← Topic 1 · Algebra
AMC 10/12 · Cheatsheet
Topic 1 · Algebra
Chapter 3 · Inequalities & logarithms
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AM–GM
a
+
b
2
≥
a
b
,
=
⟺
a
=
b
\tfrac{a+b}{2} \ge \sqrt{ab},\; = \iff a=b
2
a
+
b
≥
ab
,
=
⟺
a
=
b
Log laws
log
(
m
n
)
=
log
m
+
log
n
,
log
m
k
=
k
log
m
\log(mn)=\log m+\log n,\; \log m^k = k\log m
lo
g
(
mn
)
=
lo
g
m
+
lo
g
n
,
lo
g
m
k
=
k
lo
g
m
Change of base
log
b
x
=
log
c
x
log
c
b
\log_b x = \tfrac{\log_c x}{\log_c b}
lo
g
b
x
=
l
o
g
c
b
l
o
g
c
x
Abs as distance
∣
x
−
a
∣
=
dist
(
x
,
a
)
|x-a| = \text{dist}(x,a)
∣
x
−
a
∣
=
dist
(
x
,
a
)
Two-point min
min
(
∣
x
−
a
∣
+
∣
x
−
b
∣
)
=
∣
a
−
b
∣
\min(|x-a|+|x-b|) = |a-b|
min
(
∣
x
−
a
∣
+
∣
x
−
b
∣
)
=
∣
a
−
b
∣
Median rule
min
∑
i
=
1
n
∣
x
−
a
i
∣
\min\sum_{i=1}^n |x-a_i|
min
∑
i
=
1
n
∣
x
−
a
i
∣
at
x
=
median
(
a
i
)
x = \operatorname{median}(a_i)
x
=
median
(
a
i
)
.
Triangle inequality
∣
a
+
b
∣
≤
∣
a
∣
+
∣
b
∣
|a+b|\le |a|+|b|
∣
a
+
b
∣
≤
∣
a
∣
+
∣
b
∣