HMMT 十一月 2014 · 冲刺赛 · 第 36 题
HMMT November 2014 — Guts Round — Problem 36
题目详情
- [ 20 ] Pick a subset of at least four of the following geometric theorems, order them from earliest to latest by publication date, and write down their labels (a single capital letter) in that order. If a theorem was discovered multiple times, use the publication date corresponding to the geometer for which the theorem is named. BD CE AF C. ( Ceva ) Three cevians AD , BE , CF of a triangle ABC are concurrent if and only if = 1. DC EA F B 2 E. ( Euler ) In a triangle ABC with incenter I and circumcenter O , we have IO = R ( R − 2 r ), where r is the inradius and R is the circumradius of ABC . √ 1 H. ( Heron ) The area of a triangle ABC is s ( s − a )( s − b )( s − c ), where s = ( a + b + c ). 2 BD CE AF M. ( Menelaus ) If D , E , F lie on lines BC , CA , AB , then they are collinear if and only if = DC EA F B − 1, where the ratios are directed. P. ( Pascal ) Intersections of opposite sides of cyclic hexagons are collinear. S. ( Stewart ) Let ABC be a triangle and D a point on BC . Set m = BD , n = CD , d = AD . Then man + dad = bmb + cnc . V. ( Varignon ) The midpoints of the sides of any quadrilateral are the vertices of a parallelogram. If your answer is a list of 4 ≤ N ≤ 7 labels in a correct order, your score will be ( N − 2)( N − 3). Otherwise, your score will be zero.
解析
- [ 20 ] Pick a subset of at least four of the following geometric theorems, order them from earliest to latest by publication date, and write down their labels (a single capital letter) in that order. If a theorem was discovered multiple times, use the publication date corresponding to the geometer for which the theorem is named. BD CE AF C. ( Ceva ) Three cevians AD , BE , CF of a triangle ABC are concurrent if and only if = 1. DC EA F B 2 E. ( Euler ) In a triangle ABC with incenter I and circumcenter O , we have IO = R ( R − 2 r ), where r is the inradius and R is the circumradius of ABC . √ 1 H. ( Heron ) The area of a triangle ABC is s ( s − a )( s − b )( s − c ), where s = ( a + b + c ). 2 BD CE AF M. ( Menelaus ) If D , E , F lie on lines BC , CA , AB , then they are collinear if and only if = DC EA F B − 1, where the ratios are directed. P. ( Pascal ) Intersections of opposite sides of cyclic hexagons are collinear. S. ( Stewart ) Let ABC be a triangle and D a point on BC . Set m = BD , n = CD , d = AD . Then man + dad = bmb + cnc . V. ( Varignon ) The midpoints of the sides of any quadrilateral are the vertices of a parallelogram. If your answer is a list of 4 ≤ N ≤ 7 labels in a correct order, your score will be ( N − 2)( N − 3). Otherwise, your score will be zero. Answer: HMPCVSE The publication dates were as follows. • Heron: 60 AD, in his book Metrica . • Menelaus: We could not find the exact date the theorem was published in his book Spherics , but because Menelaus lived from 70AD to around 130AD, this is the correct placement. • Pascal: 1640 AD, when he was just 17 years old. He wrote of the theorem in a note one year before that. Guts Round • Ceva: 1678 AD, in his work De lineis rectis . But it was already known at least as early as the 11th century. • Varignon: 1731 AD. • Stewart: 1746 AD. • Euler: 1764 AD, despite already being published in 1746. Guts Round