The Parabola of the Prodigy
Lena had always feared geometry. To her, circles were just loops, and triangles were three stubborn lines refusing to get along. But when her math coach slid a worn USB drive across the table and said, “It’s the 2021 edition. Problem 37 will humble you,” she knew she was in trouble.
The file was titled: Andreescu_106_Geometry_2021.pdf.
She opened it on her laptop late that night. The first ten problems were gentle—angle chases and cyclic quadrilaterals she could handle with coffee and grit. By Problem 22, she was drawing auxiliary lines like a surgeon. By Problem 31, she had filled three notebooks.
Then came Problem 37.
“Let ABC be a triangle with incenter I. Prove that the circumcircle of BIC passes through the midpoint of arc BC not containing A, and also through the excenter opposite A.”
Lena stared. The words seemed simple, but the configuration was a hydra—every time she drew one circle, three more appeared. She sketched, erased, swore softly, and sketched again. Hours passed. Her cat abandoned her. titu andreescu 106 geometry problems pdf 2021
She scrolled to the hints section. Titu Andreescu’s voice in text: “Reflect I across the angle bisector.”
She tried it. Nothing. She tried inverting around the incircle. Still nothing. At 3 a.m., she lay on the floor, the PDF glowing on her screen like a distant star. “Why do you hate me, Problem 37?” she whispered.
But she didn’t close the file. Instead, she scrolled to the end—the solutions section. She covered the answer with a sticky note and read only the first line: “Consider the A-excircle.”
Click.
She sat up. Of course. The excenter. That rogue point outside the triangle, tangent to one side and the extensions of the others. If the incenter I and the excenter E_A are symmetric across the angle bisector, then the circle through B, I, C must also pass through… She drew it. The arc. The midpoint. The proof unfolded like a blooming flower.
She laughed out loud. Her cat hissed.
By dawn, she had solved Problems 38 through 42 without breaking stride. The PDF became her bible. Problem 55 taught her spiral similarity. Problem 81 introduced her to the beauty of radical axes. And Problem 106—the final boss—was a configuration from an IMO Shortlist that took her an entire week.
But here was the secret Titu Andreescu had planted in the 2021 edition: the problems weren’t just exercises. They were a ladder. Each rung built intuition, each diagram whispered a theorem she hadn’t known she knew.
By the time Lena finished the last proof, her desk was a mountain of coffee cups and trigonometrical runes. She closed the PDF and smiled. She wasn’t afraid of geometry anymore. She was fluent.
That spring, at the national olympiad, the final problem featured a triangle, an incenter, and a mysterious circle passing through the midpoint of an arc. Lena finished the proof in seven lines, then doodled a small “106” in the margin.
The grader didn’t notice. But Lena knew: she had passed through the circle, and come out the other side.
You might think geometry books don't need updates. Wrong. The 2021 edition of 106 Geometry Problems introduced three major improvements: The Parabola of the Prodigy Lena had always
Thus, searching for "titu andreescu 106 geometry problems pdf 2021" specifically (instead of a generic older PDF) is the correct strategy for serious learners.
In triangle $ABC$, points $D,E,F$ lie on $BC,CA,AB$ such that $AD,BE,CF$ concur at $P$. Prove that the circumcircles of $AEF$, $BFD$, and $CDE$ are coaxial.
Most solvers use Menelaus in 3D or radical axis theory. The 2021 solution shows a neat complex numbers proof.
Each problem demands between 30 minutes and 3 hours of focused work. The average solution length is 1.5 pages—no one-liners here.
Let $ABC$ be an acute triangle with orthocenter $H$. Let $M$ be the midpoint of $BC$. The circle with diameter $AH$ meets the circumcircle of $ABC$ again at $N$. Prove that $M$, $N$, and the midpoint of $AH$ are collinear.
This requires recognizing that $N$ is the antipode of something, then employing nine-point circle properties. You might think geometry books don't need updates
The book is methodically divided into three distinct sections, making it accessible to a wide range of skill levels:
If you struggle with standard problems from AoPS (Art of Problem Solving) Volume 2, complete that first. 106 Geometry Problems will frustrate you otherwise.