\r\n" );
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document.write( "    2a^2 + 2b^2 + 2c^2 = 2ab + 2ac + 2bc.    (1)\r\n" );
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document.write( "Next, you can re-write and transform (1) in this way\r\n" );
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document.write( "    (a^2-2ab+b^2) + (a^2-2ac+c^2) + (b^2-2bc+c^2) = 0,\r\n" );
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document.write( "or, equivalently,\r\n" );
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document.write( "    (a-b)^2 + (a-c)^2 + (b-c)^2 = 0.          (2)\r\n" );
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document.write( "But the squares of real numbers are always non-negative, and the sum of non-negative numbers \r\n" );
document.write( "can be equal to 0 (zero) if and only if each of these non-negative numbers is zero.\r\n" );
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document.write( "So, we conclude from (2) that  a = b = c,  and the statement is PROVED.\r\n" );
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document.write( "    |   Now, let's apply this statement to our case.   |\r\n" );
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document.write( "We are given that a+b+c = 3  and  a^2+b^2+c^2 = 3.\r\n" );
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document.write( "So, in our case  3*(a^2 + b^2 + c^2) = 3*3 = 9 = 3^2 = (a + b + c)^2.\r\n" );
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document.write( "Thus we can apply this theorem, proved above, to conclude that a = b = c.\r\n" );
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document.write( "Next, since a+b+c = 3, it means that  a = b = c = 1.\r\n" );
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document.write( "And finally, we obtain  a^2022 + b^2022 + c^2022 = 1^2022 + 1^2022 + 1^2022 = 1 + 1 + 1 = 3.\r\n" );
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document.write( "ANSWER.  Under given conditions,  a^2022 + b^2022 + c^2022 = 3.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Solved.\r
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\n" ); document.write( "\n" ); document.write( "It is well known problem of the school Math circle level.\r
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\n" ); document.write( "\n" ); document.write( "I corrected my typo which was noticed by tutor @math_tutor2020.\r
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\n" ); document.write( "\n" ); document.write( "Thanks for noticing it !\r
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\n" ); document.write( "\n" ); document.write( "Now you see the corrected version.\r
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