IMO 2005 Mexico
amocana 1. ABC tolgverda samkuTxedis gverdebze arCeulia 6 wertili: A_1 da A_2 BC-ze ,
B_1 da B_2 CA-ze, xolo C_1 da C_2 arian AB-ze ise, rom isini arian amozneqili
A_1A_2B_1B_2C_1C_2 eqvskuTxedis wveroebi, rpmelsac aqvs toli gverdebi.
daamtkiceT, rom A_1B_2, B_1C_2 da C_1A_2 wrfeebi erT wertilSi ikveTeba. amocana 2. vTqvaT (a_n) (n>=1) aris mTel ricxvTa mimdevroba sadac Sedis usasrulod bevri dadebiTi da uaryofiTi wevrebi. davuSvat, nebismieri naturaluri n ricxvisaTvis ricxvebi a_1, a_2, ...,a_n
toveben n cal gansxvavebul naSTs n-ze gayofisas. daamtkiceT, rom yoveli mTeli ricxvi Sedis am mimdevrobaSi zustad erTxel.
amocana 3. vTqvaT, x,y da z iseTi dadebiTi namdvili ricxvebia, rom xyz>=1. daamtkiceT, rom
(x^5-x^2) / (x^5+y^2+z^2) + (y^5-y^2) / (x^2+y^5+z^2) + (z^5-z^2) / (x^2+y^2+z^5) >= 0.
amocana 4. ipovet yvela iseTi naturaluri ricxvi, romelic Tanamartivia yvela wevrTan usasrulo mimdevrobisa a_n = 2^n + 3^n + 6^n - 1.
amocana 5. vTqvaT ABCD aris amozneqili oTxkuTxedi, sadac BC=DA da BC ar aris paraleluri
DA-si. Vtqvat ori cvladi wertili E da F mdebareoben Sesabamisad BC da DA gverdebze ise, rom BE=DF. AC da BD wrfeebi ikveTebian P wertilSi, BD da EF ikveTebian Q-Si, xolo EF da AC ikveTebian R wertilSi.
daamtkiceT, rom PQR samkuTxedebze Semoxazul wrewirebs, roca E da F icvlebian, aqvT saerTo wertili gansxvavebuli P-sgan.
amocana 6. maTematikur Sejibrebaze, sadac 6 amocana iyo SerCeuli monawileTaTvis, TiToeuli 2
amoixsna monawileTa 2/5-ze metis mier. amasTan, vercerTma monawilem amoxsna 6-ive amocana. aCveneT, rom arsebobs sul mcire ori monawile, romelTac amoxsnes zustad 5-5 amocana.
