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Dundgovi20115
1.
Ug.,zgm dgui Y^Flpnay
ytjtbrnm bgm-bgmn[nay klnb'ng;* Bgy;gluk.,* 2011^09^24 2@j gyun Dk;kh heugwgg 90 bnyem G G H 1^ + G H G ovtum mgg.jsu dk;^ H G G H F F H 2^ 20 hkv kabvsu 3 eemgy; he.ggr hnartt^ 1@j eemgy; 3 hkv* 4 fjffvfy kabv* 2@j eemgy; 1 hkv* 15 fjffvfy kabv dgavgy dkl 3@j eemgy; ht;ty fjffvfy kabv dgavgy dt& (Doh dklkbrnm hgjneu kl^) 3^ Hkkjky;kk 5 ygvys pfjoomta 2 hoomta ytuty gg. dgartt^ 2006 ky; gg.sy ygv 2 hoounay ygvmga ynaltt; 48 dga.^ Hgjny 2011 ky; gg.sy ygv 2 hoounay ygvys ynaldtjttv 2 ;ghny nh dklk.^ Mkb hoo y, mfjfhf; gg. ht;ty ygvmga dgavgy dt& 4^ (ytjtbrnm) Ptjutl;tt 2 ;euea; dnxnu;vty mkkyee;sy zlug.gj 2 /bee 3 dgahggj* ;eueayee;g; 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 mkkyee;su dnx^ Ug.,zgm dgui Y^Flpnay ytjtbrnm bgm-bgmn[nay klnb'ng;* Bgy;gluk.,* 2011^09^24 3@j gyun Dk;kh heugwgg 90 bnyem 1^ Hkkjky;kk 5 ygvys pfjoomta 2 hoomta ytuty gg. dgartt^ 2006 ky; gg.sy ygv 2 hoounay ygvmga ynaltt; 48 dga.^ Hgjny 2011 ky; gg.sy ygv 2 hoounay ygvys ynaldtjttv 2 ;ghny nh dklk.^ Mkb hoo y, mfjfhf; gg. ht;ty ygvmga dgavgy dt& 2^ 16 × 16 htbrttmta [.g;jgmsu 4 injhtu 1 × 1 htbrttmta* 8 injhtu 2 × 2 htbrttmta* 12 injhtu 3 × 3 htbrttmta* 7 injhtu 4 × 4 htbrttmta [.g;jgmee; dkluky he.gg^ 3^ Dgm iguyglsy bfyuffjff ;tluoojttv 12 ;t.mtj* 15 optu tv.tl 30 optu* 10 hgjgy;gg tv.tl 45 hgjgy;gg* 13 ;t.mtj he;gl;gy g.x xg;yg^ Dgm pf.hfy hgjgy;gg he;gl;gy g.vgy dkl ht;ty hgjgy;gg he;gl;gy g.gh .t& 4^ (ytjtbrnm) Pejgum opoolvty ;ojvnau 2 mtywoo htvtu dkluky he.gg^
2.
Ug.,zgm dgui Y^Flpnay
ytjtbrnm bgm-bgmn[nay klnb'ng;* Bgy;gluk.,* 2011^09^24 4@j gyun (inlrnlmnay) Dk;kh heugwgg 120 bnyem 1^ 16 × 16 htbrttmta [.g;jgmsu 4 injhtu 1 × 1 htbrttmta* 8 injhtu 2 × 2 htbrttmta* 12 injhtu 3 × 3 htbrttmta* 7 injhtu 4 × 4 htbrttmta [.g;jgmee; dkluky he.gg^ 2^ 100 hkv kabvsu 3 eemgy; he.ggr hnartt^ 1@j eemgy; 32 hkv* 6 fjffvfy kabv* 2@j eemgy; 35 hkv* 31 fjffvfy kabv dgavgy dkl 3@j eemgy; ht;ty fjffvfy kabv dgavgy dt& (Doh dklkbrnm hgjneu kl^) 3^ 4 dg 7@; he.ggu;;guuoa doh ygmejgl mkku fvfh tjtbdttj rguvggr dnxrtt^ Rguvgglmsy 2011@jm zbgj mkk dnxnu;th .t& 4^ (ytjtbrnm) Dgm* :kjr* Wtwtu uejg. dguinay utjnay ;gglug.gjm fuvfy dk;lkuee;ggv mev doj 11 dk;lkuk dk;rtt^ Dguinay fuvfy dk;lkuk dojnau mt;yna hty ytu y, dk;vky dfuff; uej.eelgg dk;vky dk;lkusy mkk y, mt;yna pf.hfy ytu y, dk;vky dk;lkusy mkkykkv 5@ggj wffy dga.