Olympiad problems

2019 AMC 12/AHSME, 3

Which one of the following rigid transformations (isometries) maps the line segment $\overline{AB}$ onto the line segment $\overline{A'B'}$ so that the image of $A(-2,1)$ is $A'(2,-1)$ and the image of $B(-1,4)$ is $B'(1,-4)?$ $\textbf{(A) } $ reflection in the $y$-axis $\textbf{(B) } $ counterclockwise rotation around the origin by $90^{\circ}$ $\textbf{(C) } $ translation by 3 units to the right and 5 units down $\textbf{(D) } $ reflection in the $x$-axis $\textbf{(E) } $ clockwise rotation about the origin by $180^{\circ}$

1999 Tuymaada Olympiad, 1

Tags: geometric transformation , reflection , trigonometry , geometry

In the triangle $ABC$ we have $\angle ABC=100^\circ$, $\angle ACB=65^\circ$, $M\in AB$, $N\in AC$, and $\angle MCB=55^\circ$, $\angle NBC=80^\circ$. Find $\angle NMC$. [i]St.Petersburg folklore[/i]

2016 India PRMO, 6

Tags: geometry , geometric transformation , reflection

Suppose a circle $C$ of radius $\sqrt2$ touches the $Y$ -axis at the origin $(0, 0)$. A ray of light $L$, parallel to the $X$-axis, reflects on a point $P$ on the circumference of $C$, and after reflection, the reflected ray $L'$ becomes parallel to the $Y$ -axis. Find the distance between the ray $L$ and the $X$-axis.

1988 IMO Longlists, 40

Tags: ratio , rectangle , geometric transformation , homothety , circumcircle , symmetry , geometry

[b]i.)[/b] Consider a circle $K$ with diameter $AB;$ with circle $L$ tangent to $AB$ and to $K$ and with a circle $M$ tangent to circle $K,$ circle $L$ and $AB.$ Calculate the ration of the area of circle $K$ to the area of circle $M.$ [b]ii.)[/b] In triangle $ABC, AB = AC$ and $\angle CAB = 80^{\circ}.$ If points $D,E$ and $F$ lie on sides $BC, AC$ and $AB,$ respectively and $CE = CD$ and $BF = BD,$ then find the size of $\angle EDF.$

2003 Bundeswettbewerb Mathematik, 1

Tags: function , geometry , geometric transformation , rotation , algebra

The graph of a function $f: \mathbb{R}\to\mathbb{R}$ has two has at least two centres of symmetry. Prove that $f$ can be represented as sum of a linear and periodic funtion.

Estonia Open Senior - geometry, 2018.1.5

Tags: geometry , geometric transformation , reflection

The midpoints of the sides $BC, CA$, and $AB$ of triangle $ABC$ are $D, E$, and $F$, respectively. The reflections of centroid $M$ of $ABC$ around points $D, E$, and $F$ are $X, Y$, and $Z$, respectively. Segments $XZ$ and $YZ$ intersect the side $AB$ in points $K$ and $L$, respectively. Prove that $AL = BK$.