第一天
1.求出所有的正整数n,使得关于x,y的方程 ＋ ＝恰有组满足x≤y的正整数解(x,y)
2.如图,在四边形ABCD的对角线AC与BD相交于点E,边AB、CD的中垂线相交于点F，点M、N分别为边AB、CD的中点，直线EF分别与边BC、AD相交于点P、Q，若MF×CD=NF×AB,DQ×BP=AQ×CP,求证:PQ垂直于BC.
3.设正数a,b,c,d满足abcd=1,求证: ＋＋＋1/d＋≥
4.有n(n≥3)名乒乓球选手参加循环赛,每两名选手之间恰比赛一次(比赛无平局).赛后发现,可以将这些选手排成一圈,使得对于任意三名选手A,B,C,若A,B在圈上相邻,则A,B中至少有一人战胜了C,求n的所有可能值.
第二天
5.给定非负实数a,求最小实数f＝f(a),使得对任意复数,Z1,Z2和实数x(0≤x≤1),若|Z1|≤a|Z1－Z2|,则|Z1－xZ2|≤f|Z1－Z2|
6.是否存在正整数m,n,使得m20+11n是完全平方数?请予以证明.
7.从左到右编号为B[1],B[2],...,B[n]的n个盒子共装有n个小球,每次可以选择一个盒子B[k],进行如下操作:若k=1且B[1]中至少有1个小球(则可从B[1]中移1个小球至B[2]中;若k＝n,且B[n]中至少有1个小球,则可从B[n]中移1个小球至B[n-1]中,若2≤k≤n－1且B[k]中至少有2个小球,则可从B[k]中分别移1个小球至B[k－1]和B[k＋1]中,求证:无论初始时这些小球如何放置,总能经过有限次操作使得每个盒子中恰有1个小球
8.如图,⊙O为△ABC中BC边上的旁切圆⑾点D、E分别在线段AB、AC上，使得DE平行于BC。⊙O[1]为△ADE的内切圆，O[1]B交DO于点F,O[1]C交EO于点G.⊙O切BC于点M.⊙O[1]切DE于点N.求证:MN平分线段FG.
Day 1
1 Find all positiveintegers
such that theequation
has exactly
positiveinteger solutions
where
.
2 The diagonals
of thequadrilateral
intersectat
. Let
be themidpoints of
respectively.Let the perpendicular bisectors of the segments
meet at
. Supposethat
meets
at
respectively.If
and
,prove that
.
3 The positive reals
satisfy
.Prove that
.
4 A tennistournament has
playersand any two players play one game against each other (ties are not allowed).After the game these players can be arranged in a circle, such that for anythree players
, if
areadjacent on the circle, then at least one of
wonagainst
. Find allpossible values for
.
Day 2
1 A real number
isgiven. Find the smallest
, such that for anycomplex numbers
and
,if
,then
.
2 Do there existpositive integers
, suchthat
isa square number?
3 There are
boxes
fromleft to right, and there are
balls inthese boxes. If there is at least
ball in
, wecan move one to
. Ifthere is at least
ball in
, wecan move one to
.If there are at least
balls in
,
wecan move one to
,and one to
.Prove that, for any arrangement of the
balls, we canachieve that each box has one ball in it.
4 The
-excircle
of
touches
at
. The points
lie onthe sides
respectivelysuch that
.The incircle
of
touches
at
. If
and
,prove that the midpoint of
lies on
.