^ Dgui ;gglug.gjm ht;ty dk;lkuk fuvfy dt& Ug.,zgm dgui Y^Flpnay ytjtbrnm bgm-bgmn[nay klnb'ng;* Bgy;gluk.,* 2011^09^24 4@j gyun Dk;kh heugwgg 120 bnyem 1^ 100 hkv kabvsu 3 eemgy; he.ggr hnartt^ 1@j eemgy; 32 hkv* 6 fjffvfy kabv* 2@j eemgy; 35 hkv* 31 fjffvfy kabv dgavgy dkl 3@j eemgy; ht;ty fjffvfy kabv dgavgy dt& (Doh dklkbrnm hgjneu kl^) 2^ Pejgum opoolvty ;ojvnau 2 htvtum he.ggr* he.gglmsy htvuoo;tt t.loolty uej.glrny hna^ 3^ Hovytumt; dgaugg 2 × 3 htbrttmta dkvkk* ht.mtt htldtjmta zbgj x mtui fywfum ;kmkj zu 2 injhtu yo; de;gu;vgy dgahggj 6 × 6 htbrttmta hovytumnay pgjnb yo;nau de;^ 4^ (ytjtbrnm) Dgm* :kjr* Wtwtu uejg. dguinay utjnay ;gglug.gjm fuvfy dk;lkuee;ggv mev doj 11 dk;lkuk dk;rtt^ Dguinay fuvfy dk;lkuk dojnau mt;yna hty ytu y, dk;vky dfuff; uej.eelgg dk;vky dk;lkusy mkk y, mt;yna pf.hfy ytu y, dk;vky dk;lkusy mkkykkv 5@ggj wffy dga.^ Dgui ;gglug.gjm ht;ty dk;lkuk fuvfy dt&
3.
Ug.,zgm dgui Y^Flpnay
ytjtbrnm bgm-bgmn[nay klnb'ng;* Bgy;gluk.,* 2011^09^24 7@j gyun Dk;kh heugwgg 120 bnyem 1^ Dgm [.g;jgmsu mev doj 7 vb '-jnb-mjmta mtui fywfumoo;t; he.ggrtt^ Hgjny Wtwtu bfy mnab htbrttmta [.g;jgmsu mev doj 8 vb '-jnb-mjmta mtui fywfumoo;t; he.ggrtt^ Wtwtunay he.gglmsy mtui fywfumnay mkk Dgmsy he.gglmsy mtui fywfumnay mkkykkv klky dgar dklkh ee& 2^ 3 kjwmka ytuty dgajgy; 300@ggv wffy mkkys hgglug dgartt^ 3@j kjwys hgglugyee;sy ;euggjsy ynam wnqjnay mkk 2@j kjwys hgglugyee;sy ;euggjsy ynam wnqjnay mkkykkv 20%@ggj nh dgavgy dkl tyt dgajys ytu kjwky; ht;ty hgglug dgavgy dt& 3^ 12 wgum ytuty ptjtu Elggydggmgjggv Bgy;gluk., hkm jee vee;lsy bginy* Bgy;gluk.nkv Elggydggmgj hkm jee g.mkdev ugjxtt^ 14 wgum vee;lsy bginy Elggydggmgj hkm dg g.mkdevys zu ;ey; dgavgy dkl vee;lsy bginy ht;ty wgum Bgy;gluk., hkm dg g.mkdevys zu ;ey; dgah .t& 4^ (ytjtbrnm) Ikkys 8 kjka ;ttj zluggmga dohtl mkkyee;su* gl, x kjka ;ttj dnxnu;vty mkk y, eu kjkamka hfji 3 kjka ;ttj dnxnu;vty mkkyee;sy ynaldtjmta mtywoo dgahggj dnxnr dklkh ee& (Ikkys ytu mgl ;ttj kjinh 2 kjkau hfji utr mkkwyk^) Ug.,zgm dgui Y^Flpnay ytjtbrnm bgm-bgmn[nay klnb'ng;* Bgy;gluk.,* 2011^09^24 8@j gyun Dk;kh heugwgg 150 bnyem 1^ Ygmejgl ������ mkkys wnqjoo;nay ynaldtjnau ������(������) utr mtb;tult-^ ������ + ������ ������ = 2011 ������ + ������ ������ = 2011 dgah doh ygmejgl ������, ������ mkkyee;su kl^ 2^ Ikkys 6 mglv ;ttj ytu* ytu ygmejgl mkk dnxvty dga.