2015IMO试题(官方版)

?r???②????②???????

Pr?????????s?②t??t????t?s?t S??????ts??t????????s?????????????r??②t??????r??t ????ts A???B??S?t??r??s?????t C??S s???t??t AC=BC???s?②t??t S?s???tr???r??????r??②t?r??????r??t????ts A?B???C??S?t??r??s??????t P??S s???t??t P A=P B=P C????????t??t??r?????t???rs n 3?t??r??①?sts?????????s?t???s?st?????n????ts?

?????t?r?????????t???rs n 3??r?????t??r??①?sts????????????tr???r??s?t???s?st?????n ????ts?

Pr?????????t?r???????tr????s(a,b,c)????s?t?????t???rs s???t??t??????t???????rs

ab?c,bc?a,ca?b

?s?????r??2?

??????r??2?s????t???r??t????r?2n????r?n?s?????????t?????t???r??

Pr???????▲?t ABC???????t?tr????????t?AB>AC?▲?tΓ???ts??r?????r????H?ts ?rt?????tr?????F t?????t??t????t?t????r??A?▲?t M??t?????????t??BC?▲?t Q??t??????t??Γs???t??t∠HQA=90???????t K??t??????t??Γs???t??t∠HKQ=90???ss???t??t t??????ts A?B?C?K???Q?r????????r??t?????????Γ??t??s?r??r?

Pr???t??t t????r?????r???s??tr??????s KQH???F KM?r?t?????t t??????t??r?

??t?r??②????②???????

Pr????????r??????ABC??s??r?????r?????????r??????tr?O????r???Γ??t????tr?A ??t?rs??ts t??s?????t BC?t????ts D???E?s???t??t B?D?E???C?r????????r??t????????????BC??t??s?r??r?▲?t F???G??t??????ts????t?rs??t?????Γ?????s???t??t A?F?B?C???G????????t??s?r??r?▲?t K??t??s?????????t????t?rs??t?????t????r?????r?????tr??????BDF???t??s?????t AB?▲?t L??t??s?????????t????t?rs??t?????t????r?????r?????tr??????CGE???t??s?????t CA?

?????s?t??t t??????s F K???GL?r?????r??t?????t?rs??t?t t??????t X?Pr???t??t X???s??t??????AO?

Pr???????▲?t R??t??s?t??r????????rs???t?r???????????t???s f:R→R s?t?s?②???t???q??t???

f x+f(x+y) +f(xy)=x+f(x+y)+yf(x)

??r???r????????rs x???y?

Pr??????????s?q?????a1,a2,...????t???rs s?t?s??s t????????????????t???s?

???1 a j 2015??r???j 1?

????k+a k=?+a???r???1 k

Pr???t??t t??r??①?st t????s?t?????t???rs b???N s???t??t

n

j=m+1

(a j?b)

10072

??r?????t???rs m???n s?t?s?②???n>m N?