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# Number Theory

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Let  $$a,b,c,d$$   be nonnegative real numbers such that $$a + b + c + d = 1$$ Find the maximum value of

$$a^2 + b^2 + c^2 + d^2$$

Jul 31, 2019

#1
+6045
+3

The maximum, 1, will occur when one of the variable is 1 and the rest 0

$$(a+b+c+d-1)=0\\ (a+b+c+d-1)^2=0\\ a^2+b^2+c^2+d^2 - 2(a+b+c+d)+2(ab+ac+ad+bc+bd+cd)+1 = 0\\ a^2+b^2+c^2+d^2+2(ab+ac+ad+bc+bd+cd)= 1\\ \text{It should be clear that to maximize the sum of the squares we want the second set of terms to be 0}\\ \text{This occurs, as noted above when only 1 element is non-zero}$$

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Jul 31, 2019
#2
+133
+2

Sorry for thanking you late

Aug 3, 2019