^ Ikkys 8 kjka mev doj ;ttj eu kjkau gueelvgy 3 mglv ;ttj dnxnu;vty mkkyee;sy ojr.tjnau dnxrtt^ Ikkys kjkayee; ;ttj dnxnu;vty mkkyee;sy ynaldtj 2009 dkl ikkys mglvee; ;ttj dnxnu;vty mkkyee;sy ynaldtj ht; .t& 3^ Klnb'ng;g; :ey;uk.,* Fbyfuk.,* :kjykuk., gabuee;ggv ynam 120 vejgux kjklwvky dfuff; vejgux;su 5 gyun; he.ggr veelugrtt^ Gl, x 2 gyun; :ey;uk., gabunay vejgux 30@ggv wffy* gl, x 3 gyun; Fbyfuk., gabunay vejgux 20@kkv wffy mkkmka dgavgy dkl gl, ytu 4 gyun; :kjykuk., gabunay vejgux 15@ggv wffyuoa dna ut;unau dgmgl^ 4^ (ytjtbrnm) Injttyna m-yynvyna mtbwtty; 8 dg 9@j gyunay vejguxn; kjklwkr hkkjky;kk mkajukkj mkulkrtt^ 8@j gyunay vejguxn; 9@j gyunay vejgux;ggv 2,5 ;ghny wffy dgavgy dklk.x* 8@j gyunay vejgux;sy hkrlsy ynam mkk 9@j gyunay vejgux;sy hkrlsy ynam mkkmka mtywoo dga.^ Mtbwtty; hgbunay klky;kk ht;ty vejgux kjklwvky dt&
4.
Ug.,zgm dgui Y^Flpnay
ytjtbrnm bgm-bgmn[nay klnb'ng;* Bgy;gluk.,* 2011^09^24 9@j gyun (inlrnlmnay) Dk;kh heugwgg 150 bnyem 1^ Ikkys 6 mglv ;ttj ytu* ytu ygmejgl mkk dnxvty dga.^ Ikkys 8 kjka mev doj ;ttj eu kjkau gueelvgy 3 mglv ;ttj dnxnu;vty mkkyee;sy ojr.tjnau dnxrtt^ Ikkys kjkayee; ;ttj dnxnu;vty mkkyee;sy ynaldtj 2009 dkl ikkys mglvee; ;ttj dnxnu;vty mkkyee;sy ynaldtj ht; .t& 2^ :tv ;gjgglvgy 18 injhtu uej.gy kjkymka mkkys gl, ytu y, ffjnay wnqjnay ynaldtjm he.ggu;gyg ut;unau dgmgl^ 1 2 2009 2010 3^ , ,⋯, , mkkyee;ggv ojr.tj y, 1 dgah 3 injhtu mkku vkyukr dklkhuoa ut;unau 2010 2009 2 1 dgmgl^ 4^ (ytjtbrnm) Bginysu 0001, 0002, ⋯ , 9999 utr ;euggjlg;gu^ Gl, ytu 2 wnqjnay ynaldtj y, ol;vty 2 wnqjnay ynaldtjmta mtywoo dgah ;euggjsu #gpmga ;euggj# utr ytjltyt^ Gpmga ;euggjsy mkku kl^ Ug.,zgm dgui Y^Flpnay ytjtbrnm bgm-bgmn[nay klnb'ng;* Bgy;gluk.,* 2011^09^24 9@j gyun Dk;kh heugwgg 180 bnyem 1^ ������ + ������ + ������ + ������ = 1 dkl 1 ������ + ������ ������ + ������ + 2(������������ + ������������) ≤ 2 mtywtmutl dni dn-lyt utr dgmgl^ 1 1 1 1 2^ ������ + ������ + (������,������) + [������,������] = 1 dgah doh ygmejgl ������, ������ mkkyee;su kl^ (Ty; (������, ������) @ttj ������, ������ mkkyee;sy hgbunay nh -jfyhna he.gguxnau* [������, ������]@ttj ������, ������ mkkyee;sy hgbunay dgug -jfyhna he.ggu;guxnau mtb;tult.^) 3^ ∠������������������ = 72° kjkay fywfumta g;nl hgreem ������������������ uej.glrys ������������ veej, ;ttj ������, ������ wtuoo;* hgree ������������, ������������ mglee; ;ttj hgjuglpgy ������, ������ wtuoo;nau ������������ + ������������ = ������������ + ������������ = ������������ dgahggj vkyuky g.xtt^ ������������, ������������ ieleeyee; ������ wtum kumlklw;ku dkl ∠������������������ fywunau kl^ 4^ (ytjtbrnm) Klnb'ng;g; 16 vejgux kjklwkr* 9 dk;lkuk dk;rtt^ Ioouxn; htjt. dk;lkusu ������ vejgux dk;kr xg;gguoa dkl eu dk;lkuk; ������ kykk fux ;ouytrtt^ Tyt ;ouytlmttj hgbunay klky dk;lkuk dk;vky vejguxnay ynaldtj kykk hgbunay wffy dgar dklkh ee&
5.
