HMMT 二月 2007 · TEAM2 赛 · 第 10 题

HMMT February 2007 — TEAM2 Round — Problem 10

[ 20 ] A positive real number x is such that √ √ 3 3 3 3 1 − x + 1 + x = 1 . 2 Find x . Adult Acorns - Gee, I’m a Tree! [ 200 ] In this section of the team round, your team will derive some basic results concerning tangential quadri- laterals. Tangential quadrilaterals have an incircle , or a circle lying within them that is tangent to all four sides. If a quadrilateral has an incircle, then the center of this circle is the incenter of the quadrilateral. As you shall see, tangential quadrilaterals are related to cyclic quadrilaterals. For reference, a review of cyclic quadrilaterals is given at the end of this section. Your answers for this section of the team test should be proofs. Note that you may use any standard facts about cyclic quadrilaterals, such as those listed at the end of this test, without proving them. Additionally, you may cite the results of previous problems, even if you were unable to prove them. For these problems, ABCD is a tangential quadrilateral having incenter I. For the first three problems, the point P is constructed such that triangle P AB is similar to triangle IDC and lies outside ABCD . 1

解析

[ 20 ] A positive real number x is such that √ √ 3 3 3 3 1 − x + 1 + x = 1 . 2 Find x . √ 3 28 Answer: . Cubing the given equation yields 3 ( ) √ √ √ √ 3 3 3 3 3 3 3 3 3 3 6 1 = (1 − x ) + 3 (1 − x )(1 + x ) 1 − x + 1 + x + (1 + x ) = 2 + 3 1 − x . √ √ 3 3 − 1 − 1 6 6 28 2 28 6 Then = 1 − x , so = 1 − x and x = and x = . 3 27 27 3 Adult Acorns - Gee, I’m a Tree! [ 200 ] In this section of the team round, your team will derive some basic results concerning tangential quadri- laterals. Tangential quadrilaterals have an incircle , or a circle lying within them that is tangent to all four sides. If a quadrilateral has an incircle, then the center of this circle is the incenter of the quadrilateral. As you shall see, tangential quadrilaterals are related to cyclic quadrilaterals. For reference, a review of cyclic quadrilaterals is given at the end of this section. Your answers for this section of the team test should be proofs. Note that you may use any standard facts about cyclic quadrilaterals, such as those listed at the end of this test, without proving them. Additionally, you may cite the results of previous problems, even if you were unable to prove them. For these problems, ABCD is a tangential quadrilateral having incenter I. For the first three problems, the point P is constructed such that triangle P AB is similar to triangle IDC and lies outside ABCD .