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Difference between revisions of "1975 AHSME Problems/Problem 3"

(Created page with "Which of the following inequalities are satisfied for all real numbers <math>a, b, c, x, y, z</math> which satisfy the conditions <math>x < a, y < b</math>, and <math>z < c</m...")
 
 
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==Problem==
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Which of the following inequalities are satisfied for all real numbers <math>a, b, c, x, y, z</math> which satisfy the conditions <math>x < a, y < b</math>, and <math>z < c</math>?  
 
Which of the following inequalities are satisfied for all real numbers <math>a, b, c, x, y, z</math> which satisfy the conditions <math>x < a, y < b</math>, and <math>z < c</math>?  
  
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Notice if <math>a</math>, <math>b</math>, and <math>c</math> are <math>0</math>, then we can find <math>x</math>, <math>y</math>, and <math>z</math> to disprove <math>\text{I}</math> and <math>\text{II}</math>. For example, if <math>(a, b, c, x, y, z) = (0, 0, 0, -1, -1, -1)</math>, then <math>\text{I}</math> and <math>\text{II}</math> are disproved. If <math>(a, b, c, x, y, z) = (0, 1, 2, -1, -1, 1)</math>, then <math>\text{III}</math> is disproved. Therefore the answer is <math>\boxed{\textbf{(A) } \text{None are satisfied}}</math>.
 
Notice if <math>a</math>, <math>b</math>, and <math>c</math> are <math>0</math>, then we can find <math>x</math>, <math>y</math>, and <math>z</math> to disprove <math>\text{I}</math> and <math>\text{II}</math>. For example, if <math>(a, b, c, x, y, z) = (0, 0, 0, -1, -1, -1)</math>, then <math>\text{I}</math> and <math>\text{II}</math> are disproved. If <math>(a, b, c, x, y, z) = (0, 1, 2, -1, -1, 1)</math>, then <math>\text{III}</math> is disproved. Therefore the answer is <math>\boxed{\textbf{(A) } \text{None are satisfied}}</math>.
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==See Also==
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{{AHSME box|year=1975|num-b=2|num-a=4}}
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{{MAA Notice}}

Latest revision as of 16:51, 19 January 2021

Problem

Which of the following inequalities are satisfied for all real numbers $a, b, c, x, y, z$ which satisfy the conditions $x < a, y < b$, and $z < c$?

$\text{I}. \ xy + yz + zx < ab + bc + ca \\ \text{II}. \ x^2 + y^2 + z^2 < a^2 + b^2 + c^2 \\ \text{III}. \ xyz < abc$

$\textbf{(A)}\ \text{None are satisfied.} \qquad \textbf{(B)}\ \text{I only} \qquad \textbf{(C)}\ \text{II only} \qquad \textbf{(D)}\ \text{III only} \qquad \textbf{(E)}\ \text{All are satisfied.}$


Solution

Solution by e_power_pi_times_i


Notice if $a$, $b$, and $c$ are $0$, then we can find $x$, $y$, and $z$ to disprove $\text{I}$ and $\text{II}$. For example, if $(a, b, c, x, y, z) = (0, 0, 0, -1, -1, -1)$, then $\text{I}$ and $\text{II}$ are disproved. If $(a, b, c, x, y, z) = (0, 1, 2, -1, -1, 1)$, then $\text{III}$ is disproved. Therefore the answer is $\boxed{\textbf{(A) } \text{None are satisfied}}$.

See Also

1975 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 2 		Followed by
Problem 4
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

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