Ug.,zgm dgui Y^Flpnay
ytjtbrnm bgm-bgmn[nay klnb'ng;* Bgy;gluk.,* 2011^09^24 10@j gyun Dk;kh heugwgg 180 bnyem 1^ ������ + ������ + ������ + ������ = 1 dkl 1 ������ + ������ ������ + ������ + 2(������������ + ������������) ≤ 2 mtywtmutl dni dn-lyt utr dgmgl^ 2^ ������ × ������ htbrttmta hflunay yo;yoo;nau hgj dg wguggy fyuffj igmjsy hflfu inu glguleelgy de;vgy dga.^ Ytu oal;lttj hflunay hgg ytumtt 2 × 2 htbrttmta [.g;jgm vkyuky g.x* eu [.g;jgmsy 4 yo;yna fyunau ytuty ptjtu tvjtuttj y, vkl,r dklyk^ Zbgj ygmejgl ������ mkkys he.,; tyt oal;lnay mevlgbrmgauggj hflunay doh yo;nau nrnl fyufmta dklukr xg;gh .t& 3^ ������������������ uej.glrys ������ kjkaukkv mgmvgy fy;fj ������������ dga.^ Uej.glrys ������������, ������������ mglee;sy ;ey;gr wtuoo; y, hgjuglpgy ������, ������ dfuff; pf. ������������������, ������������������ uej.glryee;su ������������������ uej.glrys ug;yg mgl; pejgum opoolvyttj dgaueelrtt^ ∠������������������ = 90° ut;unau dgmgl^ 4^ (ytjtbrnm) Ygmejgl ������, ������, ������ mkkyee; fufu;rtt^ ������ 2 ������ 2 + ������ ������ 2 ������ 2 + ������ (������ 2 ������ 2 + ������) ojr.tj y, domty [.g;jgm dgah ygmejgl ������ mkk kl;kyk utr dgmgl^ Ug.,zgm dgui Y^Flpnay ytjtbrnm bgm-bgmn[nay klnb'ng;* Bgy;gluk.,* 2011^09^24 11@j gyun Dk;kh heugwgg 210 bnyem 1^ T-jtu ������, ������, ������ mkkyee;sy he.,; ������������ ������������ ������������ + + ≥1 ������ 2 + ������������ + ������������ ������ 2 + ������������ + ������������ ������ 2 + ������������ + ������������ mtywtmutl dni dn-lyt ut;unau dgmgl^ 2^ Ygmejgl ������ > 1 mkkys he.,; 1!, 2!, ⋯ , ������ − 1 !, ������! mkkyee;su ������@; he.gghg; zluggmga ol;tu;loo; ugj;gu dga.^ ������ y, gyhys mkk utr dgmgl^ 3^ Ygmejgl ������, ������, ������ mkkyee; fufu;rtt^ ������ 2 ������ 2 + ������ ������ 2 ������ 2 + ������ (������ 2 ������ 2 + ������) ojr.tj y, domty [.g;jgm dgah ygmejgl ������ mkk kl;kyk utr dgmgl^ 4^ (ytjtbrnm) ������������������ uej.glrys ������, ������ kjkau ;gajvgy mkajku uej.glrys ������������, ������������ mglee;su hgjuglpgy ������, ������ wtum kumklyk^ ������������������, ������������������ uej.glryee;g; dgumvgy mkajunay mf.oo; ������, ������ dfuff; ������������ ieleey ������������, ������������ mglee;su ������, ������ wtum kumkl;ku dkl ������������ = ������������ utr dgmgl^
6.
