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110-geometry-problems-from-imo-soft-look-inside
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- Nursing 220 Final Exam Review
110-geometry-problems-from-imo-soft-look-inside
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PrefaceThis book is an unofficial sequel to the first two geometry books pub-
lished by XYZ Press, namely 106 Geometry Problems from the AwesomeMath
Summer Program and 107 Geometry Problems from the AwesomeMath Year
Round Program. Assuming the background presented in these two books,
110 comes as a collection of problems designed for passionate geometers and
students preparing for the difficult geometry questions of the IMO.
We know by now that the key to problem solving in geometry is mastering
the basics and being able to apply them efficiently. Nonetheless, not all geom-
etry problems can be solved by simply building up from nineteenth century
geometry. Some results require deep insight into the configurations, as well as
insight that goes well beyond the naive pursuit of common sense techniques.
The book we have written is a collection of our favorite such problems.
The solutions and the commentaries that usually follow them represent
however the heart of our work. Since a non-trivial portion of the problems
were in fact proposed by us in various contests around the world, we often
chose to reproduce the way we initially thought about them as the way to
present the proofs. This is not only meant to give the reader a glimpse into
how authors come up with their problems, but also to induce a structural way
of thinking when it comes to intricate configuration debugging. It is also the
way we came up with most of the solutions for the non-personal problems that
you will encounter in this book.
We wish you a pleasant reading.
Titu Andreescu and Cosmin Pohoata
,Abbreviations and Notation
Notation of geometrical elements
∠BAC convex angle by vertex A
∠(p, q) directed angle between lines p and q
∠BAC ≡ ∠B 0 AC 0 angles BAC and B 0 AC 0 coincide
AB line through points A and B, distance between
point A and B
AB directed segment from point A to point B
X ∈ AB X lies on the line AB
X = AC ∩ BD X is the intersection of the lines AC and BD
4ABC triangle ABC
[ABC] area of 4ABC
[A1 . . . An ] area of polygon A1 . . . An
(ABC) circumcircle of triangle ABC
(A1 . . . An ) area of cyclic polygon A1 . . . An
ABkCD lines AB and CD are parallel
AB ⊥ CD lines AB and CD are perpendicular
p(X, ω) power of point X with respect to circle ω
4ABC ∼ = 4DEF triangles ABC and DEF are congruent
(in this order of vertices)
4ABC ∼ 4DEF triangles ABC and DEF are similar
(in this order of vertices)
H(H, k) homothety with center H and factor k
Ψ(P, r2 ) inversion with center P and power r2
S(S, k, ϕ) spiral similarity with center S, dilation
factor k, and angle of rotation ϕ
, viii Abbreviations and Notation
Notation of triangle elements
a, b, c sides or side lengths of 4ABC
∠A, ∠B, ∠C angles by vertices A, B, and C of 4ABC
s semiperimeter
x, y, z expressions 12 (b + c − a), 21 (c + a − b), 12 (a + b − c)
r inradius
R circumradius
K area
ha , hb , hc altitudes in 4ABC
ma , mb , mc medians in 4ABC
la , lb , lc angle bisectors (segments) in 4ABC
ra , rb , rc exradii in 4ABC
Abbreviations
USAMO United States of America Mathematical Olympiad
USAJMO USA Junior Mathematical Olympiad
IMO International Mathematical Olympiad
Romania IMO TST Romania IMO Team Selection Test
USA IMO TST United States of America IMO Team Selection Test
RMM Romanian Masters of Mathematics
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