IMO 2007:
amocana 1. mocemulia a_1, a_2,...,a_n namdvili ricxvebi. TiToeuli i-tvis, (1<=i<=n), ganvsazRvroT
d_i = max { a_ j | 1<=j<=i } - min { a_ j | i<=j<=n } da vTqvaT d = max { d_i | 1<=i<=n }.
(a) daamtkiceT, rom nebismieri namdvili x_1 <= x_2 <= ... <= x_n ri cxvebisaTvis
max { |x_i - a_i| | 1<=i<=n } >= d/2. (*)
(b) daamtkiceT, rom arseboben iseTi namdvili x_1<=x_2<=...<=x_n ricxvebi, romelTaTvisac (*)-Si toloba sruldeba. amocana 2. sibrtyeze mocemulia 5 A,B,C,D and E wertili, romelTaTvisac ABCD paralelogramia, xolo BCED- Caxazuli oTxkuTxedi. vTqvaT l aris wrfe, romelic gadis A wertilze. davuSvatt, l kveTs DC monakveTs F wertilSi da kveTs BC wrfes G wertilSi. davuSvaT aseve, rom EF=EG=EC. daamtkiceT, rom l aris DAB kuTxis biseqtrisa.
amocana 3. maTematikur Sejibrebaze zogi monawileebi ar megobrebi. megobroba yovelTvis ormxrivia. monawileTa jgufs vuwodoT "kliki", Tu TiToeuli ori monawile am jgufSi erTmaneTis megobaria. (amasTan, nebismieri jgufi, sadac Sedis orze naklebi monawile aseve "klikia").
monawileTa raodenobas klikSi vuwodoT am klikis "zoma".
mocemulia, rom am Sejibrebaze udidesi zomis klikis zoma luwia. daamtkiceT, rom SesaZlebelia am monawileTa gadanawileba or oTaxSi ise, rom udidesi zomis klikis zoma pirvel
oTaxSi igivea, rac udidesi zomis klikis zoma meore oTaxSi.
amocana 4. ABC samkuTxedSi BCA kuTxis biseqtrisa kveTs ABC samkuTxedze Semoxazul wrewirs xelmeored R wertilSi, BC gverdis SuamarTobs P-Si, xolo AC gverdis SuamarTobs-Q
wertilSi. BC monakveTis Suawertilia K, xolo CA gverdis Suawertilia L. daamtkiceT, rom RPK da RQL samkuTxedebs aqvT toli farTobebi.
amocana 5. vTqvaT a da b dadebiTi mTeli ricxvebia. daamtkiceT, rom Tu (4ab-1) yofs (4a^2-1)^2-s, maSin a=b.
amocana 6. vTqvaT, n mTeli dadebiTi ricxvia. ganvixiloT
S= { (x,y,z) | x,y,z ekutvnis {0,1,2,...,n}, x+y+z>0}, rogorc simravle (n+1)^3-1 cali wertilebisa sam ganzomilebian sivrceSi. gansazRvret umciresi SesaZlo raodenoba iseTi sibrtyeebisa, romelTa gaerTianeba Seicavs S-s, magram ar Seicavs (0,0,0)-s.
