证明题(1922年东南大学

联四边对边中点之两直线,必互为二等分,试证之.

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已知直线 y = kx + b (k > 0) 与圆 x2 + y2 = 1 和圆 (x − 4)2 + y2 = 1 均相切, 则 k = _______, b = _______.

There are 4n pebbles of weights 1,2,3,…,4n. Each pebble is coloured in one of n colours and there are four pebbles of each colour. Show that we can arrange the pebbles into two piles so that the following two conditions are both satisfied:● The total weights of both piles are the same.● Each pile contains two pebbles of each colour.有 4n 枚石子,重量分别为 1 , 2 , 3 , … , 4n .每一枚小石子都染了n种颜色之一,使得每种颜色的小石子恰有四枚.证明:可以把这些小石子分成两堆,且满足以下两个条件:● 两堆小石子的总重量相同;● 每堆中每种颜色的小石子各有两枚.(匈牙利供题)

如图,AB是半圆的直径,C是半圆上一点,直线MN切半圆于C点,AM⊥MN于M点,BN⊥MN于N点,CD⊥AB于D点 . 求证:(1) CD=CM=CN;(2) CD2=AM•BN.

设 CEDF 是一个已知圆的内接矩形,过 D 作该圆的切线与 CE 的延长线相交于点 A ,与 CF 的延长线相交于点 B . 求证:BF/AE=BC3/AC3 .

如图,小圆圈表示网络的结点,结点之间的连线表示它们有网线相连.连线标注的数字表示该段网线单位时间内可以通过的最大信息量.现从结点A向结点B传递信息,信息可以分开沿不同的路线同时传递,则单位时间内传递的最大信息量为【 】

Suppose a convex pentagon ABCDE such that BC=DE.If there exists a point T inside ABCDE suchthat TB=TD TC=TE and ∠ABT=∠TEA. AB meet CD and CT at point P and Q respectively, withP,B,A,Q in this order on the same line. AE meet CD and DT at point R and S respectively, with R,E,A,S in this order on the same line.Prove that P,S,Q,R are on the same circle.译文:设凸五边形ABCDE满足BC=DE.若在ABCDE内存在一点T使得TB=TD,TC=TE且∠ABT= ∠TEA.直线AB分别与直线CD和CT交于点P和Q,且P,B,A,Q在同一直线上按此顺序排列;直线AE分别与直线CD和DT交于点R和S,且R,E,A,S在同一直线上按此顺序排列.证明:P,S,Q,R 四点共圆.

Let n be a positive integer. A“Northern European Square Matrix (NESM) is an n×n square containing all the integers from 1 to n²,so that there is exactly one number in each grid.The two different grids are neighbours if they share a common edge.A grid is called a "valley”if the integer in it in smaller than the integers in all the neighbours of the grid. An "uphill path”is a sequence containing one or more grids satisfying:(i)the frist grid of the sequence is a valley,(ii) each subsequent grid in the sequence is the neighbour of its previous grid,(iii) the integers in the girds of the sequence is incremented.Figure out the minimum possible value of the number of uphill paths in a NESM which should be represented by a function of n.译文:令n为一个正整数,一个“北欧方阵”是一个包含1至n²所有整数的n×n的方格表,使得每个方格中恰有一个数字。两个相异方格如果有公共边,称它们是相邻的。如果一个方格内的数字比所有相邻方格内的数字都小,称其为“山谷”。一条“上坡路径”是一个包含一或多个方格的序列,满足:(1)序列的第一个方格是山谷;(2)序列中随后的每个方格都和前一个方格相邻;(3)序列中方格所写的数字递增。试求一个北欧方阵中山坡路径的最小可能值,以n的函数表示之。

如图所示,在△ABC中,H是垂心.以H为圆心,过点A的圆与边AC,AB分别相交于不同于A的另外两点D,E.△ADE的垂心是H',AH'的延长线与DE相交于点F.点P在四边形BCDE内部,满足△PDE∽△PBC(顶点按对应顺序排列).设直线HH',PF相交于点K,证明:A,H,P,K四点共圆.

小陶同学玩如下游戏:取定大于1的常数v;对正整数m,第m轮与第m+1轮间隔为2-m单位时长;其中第m轮是在平面上取一个半径为2-m+1的圆形安全区域(含边界,取圆时间忽略不计);取定后,该圆形安全区域将在整个游戏剩余时间内保持圆心不动,半径以速率v匀速减小,直至半径为零时,去掉该圆形安全区域.若小陶可在第100轮之前(含第100轮)的某轮将圆形安全区域完全取在已有的安全区域内,求[1/(v-1)]的最小值([x]表示不超过x的最大整数).

如图所示,四边形ABCD内接于圆,(AB) ̅=5,(AC) ̅=3√5,(AD) ̅=7,∠BAC=∠CAD,则圆的半径为【 】