geometry problems from Olympiade Mathématique Belge (OMB) from Belgium with aops links
collected inside aops here
2004 - 2021 Maxi / Seniors
Let $\triangle ABC$ with $c>b\ ,$ $D\in (AC$ so that $BD=CD$ and $E\in (BC)\ ,$ $F\in AB$ so that $EF\parallel BD$ and $G\in AE\cap BD\ .$ Prove that $\widehat{BCF}\equiv\widehat{BCG}\ .$
In the triangle $ABC$, the lines $AE$ and $CD$ are the interior bisectors of the angles $\angle BAC$ and $\angle ACB$ respectively, $E$ lies on $BC$ and $D$ lies on $AB$. For what measured of the angle $\angle ABC$ do we certainly have that
(a) $|AD|+|EC|=|AC|$ ?
(b) $|AD|+|EC|>|AC|$ ?
(c) $|AD|+|EC|<|AC|$ ?
Consider a parallelogram $ABCD$. On $DC$ and $BC$, we construct the parallelograms $DCFE$ and $BCHG$, such that A,B and G are aligned, as well as $A, D$ and $E$. Show that $EH, FG$ and $AC$ are concurrent.
The triangle $ABC$ is equilateral. The ray $[BE$ intersects the segment $[AC]$ at $D$, such that $\angle CBE= 20^o$ and $|DE|=|AB|$. What is the measure of the angle $\angle BEC$ ?
Let $r$ be the radius of the inscribed circle , $R$ be the radius of the circumscribed circle, $p$ the perimeter and $c$ the length of the hypotenuse of a right triangle.
(a) Show that $\frac{p}{c}-\frac{r}{R}=2$
(b) Of all the right triangles, what is the greatest value the ratio $\frac{r}{R}$ can take ? For which right triangles is this maximum reached?
Let $M$ and $N$ , respectively, be the points of the sides $[AB]$ and $[BC]$ of a rectangle $ABCD$ . Let $a,b,c$ be the area of the triangle $AMD,MBN, NCD$ respectively. Express in terms of $a,b$ and $c$ , the area of the triangle $DMN$ .
Two circles $C_1$ and $C_2$ intersect at two distinct points $P$ and $Q$ . Points $R,S$ lies on $C_1,C_2$ respectively such that $R,Q$ and $S$ are aligned. The lines $RP$ and $SP$ intersect $C_2$ and $C_1$ at $N$ and $M$ respectively . Let $T$ be the intersection of $RM$ and $SN$. Show that the triangle $TMN$ is equilateral if and only if $MN$ is tangent to the two circles.
(a) Prove that if, in a triangle $ABC$ , $\angle A = 2 \angle B$ , then $|BC|^2=|AC|^2+ |AC|\cdot |AB|$
(b) Is the converse true?
In the triangle $ABC$ , let $H$ be the foot of the altitude from $A $. Let $E$ be the point of intersection of the bisector from $B$ with the side $AC$. Knowing that $\angle BEA=45^o$, determine $\angle EHC$.
Consider a rectangle $ABCD$. The points $K,L,M$ and $N$ are chosen on the sides $]AB[, ]BC[,]CD[$ and $]DA[$ respectively . The point of intersection of $KM$ and $LN$ is denoted as $P$.
(a) If $KM \perp LN$ and $P\in BD$ , can we conclude that $KL\parallel MN$?
(b) If $P\in BD$ and $KL\parallel MN$ can we conclude that $KM \perp LN$ ?
(c) If $KL\parallel MN$ and $KM \perp LN$ can we conclude that $P\in BD$?
Consider a rectangular parallelepiped $ABCDEFGH$.
(a) If this parallelepiped is a cube, what is the measure of the angle $\angle FCH$ ?
(b) Can the lengths of the edges of this parallelepiped be chosen so that the angle $\angle FCH =45^o$?
(c) What are all the values that the measure of the angle $\angle FCH$ can take ?
An open disc (that is to say without its edge) of radius $1$ is divided into open regions by two lines.
(a) If the two lines are parallel, is there still an open disc of radius $1/3$ which is entirely contained in one of these regions?
(b) If the two lines are perpendicular, is there still an open disc of radius $1/3$ which is entirely contained in one of these regions?
