solid geometry - Triangle Inequality

SOURCE:COMPETITION
Number of Problems: 5. : (Book)

1.

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The sides of a triangle with positive area have lengths $4$ , $6$ , and $x$ . The sides of a second triangle with positive area have lengths $4$ , $6$ , and $y$ . What is the smallest positive number that is not a possible value of $|x-y|$ ?

$mathrm{(A)} 2 qquadmathrm{(B)} 4 qquadmathrm{(C)} 6 qquadmathrm{(D)} 8 qquadmathrm{(E)} 10$

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2.

How many non-congruent triangles with perimeter $7$ have integer side lengths?

$mathrm{(A) } 1qquad mathrm{(B) } 2qquad mathrm{(C) } 3qquad mathrm{(D) } 4qquad mathrm{(E) } 5$

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How many non-congruent triangles with perimeter $7$ have integer side lengths?

$mathrm{(A) } 1qquad mathrm{(B) } 2qquad mathrm{(C) } 3qquad mathrm{(D) } 4qquad mathrm{(E) } 5$

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3.

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In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle?

$mathrm{(A) } 43qquad mathrm{(B) } 44qquad mathrm{(C) } 45qquad mathrm{(D) } 46qquad mathrm{(E) } 47$

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4.

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In quadrilateral $ABCD$ , $AB = 5$ , $BC = 17$ , $CD = 5$ , $DA = 9$ , and $BD$ is an integer. What is $BD$ ?

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$extbf{(A)} 11 qquad extbf{(B)} 12 qquad extbf{(C)} 13 qquad extbf{(D)} 14 qquad extbf{(E)} 15$ '

5.

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Let $T_1$ be a triangle with sides $2011, 2012,$ and $2013$ . For $n ge 1$ , if $T_n = riangle ABC$ and $D, E,$ and $F$ are the points of tangency of the incircle of $riangle ABC$ to the sides $AB, BC$ and $AC,$ respectively, then $T_{n+1}$ is a triangle with side lengths $AD, BE,$ and $CF,$ if it exists. What is the perimeter of the last triangle in the sequence $( T_n )$ ?

$extbf{(A)} frac{1509}{8} qquad extbf{(B)} frac{1509}{32} qquad extbf{(C)} frac{1509}{64} qquad extbf{(D)} fra...$

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