Exercices - IMOSuisse

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Olympiades Suisses de Mathématiques
ABC AB =AC ∠ABC
AC ABC
ABC AB > AC C
∠ABC D BC D CA L AB M
LM =BM −CL
A B C D ∠DBA
∠DCA = 56◦
∠CBD = 39◦∠ADC = 121◦
∠CBA = 91◦∠CAD = 13◦
∠ADB = 41◦∠DCB = 103◦
∠BAD = 140◦∠ACB = 17◦
ABC P ∠BAC
ABC BP C
ABCD ∠BAD = 131◦∠DBA = 17◦∠ACB = 32◦
∠DCA

ABC k O t k
A s AB t s
ABO
A B k AB
AB 90◦
AB
ABC I CI
ABI D AI BCI
E
D E B
ABC M AB AM =BM =
CM
ABC M AB H
C W AB ∠ACB
∠HCW =∠W CM
AA0BB0CC0ABC ABC
A0B0C0
A0B0C0ABC
S AA0A0B0C0
ABC H HAHBHC
AHCHHBBCHBHC
ABC D E F BC CA AB
P F BD
DCE AF P E

ABCD M AB P C MD
P M D CP M D
P BC
ABC H D E F
H DEF
ABCD
n≤180◦P
P AB BC CD
DA
ABC H M AH N
BC
ABC
MN
ABC
ABC AH BH CH
9
A B k C k
B AB =AC D ∠ABC
AC D k ∠ABC > 72◦
k1k2M1M2A B
M1B k2F6=B M2B k1E6=B EF B
k1k2P Q
B AEF
P Q =AE +AF
g k1k2
g g A B k3
k1D k2C
ABCD
BC AD k3

ABC ∠CAB = 90◦L
BC ABL AC M
CAL AB N N
AB M AC L M N
A B k l
K L M N k
l A A M N
D MK NL M N B D
M1M2k1k2
P∠M1P M2= 90◦
M1M2k1Q M1M2M2P Q
k2
k1k2M1M2
A B k1A k2
P M1B k2Q Q
k1P6=Q P Q M1M2