(c) Is there always an open disc of radius $1/3$ which is entirely contained in one of these regions, regardless of the position of the two lines?
Let $ABCD$ be a square of side $1$ and $A'B'C'D'$ its image by a rotation $r$ of center $D$ . The intersection of these two squares is the quadrilateral $DALC'$. By $C'$, we draw the parallel to $AD$, which intersects $AB$ in $E$ .
(a) What is the area of the quadrilateral $DALC'$ if $E$ is the midpoint of $[AB]$?
(b) Find the position of $E$ in $AB$ so that the area of $DALC'$ is half the area of the initial square.
(c) Determine the position of $E$ in $AB$ for the area $DALC'$ to be $1/3 ,1/4, ...,1/m$, the area of the initial square ($m$ being a nonzero natural). .
Consider a sphere $S$ of radius $R$ tangent to a plane $\pi$. Eight other spheres of the same radius $r$ are tangent to $\pi$ and tangent to $S$, they form a "collar" in which each is tangent to its two neighbors. Express $r$ in terms of $R$.
In the triangle $ABC$, the point $M$ is the midpoint of $[AB]$ and $D$ is the foot of the bisector of the angle $\angle ABC$. In addition $MD \perp BD$.
a) Indicate how to construct (with a ruler and a compass) such a triangle.
b) Prove that $|AB|=3|BC|$.
c) Is the converse also true, i.e., the condition $MD\perp BD$ necessary for $|AB|=3|BC|$ ?
A sheet of paper $ABCD$ that is rectangular with width $|AB|=4$ and length $|BC|=6$ is folded so that the lower right corner $B$ to tuck into $S$ over the left edge $[AD]$. The fold follows the segment $[RT]$, with $R$ on the bottom edge [AB] of the sheet and $T$ on its right edge $[BC]$. Let $x=[RB]$ and $y=[BT]$ .
a) What are the minimum and maximum values of $x$?
b) Find a relationship between $x$ and $y$.
c) Find the value of $x$ for which the area of the folded part (the triangle $RST$ ) is minimum. What then is the nature of the triangle $BST$ ?
In the plane, the triangles $ABD$ and $BCE$ are outside the triangle $ABC$ and are also isosceles triangles right at $D$ and $C$ respectively. The point $H$ is the intersection of $CD$ and $AE$. Show that $BH$ and $AE$ are perpendicular.
The quadrilateral $ABCD$ is inscribed in a semicircle of diameter $[AB]$ with $|BC|=|CD|=a$, $|DA|=b\ne a$ and $|AB|=c$ . If a$,b,c$ are natural numbers, show that $c$ can not be a prime number.
In a convex quadrilateral $ABCD$, the angles $B$ and $C$ are greater than $120^o$.
Prove that $|AC|+|BD|>|AB|+|BC|+|CD|$ .
In the acute triangle $ABC$, $|AB|\ne |AC|$ . The points $P$ and $Q$ are the feet of the altitudes from $B$ and $C$ respectively. The line $PQ$ intersects the line $BC$ at $R$ . Show that the bisectors of the angles $\angle BAC$ and $\angle PRB$ are perpendicular.
Let $ABC$ be a triangle acute with $\angle A=60^o$. The bisector of the angle $B$ meets the side $AC$ at $M$ and the bisector of the angle $C$ meets the side $AB$ at $N$. Prove that exists is a point $P$ on the side $BC$ such that the triangle $MNP$ is equilateral.
The triangle $ABC$ has three acute angles. Its altitudes $AD, BE$ and $DF$ are extended to meet the circle circumscribed to the triangle $ABC$, respectively at $X, Y$ and $Z$. What is the value of $\frac{|AX|}{|AD|}+\frac{|BY|}{|BE|}+\frac{|CZ|}{|CF|}$:
a) If the triangle $ABC$ is equilateral?
b) If the triangle $ABC$ has any three acute angles?
The circle $\omega$ with center $I$ and radius $r$ is inscribed with the triangle $ABC$. The sides $[BC]$ , $[AC]$ and $[AB]$ have respectively lengths $a,b,c$ and are tangent to $\omega$ at the points $X, Y$ and $Z$ . What is the length of the vector $a\overrightarrow{IX}+b\overrightarrow{IY}+b\overrightarrow{IZ}$?
source: http://omb.sbpm.be/
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