❙♦✉t❤ ❆❢r✐❝❛♥ ▼❛t❤❡♠❛t✐❝❛❧ ❖❧②♠♣✐❛❞
❚❤✐r❞ ❘♦✉♥❞ ✷✵✵✷
❚✐♠❡✿ ✹ ❤♦✉rs
✶✳ ●✐✈❡♥ ❛ q✉❛❞r✐❧❛t❡r❛❧ABCDs✉❝❤ t❤❛tAB2+CD2=BC2+AD2✱ ♣r♦✈❡ t❤❛tAC⊥BD.
✷✳ ❋✐♥❞ ❛❧❧ tr✐♣❧❡s ♦❢ ♥❛t✉r❛❧ ♥✉♠❜❡rs (a, b, c) s✉❝❤ t❤❛t a, b ❛♥❞ c ❛r❡ ✐♥ ❣❡♦♠❡tr✐❝
♣r♦❣❡ss✐♦♥✱ ❛♥❞a+b+c=111.
✸✳ ❆ s♠❛❧❧ sq✉❛r❡ PQRS✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ❜✐❣ sq✉❛r❡✳ Pr♦❞✉❝❡ PQ t♦ A, QRt♦B, RS t♦ C❛♥❞ SPt♦Ds♦ t❤❛tA, B, C❛♥❞ D❧✐❡ ♦♥ t❤❡ ❢♦✉r s✐❞❡s ♦❢ t❤❡ ❧❛r❣❡ sq✉❛r❡ ✐♥ ♦r❞❡r✱
♣r♦❞✉❝❡❞ ✐❢ ♥❡❝❡ss❛r②✳
Pr♦✈❡ t❤❛t AC=BD❛♥❞ AC⊥BD.
✹✳ ❍♦✇ ♠❛♥② ✇❛②s ❛r❡ t❤❡r❡ t♦ ❡①♣r❡ss ✶ ✵✵✵ ✵✵✵ ❛s ❛ ♣r♦❞✉❝t ♦❢ ❡①❛❝t❧② t❤r❡❡ ✐♥t❡❣❡rs ❣r❡❛t❡r t❤❛♥ ✶❄ ✭❋♦r t❤❡ ♣✉r♣♦s❡ ♦❢ t❤✐s ♣r♦❜❧❡♠✱ abc✐s ♥♦t ❝♦♥s✐❞❡r❡❞ ❞✐☛❡r❡♥t ❢r♦♠ bac, ❡t❝✳✮
✺✳ ■♥ ❛❝✉t❡✲❛♥❣❧❡❞ tr✐❛♥❣❧❡ ABC,❛ s❡♠✐❝✐r❝❧❡ ✇✐t❤ r❛❞✐✉s ra✐s ❝♦♥str✉❝t❡❞ ✇✐t❤ ✐ts ❜❛s❡ ♦♥ BC ❛♥❞ t❛♥❣❡♥t t♦ t❤❡ ♦t❤❡r t✇♦ s✐❞❡s✳ rb ❛♥❞ rc ❛r❡ ❞❡☞♥❡❞ s✐♠✐❧❛r❧②✳ r ✐s t❤❡ r❛❞✐✉s ♦❢ t❤❡ ✐♥❝✐r❝❧❡ ♦❢ ABC.
❙❤♦✇ t❤❛t 2
r = 1 ra
+ 1
rb
+ 1
rc
.
✻✳ ❋✐♥❞ ❛❧❧ r❛t✐♦♥❛❧ ♥✉♠❜❡rsa, b, c❛♥❞ d s✉❝❤ t❤❛t
8a2−3b2+5c2+16d2−10ab+42cd+18a+22b−2c−54d=42;
15a2−3b2+21c2−5d2+4ab+32cd−28a+14b−54c−52d= −22.
