geometry problems from Mexican Geometry Olympiad - La Geometrense with aops links in the names
collected inside aops here
2014, 2020-2
The incircle of triangle $ABC$ touches sides $BC$, $CA$, and $AB$ at $A'$, $B'$, and $C'$ respectively. $G$ is the intersection point of $AA '$, $BB'$ and $CC '$. The circumcircle of $GA'B'$ again touches $AC$ and $BC$ at points $C_A$ and $C_B$ respectively. Similarly are defined points $B_A$, $B_C$, $A_B$ and $A_C$ . Show that the points $C_A$, $C_B$, $B_A$, $B_C$, $A_B$ and $A_C$ all lie on the same circle.
Quadrilateral $ABCD$ is inscribed in a circle of radius $ 1$, such that the diagonal $AC$ is a diameter and $BD=AB$. Diagonals intersect at $P$. It is known that $PC=\frac25$ . What is the length of side $CD$?
Let $ABCD$ be a square. Find the locus of the points $P$ such that the circumradiii of the triangles $ABP$ and $CDP$ have the same length.
Let $ABC$ be a triangle and $G$ its centroid. A point $P$ is taken on segment $BC$. Take points $Q$ and $R$ on the sides $AC$ and $AB$ respectively, so that $PQ \parallel AB$ and $PR \parallel AC$. Show that as $P$ varies in segment $BC$, the circumcircle of triangle $AQR$ passes through a fixed point $X$.
$\bullet$ Take two points $A (x_1, y_1$) and $B (x_2, y_2)$ on the graph of the function $y = \frac{1}{x}$ such that $0 <x_1 <x_2$ and $AB = 2 OA$ (here $O = (0, 0)$). Let $C$ be the midpoint of $AB$. Prove that the angle between the $x$-axis and the ray $OA$ is equal to three times the angle between the $x$-axis and the ray $OC$ .
$\bullet$ You have the hyperbola $y = \frac{1}{x}$ drawn on the plane. Construct a $10^o$ angle with a ruler and compass.
There are several right triangles. In each of them Xavi chooses a leg and adds all those lengths (one from each triangle). Kevin keeps the remaining legs and adds them up (one from each triangle). Finally, Juan finds the sum of the hypotenuses. We know that a right triangle can be made whose side lengths are the sums found by Xavi, Kevin, and Juan. Shows that the initial triangles were all similar.
In a right triangle, two circles of the same radius are constructed so that are tangent to each other and so that each of them is tangent to the hypotenuse and to a leg (a different leg for each circle). Let $L$ and $N$ be the touchpoints of the circles with the hypotenuse. Prove that the midpoint of $LN$ lies on the bisector of the right angle.
Let $ABCD$ be a square. Points $K$ and $L$ are chosen on the segments $AB$ and $BC$ respectively such that $BK = CL$. Let $P$ be the intersection of segments $AL$ and $KC$.
Prove that $DP$ is perpendicular to $KL$.
There are $6$ points on the plane in general position. A good match is called when having a partition of the points into $3$ pairs so that if $A, B$ and $C, D$ are two different pairs of points, then the segments $AB$ and $CD$ do not intersect. Find
$\bullet$ The minimum number of good matches the set can have.
$\bullet$ The maximum number of good matches the set can have.
Show that every worm of length $ 1$ can be covered by some semicircle of radius $\frac12$.
Note: the worm can be represented by a curve that can be closed or open and can also self-intersect.
original wording:
Demuestra que todo gusano de longitud 1 puede ser cubierto por algun semicırculo (semi disco) de radio 1/2.
Nota: el gusano puede ser representado por una curva que puede ser cerrada o abierta y tambien puede autointersectarse
Let $\Gamma_1$ and $\Gamma_2$ be two circles internally tangent to a circle $\Gamma$, as the picture shows. The internal common tangents of $\Gamma_1$ and $\Gamma_2$ intersect $\Gamma$ at four points, two of which are marked $ A$ and $ B$ in the figure. The external common tangent of $\Gamma_1$ and $\Gamma_2$ closest to segment $AB$ intersects $\Gamma$ at $C$ and $D$. Prove that $CD$ is parallel to $AB$.
Let $M, N$ and $P$ be three points in the interior of a triangle $\vartriangle ABC$ such that among the six points $M$, $N$, $P$, $A$, $B$, $C$, there are not three of them that are collinear. We denote as $S_M$ to the sum of the distances from $M$ to the sides of $\vartriangle ABC$. In an analogous way they define $S_N$ and $S_P$. Prove that if $S_M = S_N = S_P$, then $\vartriangle ABC$ is an equilateral triangle.
Let $K$ be a convex figure in the plane and let $T$ be a triangle inscribed in $K$ (that all its vertices are on the boundary of $K$). Let's consider the triangle $T'$ which is circumscribed to $K$ (that all its sides are tangent to $K$) and it is homothetic to $T$ (with a positive homothetic ratio). Prove that $[A (K)]^2 \ge A (T) A (T')$, where $A (F)$ denotes the area of figure $F$.
Let $T_1$ and $T_2$ be two triangles such that the distance between any two vertices, taken from a different triangle, it is less than or equal to $ 1$. Prove that one of these triangles is contained in a circle of radius smaller than $\sqrt{\frac23}$.
