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Inequality_Problem_Set_I
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- Modular_Arithmetic_in_the_AMC_and_AIME.
Inequality_Problem_Set_I
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Inequality Problem Set IMaths Olympiad Preparation
29th January 2024
1 Problems
1. Let x, y, z be positive real numbers that satisfy xy + yz + zx = 1.
Prove the following:
1 − xy 1 − yz 1 − zx 3
(i) + + ≥ .
1 + xy 1 + yz 1 + zx 2
1 + xy 1 + yz 1 + zx
(ii) + + ≥ 6.
1 − xy 1 − yz 1 − zx
1 + xy 1 + yz 1 + zx
(iii) · · ≥ 8.
1 − xy 1 − yz 1 − zx
2. In ∆ABC with circumradius R and medians ma , mb & mc . Prove
the following:
(i) 4R(ama + bmb + cmc ) ≥ a2 (b + c) + b2 (c + a) + c2 (a + b).
1 1 1 ma mb mc
(ii) 2R + + ≥ + + .
bc ca ab mb mc mc ma ma mb
(iii) 12Rma mb mc ≥ a(b + c)ma 2 + b(c + a)mb 2 + c(a + b)mc 2 .
3. Let O be the circumcentre of ∆ABC such that AO extended meets
the circumcircle of ∆BOC at the point A′ . Define the points B ′
& C ′ analogously. Prove that
AA′ BB ′ CC ′ 9
′
+ ′
+ ≥ .
OA OB OC ′ 2
4. Let G be the centroid of ∆ABC such that AG extended meets
the side BC at the point D and the circumcircle of ∆BGC at the
point A′ . Define the points E, B ′ , F & C ′ analogously. Prove that
AD BE CF 3
+ + ′ ≥ .
A′ D B ′ E CF 2
1
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2 Solutions
1. The condition xy + yz + zx = 1 implies that there exists a triangle ABC
with sides a, b, c & angles A, B & C such that x = tan A2 , y = tan B2 &
z = tan C2 .
1 + xy 1 + tan A2 tan B2
∴ =
1 − xy 1 − tan A2 tan B2
cos A2 cos B2 + sin A2 sin B2
=
cos A2 cos B2 − sin A2 sin B2
cos B−C
2
=
cos B+C
2
cos B−C2
= .
sin A2
B−C
cos
CLAIM I: sin A2 = b+c a .
2
Proof: By Law of Sines, we have,
b+c sin B + sinC
=
a sin A
2 sin B+C2 cos 2
B−C
=
2 sin A2 cos A2
cos A2 cos B−C2
=
sin A2 cos A2
cos B−C2
= .
sin A2
1+xy 1+yz
Thus, 1−xy = b+c 1+zx
a . We obtain analogous expressions for 1−yz & 1−zx .
Hence, the first inequality is just Nesbitt’s Inequality and the other two
follow by AM-GM.
QED.
Equality holds in all three iff x = y = z = √13 .
2. Tereshin’s Inequality tells us that,
4Rma ≥ b2 + c2
or, 4Rama ≥ a(b2 + c2 ) .
Building up two similar inequalities and adding all 3 up proves (i).
2 Maths Olympiad Preparation
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