Olympiad Maths Prep

Track / Stage 5 / 65 of 400 #665 of 2000

Problem 665

AIME late
Algebra Difficulty 5.2 Find the answer

Test Question D: The sum of four positive numbers is 4, and the sum of their squares is 8. Determine the maximum value of the largest of these four numbers.

Restate it as: Let a⩾b⩾c⩾d>0 a \geqslant b \geqslant c \geqslant d > 0 , and satisfy
a+b+c+d=4,a2+b2+c2+d2=8, a+b+c+d=4, \quad a^{2}+b^{2}+c^{2}+d^{2}=8,

Find max⁡a=\max a= ?

Official solution

max⁡a=u+(n−1)(nv−u2)n=4+(4−1)(4×8−16)4=1+3. \begin{aligned} \max a & =\frac{u+\sqrt{(n-1)\left(n v-u^{2}\right)}}{n} \\ & =\frac{4+\sqrt{(4-1)(4 \times 8-16)}}{4} \\ & =1+\sqrt{3} . \end{aligned}

In the above problem, taking n=4,u=4,v=8n=4, u=4, v=8, we have
max⁡a=u+(n−1)(nv−u2)n=4+(4−1)(4×8−16)4=1+3. \begin{aligned} \max a & =\frac{u+\sqrt{(n-1)\left(n v-u^{2}\right)}}{n} \\ & =\frac{4+\sqrt{(4-1)(4 \times 8-16)}}{4} \\ & =1+\sqrt{3} . \end{aligned}

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.