Ug.,zgm dgui Y^Flpnay
ytjtbrnm bgm-bgmn[nay klnb'ng;* Bgy;gluk.,* 2011^09^24 Dgug gyunay dgui Dk;kh heugwgg 150 bnyem 1^ Dgm* Wtwtu hkcj pf. mg.gy fywfum ������������������������������ htldtjmta wtwtjltum hojttlty; A eelpghggj mkhnjvky dklk.x Dgm ������ hgglugy ;ttj* Wtwtu ������ hgglugy ;ttj njvty dga.^ Mtutt; mt; dn- dn-ytt hgay hgiggu mkajky ytuty ptjtu glhgr thtlvty E B dfuff; hgjgbvglmga y, ytu nrnl xnultl; glhgr thtlrtt^ Wtwtjltum hojttltyunay hgiggys '-jnb-mj 2 [b dfuff; Dgmsy hej; 50 b/bny* Wtwtunay hej; 46 b/bny dga.^ Mt; mg.gy fywfumnay ytu mgl; ptjtu dgah o-; D C l dn- dn-ytt klr hgjgh dklkbrmka dkl glhgr thltt; ht;ty bnyemsy ;gjgg ytu y, yfuffuff gyh klr hgjgh .t& 1 1 1 1 2^ 1 + 2 + ⋯+ 2010 + 2011 > 2011 ut;unau dgmgl^ 3^ ������������������������ [.g;jgmsy ������������ mgl ;ttj ������ wtu* ������������ mgl ;ttj ������ wtunau vkyuky g.xtt^ ∠������������������ = ∠������������������ dkl ������������ = ������������ + ������������ ut;unau dgmgl^ 4^ (ytjtbrnm) Gl, x bfj* gl, x dgugyg; hgj fyunay yo; vky;uka mkkmka dgahggj 3 × 5 htbrttmta hovytumnay yo;yoo;nau hnxytty zluggmga gjuggj hgj dg wguggy fyuffj de;gr dklkh .t& Ug.,zgm dgui Y^Flpnay ytjtbrnm bgm-bgmn[nay klnb'ng;* Bgy;gluk.,* 2011^09^24 :ey; gyunay dgui Dk;kh heugwgg 210 bnyem 1^ Ygmejgl ������ > 1 mkkys he.,; 1!, 2!, ⋯ , ������ − 1 !, ������! mkkyee;su ������@; he.gghg; zluggmga ol;tu;loo; ugj;gu dga.^ ������ y, gyhys mkk utr dgmgl^ 2^ ������ × ������ htbrttmta hflunay yo;yoo;nau hgj dg wguggy fyuffj igmjsy hflfu inu glguleelgy de;vgy dga.^ Ytu oal;lttj hflunay hgg ytumtt 2 × 2 htbrttmta [.g;jgm vkyuky g.x* eu [.g;jgmsy 4 yo;yna fyunau ytuty ptjtu tvjtuttj y, vkl,r dklyk^ Zbgj ygmejgl ������ mkkys he.,; tyt oal;lnay mevlgbrmgauggj hflunay doh yo;nau nrnl fyufmta dklukr xg;gh .t& 3^ T-jtu ������, ������, ������ mkkyee;sy he.,; ������ ������ ������ ������ ������ ������ 3 + + ≥ ������3 + ������������������ ������ 3 + ������������������ ������ 3 + ������������������ 2 mtywtmutl dni dn-lyt ut;unau dgmgl^ 4^ (ytjtbrnm) Mkajkum dgumvgy ������������������������ ;fj.fy fywfumnay ������������, ������������ mglee;sy ojutlrltl ������ wtum* ������������, ������������ mglee;sy ojutlrltl ������ wtum kumlklwkh dfuff; ������������ htjxbnay ;ey;gr wtu ������ dga.^ ������������ ieleey ;fj.fy fywfumnau dgumggvgy mkajunau ������ wtuttv zluggmga ������ wtum kumkl;ku dkl ������, ������, ������, ������ wtuoo; ytu mkajku ;ttj kjinyk ut;unau dgmgl^
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