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Đề thi toán quốc tế lần thứ 48 được tổ chức tại Hà Nội - Việt Nam. Kỳ thi diễn ra từ 19 đến 31-7-2007..
■▼❖ ✷✵✵✼ ❍❛ ◆♦✐✱ ❱✐❡t♥❛♠ ❉❛② ✶ ✲ ✷✺ ❏✉❧② ✷✵✵✼ ✶ ❘❡❛❧ ♥✉♠❜❡rs ❛ ✶ ✱ ❛ ✷ ✱ ✿ ✿ ✿✱ ❛ ♥ ❛r❡ ❣✐✈❡♥✳ ❋♦r ❡❛❝❤ ✐✱ ✭✶ ✔ ✐ ✔ ♥✮✱ ❞❡☞♥❡ ❞ ✐ ❂ ♠❛①❢❛ ❥ ❥ ✶ ✔ ❥ ✔ ✐❣ ♠✐♥❢❛ ❥ ❥ ✐ ✔ ❥ ✔ ♥❣ ❛♥❞ ❧❡t ❞ ❂ ♠❛①❢❞ ✐ ❥ ✶ ✔ ✐ ✔ ♥❣✳ ✭❛✮ Pr♦✈❡ t❤❛t✱ ❢♦r ❛♥② r❡❛❧ ♥✉♠❜❡rs ① ✶ ✔ ① ✷ ✔ ✁ ✁ ✁ ✔ ① ♥ ✱ ♠❛①❢❥① ✐ ❛ ✐ ❥ ❥ ✶ ✔ ✐ ✔ ♥❣ ✕ ❞ ✷ ✿ ✭✄✮ ✭❜✮ ❙❤♦✇ t❤❛t t❤❡r❡ ❛r❡ r❡❛❧ ♥✉♠❜❡rs ① ✶ ✔ ① ✷ ✔ ✁ ✁ ✁ ✔ ① ♥ s✉❝❤ t❤❛t t❤❡ ❡q✉❛❧✐t② ❤♦❧❞s ✐♥ ✭✯✮✳ ✷ ❈♦♥s✐❞❡r ☞✈❡ ♣♦✐♥ts ❆✱ ❇✱ ❈✱ ❉ ❛♥❞ ❊ s✉❝❤ t❤❛t ❆❇❈❉ ✐s ❛ ♣❛r❛❧❧❡❧♦❣r❛♠ ❛♥❞ ❇❈❊❉ ✐s ❛ ❝②❝❧✐❝ q✉❛❞r✐❧❛t❡r❛❧✳ ▲❡t ❵ ❜❡ ❛ ❧✐♥❡ ♣❛ss✐♥❣ t❤r♦✉❣❤ ❆✳ ❙✉♣♣♦s❡ t❤❛t ❵ ✐♥t❡rs❡❝ts t❤❡ ✐♥t❡r✐♦r ♦❢ t❤❡ s❡❣♠❡♥t ❉❈ ❛t ❋ ❛♥❞ ✐♥t❡rs❡❝ts ❧✐♥❡ ❇❈ ❛t ●✳ ❙✉♣♣♦s❡ ❛❧s♦ t❤❛t ❊❋ ❂ ❊● ❂ ❊❈✳ Pr♦✈❡ t❤❛t ❵ ✐s t❤❡ ❜✐s❡❝t♦r ♦❢ ❛♥❣❧❡ ❉❆❇✳ ✸ ■♥ ❛ ♠❛t❤❡♠❛t✐❝❛❧ ❝♦♠♣❡t✐t✐♦♥ s♦♠❡ ❝♦♠♣❡t✐t♦rs ❛r❡ ❢r✐❡♥❞s✳ ❋r✐❡♥❞s❤✐♣ ✐s ❛❧✇❛②s ♠✉t✉❛❧✳ ❈❛❧❧ ❛ ❣r♦✉♣ ♦❢ ❝♦♠♣❡t✐t♦rs ❛ ❝❧✐q✉❡ ✐❢ ❡❛❝❤ t✇♦ ♦❢ t❤❡♠ ❛r❡ ❢r✐❡♥❞s✳ ✭■♥ ♣❛rt✐❝✉❧❛r✱ ❛♥② ❣r♦✉♣ ♦❢ ❢❡✇❡r t❤❛♥ t✇♦ ❝♦♠♣❡t✐t✐♦rs ✐s ❛ ❝❧✐q✉❡✳✮ ❚❤❡ ♥✉♠❜❡r ♦❢ ♠❡♠❜❡rs ♦❢ ❛ ❝❧✐q✉❡ ✐s ❝❛❧❧❡❞ ✐ts s✐③❡✳ ●✐✈❡♥ t❤❛t✱ ✐♥ t❤✐s ❝♦♠♣❡t✐t✐♦♥✱ t❤❡ ❧❛r❣❡st s✐③❡ ♦❢ ❛ ❝❧✐q✉❡ ✐s ❡✈❡♥✱ ♣r♦✈❡ t❤❛t t❤❡ ❝♦♠♣❡t✐t♦rs ❝❛♥ ❜❡ ❛rr❛♥❣❡❞ ✐♥t♦ t✇♦ r♦♦♠s s✉❝❤ t❤❛t t❤❡ ❧❛r❣❡st s✐③❡ ♦❢ ❛ ❝❧✐q✉❡ ❝♦♥t❛✐♥❡❞ ✐♥ ♦♥❡ r♦♦♠ ✐s t❤❡ s❛♠❡ ❛s t❤❡ ❧❛r❣❡st s✐③❡ ♦❢ ❛ ❝❧✐q✉❡ ❝♦♥t❛✐♥❡❞ ✐♥ t❤❡ ♦t❤❡r r♦♦♠✳ ❤tt♣✿✴✴✇✇✇✳❛rt♦❢♣r♦❜❧❡♠s♦❧✈✐♥❣✳❝♦♠✴ ❚❤✐s ☞❧❡ ✇❛s ❞♦✇♥❧♦❛❞❡❞ ❢r♦♠ t❤❡ ❆♦P❙ ▼❛t❤▲✐♥❦s ▼❛t❤ ❖❧②♠♣✐❛❞ ❘❡s♦✉r❝❡s P❛❣❡ P❛❣❡ ✶ ❤tt♣✿✴✴✇✇✇✳♠❛t❤❧✐♥❦s✳r♦✴ ■▼❖ ✷✵✵✼ ❍❛ ◆♦✐✱ ❱✐❡t♥❛♠ ❉❛② ✷ ✲ ✷✻ ❏✉❧② ✷✵✵✼ ✹ ■♥ tr✐❛♥❣❧❡ ❆❇❈ t❤❡ ❜✐s❡❝t♦r ♦❢ ❛♥❣❧❡ ❇❈❆ ✐♥t❡rs❡❝ts t❤❡ ❝✐r❝✉♠❝✐r❝❧❡ ❛❣❛✐♥ ❛t ❘✱ t❤❡ ♣❡r✲ ♣❡♥❞✐❝✉❧❛r ❜✐s❡❝t♦r ♦❢ ❇❈ ❛t P✱ ❛♥❞ t❤❡ ♣❡r♣❡♥❞✐❝✉❧❛r ❜✐s❡❝t♦r ♦❢ ❆❈ ❛t ◗✳ ❚❤❡ ♠✐❞♣♦✐♥t ♦❢ ❇❈ ✐s ❑ ❛♥❞ t❤❡ ♠✐❞♣♦✐♥t ♦❢ ❆❈ ✐s ▲✳ Pr♦✈❡ t❤❛t t❤❡ tr✐❛♥❣❧❡s ❘P ❑ ❛♥❞ ❘◗▲ ❤❛✈❡ t❤❡ s❛♠❡ ❛r❡❛✳ ✺ ▲❡t ❛ ❛♥❞ ❜ ❜❡ ♣♦s✐t✐✈❡ ✐♥t❡❣❡rs✳ ❙❤♦✇ t❤❛t ✐❢ ✹❛❜ ✶ ❞✐✈✐❞❡s ✭✹❛ ✷ ✶✮ ✷ ✱ t❤❡♥ ❛ ❂ ❜✳ ✻ ▲❡t ♥ ❜❡ ❛ ♣♦s✐t✐✈❡ ✐♥t❡❣❡r✳ ❈♦♥s✐❞❡r ❙ ❂ ❢✭①❀ ②❀ ③✮ ❥ ①❀ ②❀ ③ ✷ ❢✵❀ ✶❀ ✿ ✿ ✿ ❀ ♥❣❀ ① ✰ ② ✰ ③ ❃ ✵❣ ❛s ❛ s❡t ♦❢ ✭♥✰✶✮ ✸ ✶ ♣♦✐♥ts ✐♥ t❤❡ t❤r❡❡✲❞✐♠❡♥s✐♦♥❛❧ s♣❛❝❡✳ ❉❡t❡r♠✐♥❡ t❤❡ s♠❛❧❧❡st ♣♦ss✐❜❧❡ ♥✉♠❜❡r ♦❢ ♣❧❛♥❡s✱ t❤❡ ✉♥✐♦♥ ♦❢ ✇❤✐❝❤ ❝♦♥t❛✐♥s ❙ ❜✉t ❞♦❡s ♥♦t ✐♥❝❧✉❞❡ ✭✵❀ ✵❀ ✵✮✳ ❤tt♣✿✴✴✇✇✇✳❛rt♦❢♣r♦❜❧❡♠s♦❧✈✐♥❣✳❝♦♠✴ ❚❤✐s ☞❧❡ ✇❛s ❞♦✇♥❧♦❛❞❡❞ ❢r♦♠ t❤❡ ❆♦P❙ ▼❛t❤▲✐♥❦s ▼❛t❤ ❖❧②♠♣✐❛❞ ❘❡s♦✉r❝❡s P❛❣❡ P❛❣❡ ✷ ❤tt♣✿✴✴✇✇✇✳♠❛t❤❧✐♥❦s✳r♦✴