❙✉✐❞✲❆❢r✐❦❛❛♥s❡ ❲✐s❦✉♥❞❡✲❖❧✐♠♣✐❛❞❡
❉❡r❞❡ ❘♦♥❞❡ ✷✵✵✷
❚②❞✿ ✹ ✉✉r
✶✳ ●❡❣❡❡ ✬♥ ✈✐❡r❤♦❡❦ ABCDs♦❞❛tAB2+CD2=BC2+AD2✱ ❜❡✇②s ❞❛tAC⊥BD.
✷✳ ❱✐♥❞ ❛❧❧❡ tr✐♣❧❡tt❡ ✈❛♥ ♥❛t✉✉r❧✐❦❡ ❣❡t❛❧❧❡ (a, b, c)s♦❞❛ta, b❡♥ c✬♥ ♠❡❡t❦✉♥❞✐❣❡ r❡❡❦s
✐s ❡♥ a+b+c=111.
✸✳ ✬♥ ❑❧❡✐♥ ✈✐❡r❦❛♥t PQRS✐s ❜❡✈❛t ✐♥ ✬♥ ❣r♦♦t ✈✐❡r❦❛♥t✳ ❱❡r❧❡♥❣PQ♥❛A, QR♥❛B, RS♥❛ C❡♥ SP♥❛ Ds♦❞❛tA, B, C❡♥ D♦♣ ❞✐❡ ✈✐❡r s②❡ ✈❛♥ ❞✐❡ ❣r♦♦t ✈✐❡r❦❛♥t ❧❫❡✱ ✐♥ ✈♦❧❣♦r❞❡
❡♥ ✈❡r❧❡♥❣ ✐♥❞✐❡♥ ♥♦❞✐❣✱
❇❡✇②s ❞❛t AC=BD ❡♥ AC⊥BD.
✹✳ ❖♣ ❤♦❡✈❡❡❧ ♠❛♥✐❡r❡ ❦❛♥ ✬♥ ♠❡♥s ✶ ✵✵✵ ✵✵✵ ✉✐t❞r✉❦ ❛s ✬♥ ♣r♦❞✉❦ ✈❛♥ ♣r❡s✐❡s ❞r✐❡ ❤❡❡❧✲ ❣❡t❛❧❧❡ ❣r♦t❡r ❛s ✶❄ ✭❱✐r ❞✐❡ ❞♦❡❧ ✈❛♥ ❤✐❡r❞✐❡ ♣r♦❜❧❡❡♠ ✇♦r❞ abc♥✐❡ ❛s ✈❡rs❦✐❧❧❡♥❞ ✈❛♥ bac❜❡s❦♦✉ ♥✐❡✱ ❡♥s✳✮
✺✳ ■♥ ❞✐❡ s❦❡r♣❤♦❡❦✐❣❡ ❞r✐❡❤♦❡❦ ABC✇♦r❞ ✬♥ ❤❛❧❢s✐r❦❡❧ ♠❡t str❛❛❧ra❣❡tr❡❦ s♦❞❛t s② ❜❛s✐s ♦♣ BC ❧❫❡ ❡♥ ❤② ❞✐❡ ❛♥❞❡r t✇❡❡ s②❡ r❛❛❦✳ rb❡♥ rc✇♦r❞ ♦♣ s♦♦rt❣❡❧②❦❡ ✇②s❡ ❣❡❞❡☞♥✐❡❡r✳
r ✐s ❞✐❡ str❛❛❧ ✈❛♥ ❞✐❡ ✐♥❣❡s❦r❡✇❡ s✐r❦❡❧ ✈❛♥ABC.
❇❡✇②s ❞❛t 2
r = 1 ra
+ 1
rb
+ 1
rc
.
✻✳ ❱✐♥❞ ❛❧❧❡ r❛s✐♦♥❛❧❡ ❣❡t❛❧❧❡ a, b, c❡♥ ds♦❞❛t
8a2−3b2+5c2+16d2−10ab+42cd+18a+22b−2c−54d=42;
15a2−3b2+21c2−5d2+4ab+32cd−28a+14b−54c−52d= −22.