We say that the region of the plane between two parallel lines at distance $h$ is a strip of width $h$, furthermore, if the origin $O$ is at the same distance from the lines that enclose the strip, we say that the strip is has center on the origin. Let $S$ be a finite set of points in the plane which has the property that any two of its points can be covered with a strip of width $ 1$ and center at the origin. Prove that there is a strip with center at the origin and of width $\sqrt{3}$ that covers $S$.
Let $K$ be a convex figure in the plane, that is, a compact, convex set with non-empty interior (in other words a convex set with positive and finite area) which has perimeter equal to $ 1$. Let $n \ge 2$ be an integer and let $r <n$ be a positive integer. Prove that there is a pair of lines perpendicular to each other that divide the boundary of $K$ into arcs of lengths $\frac{r}{2n}$, $\frac{r}{2n}$, $\frac{n-r}{2n}$, $\frac{n-r}{2n}$ in that order.
Let $E$ be an ellipse in the plane and let $A$ be a point outside $E$. From $A$ we draw the lines tangent to the ellipse. A pair of parallel lines $\ell_1$ and $\ell_2$ are tangent to $E$ and intersect the tangents from $A$ to $B, C$ and $M, N$, respectively. Prove that the product of the areas $A\, (ABC) \cdot A\,(AMN)$, does not depend on the pair of parallel lines $\ell_1$ and $\ell_2$ chosen.
Let $\Gamma$ and $\Delta$ be a circle and a triangle in the same plane respectively. Prove that the area of the intersection of $\Gamma$ and $\Delta$ is at most one third of the area of $\Delta$ plus one-half of the area of $\Gamma$.
Let $\Gamma_1$ and $\Gamma_2$ be tangent circles at $A$ such that $\Gamma_2$ is inside $\Gamma_1$. Let $ B$ be a point at $\Gamma_2$, and let $C$ be the second intersection of $\Gamma_1$ with $AB$. Let $D$ be a point in $\Gamma_1$ and $P$ a point in the line $CD$. $BP$ cuts $\Gamma_2$ a second time at $Q$. Prove that $A, D, P, Q$ are concyclic.
Let $ABC$ be an isosceles triangle with $AC = BC$. Let $D$ be a point on line $BA$ such that $A$ lies between $B$ and $D$. Let $O_1$ be the circumcircle of the triangle $DAC$. $O_1$ intersects $BC$ at $E$. Let $F$ be the point on $BC$ such that $FD$ is tangent to $O_1$, and let $O_2$ be the circumcircle of $DBF$. The circles $O_1$ and $O_2$ intersect at $G \ne B$. Let $O$ be the circumcenter of $BEG$. Prove that $FG$ is tangent to circumcircle of $BEG$ if and only if $DG$ is perpendicular to $OF$.
Let $ABC$ be a triangle and let $P$ and $Q$ be points on the segments $AB$ and $AC$ respectively such that $BP = CQ$. Let $R$ be the intersection point of segments $BQ$ and $CP$ and let $S$ be the second intersection point of the circumcircles of $BPR$ and $CQR$. Prove that $S$ lies on the bisector of the angle $\angle BAC$.
Let $ABC$ be a triangle with incenter $I.$ Lines $AI$ and BC intersect at $A_1$. Similarly $B_1$ and $C_1$ are defined. Let $\ell_A$ be the line that passes through $A_1$ and is perpendicular to $AI$. Similarly $\ell_B$ and $\ell_C$ are defined. Let $\Delta$ be the triangle formed by $\ell_A$, $\ell_B$ and $\ell_C$, and let $N$ be the center of the circle that passes through the midpoints of the sides of $\Delta$ . Prove that $I$ and $N$ are isogonal conjugates with respect to $\Delta$.
Let $ABCDE$ be a regular pentagon and let $P$ and $Q$ be points on the sides $BC$ and $CD$ respectively such that $\angle APQ = 90^o$ and $\angle PAQ = 36^o$. What is the measure of the angle $\angle QED$ ?
Let $A_1A_2A_3$ be an acute triangle, and let $O$ and $H$ be the circumcenter and orthocenter of the triangle, respectively. For $1 \le i \le 3$, the points $P_i$ and $Q_i$ lie on the lines $OA_i$ and $A_{i + 1}A_{i + 2}$ (where $A_{i + 3} = A_i$), respectively, such that $OP_iHQ_i$ is a parallelogram. Prove that$$\frac{OQ_1}{OP_1}+\frac{OQ_2}{OP_2}+\frac{OQ_3}{OP_3}\ge 3$$
Let $ABC$ be an acute triangle and $D$ be the foot of the altitude from $A$. Let $E$ and $F$ be the midpoints of $BD$ and $CD$ respectively. Let $O$ and $Q$ be the circumcenters of $ABF$ and $ACE$ respectively, and let $P$ be the intersection point of $OE$ and $QF$. Prove that $PB = PC$
Let $ABC$ be an acute triangle with circumcenter $O$. Let $A'$, $B'$ and $C'$ be points on the sides $BC$, $CA$ and $AB$ respectively such that the circumcircles of $AB'C'$, $BC'A'$ and $CA'B'$ all pass through $O$. Let $ \ell_A$ be the radical axis of the circle with center $B'$ and radius $B'C$ and the circumference with center $C'$ and radius $C'B$. Similarly, $\ell_B$ and $\ell_C$ are defined. Prove that $\ell_A$, $\ell_B$ and $\ell_C$ determine a triangle whose orthocenter coincides with the orthocenter of $ABC$.
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