Belgium OMB Juniors 2004-21 23p

geometry problems from Junior (Midi) final round of Olympiade Mathématique Belge (OMB) from Belgium with aops links

collected inside aops here

2004 - 2021 Midi / juniors

2004 Belgium OMB Juniors p2

In the parallelogram $ABCD$, the point $M$ is the midpoint of $[BC]$. The point $E$ is the foot of the perpendicular from $D$ on $MA$. Prove that $|CD|=|DE|$ .

2005 Belgium OMB Juniors p1

In a rectangular billiard table $ABCD$ , a ball is thrown from the corner $A$; it bounces according to the laws of reflection to the side $CD$, then to the side $BC$, then to the side $BA$ and ends its course in the corner $D$. What is the length of the path of the center of the ball knowing that the diameter of the ball is $6$ cm and that the dimensions of the billiards are $|AD|=|BC|=156$ cm and $|AB|=|CD|=306$ cm?

2005 Belgium OMB Juniors p4

Consider a convex quadrilateral $ABCD$ having no right angle and whose four vertices belong to the same circle. Let us denote by $P$ and $Q$ the feet of the perpendiculars lowered by $D$ respectively on $AB$ and $BC$. Let us denote by $R$ and $S$ the feet of the perpendiculars lowered respectively from $B$ on $AD$ and $CD$ .

The convex quadrilateral formed by four points $P,Q,R,S$:

(a) is it necessarily a trapezoid?

(b) does it necessarily have two equal sides?

2006 Belgium OMB Juniors p1

By the vertex $A$ top of the triangle $ABC$, we draw a straight line $d$. The feet of the perpendiculars drawn from $B$ and from $C$ on d are respectively $D$ and $E$ . Let point $M$ be the midpoint of $[BC]$, prove that $|MD|=|ME|$ .

2007 Belgium OMB Juniors p3

The triangle $ABC$ is right with$\angle  ABC= 90^o$. The foot of the altitude relative to the hypotenuse is $D$ and the feet of the perpendiculars drawn from $D$ on $[BA]$ and $[BC]$ are $E$ and $F$ respectively . Let $r_1,r_2$ and $r_3$ be the radii of the circles inscribed in the triangles $AED, EDF$ and $FDC$ respectively. Is the formula $r_2=\sqrt{r_1\cdot r_3}$ correct?

2008 Belgium OMB Juniors p3

The triangle $ABC$ is inscribed in a circle of center $O$. The altitude from $A$ resulting to $BC$ intrsects it at $H$. The circle of diameter $[AH]$ cuts $AB$ at $D$ and cuts $AC$ at $E$. Prove that $OA$ is perpendicular to $DE$.

2009 Belgium OMB Juniors p2

Two right triangles $ABC$ and $ABD$ are located on the same side of their common hypotenuse $[AB]$ . Their sides $AC$ and $BD$ intersect at $I$. Let $H$ be the orthogonal projection of $I$ on $AB$ . Prove that $IH$ is the bisector of $\angle CHD$.

2010 Belgium OMB Juniors p1

Let $ABCD$ be a square. On the sides $[AB], [BC], [CD]$ and $[DA]$ we construct the points $A',B',C'$ and $D'$ so that |$AA'|=|BB'|=|CC'|=|DD'|$. Show that, for any point $P$ chosen inside the square, $$(AA'PD')+(CC'PB')=(BB'PA')+(DD'PC')$$ , where $(F)$ denotes the area of the figure $F$.

2010 Belgium OMB Juniors p4

Consider a square $ABCD$ with the center $E$. The line $CF$ is tangent to the circle of diameter $AB$ , at $F \ne B$ . What is the ratio of the areas of the triangle $BEF$ and the square $ABCD$?

2011 Belgium OMB Juniors p1

Let $C_1, C_2$ and $C_3$ three circles of the same radius $r$ , tangent two by two.

How many circles are there tangent to these three circles? What are their rays?

2011 Belgium OMB Juniors p4

In a triangle $ABC$, the bisector of the angle $B$ intersects $[AC]$ at $D$. The circumcircle of the triangle $ADB$ intersects $[BC]$ at $F$ and the circumcircle of the triangle $BDC$ intersects $[AB]$ at $E$. Prove that $|AE|=|CF|$.

2012 Belgium OMB Juniors p3

A hexagon $ABCDEF$ is said to be nice if it is inscribed in a circle and the bisectors of the angles $\angle ABC,  \angle CDE$, and $\angle EFA$ all three pass through the center $O$ of this circle.

(a) Prove that, if $ABCDEF$ is nice, then $\angle AOB=\angle BOC$.

(b) Are there any nice non-regular hexagons? If so, construct a figure by showing one. (Explain the construction performed.)

(c) Prove that the three large diagonals $[AD],[BE]$ and$[CF]$ of a nice hexagon, are concurrent.

(d) Let $B,D$ and $F$ be three non collinear points. Construct ("with a ruler and a compass") three points $A,B$ and $C$ such that the hexagon $ABCDEF$ to be nice.

2012 Belgium OMB Juniors p4

The measures of the three sides of an isosceles triangle are $29, 29$, and $40$, respectively.

(i) Calculate its perimeter and area.

(ii) Is there an isosceles triangle with the same perimeter and same area, although the measurements of its sides are different from the previous triangle? If so, give all the correct isosceles triangles.

2013 Belgium OMB Juniors p4

(a) The Reuleaux triangle, constructed from an equilateral triangle, is the union of three arcs of a circle of amplitude $60^o$ (see the figure on the left). Show that its width is the same in all directions (see the figure in the center).

(b) We want to form other curves of constant width, from a triangle $ABC$ whose sides measure $a=3, b=4$, and $c=2$ by drawing six arcs of a circle as in the figure on the right: each side of the triangle is extended in both directions; each vertex becomes the center of two arcs of a circle "on either side" whose radii, for example around $A$, are equal to $x$ and $b+z$.

i) For these values of $a,b$ and $c$, show that there exists at least one choice of values of $x,y$ and $z$ which gives a continuous curve.

ii) For these values of $a,b$ and $c$ what are all the value choices for $x,y$ and $z$ which give a continuous curve?

iii) What is the perimeter of the continuous figure thus drawn? Show that its width is the same in all directions.

2014 Belgium OMB Juniors p1

The lines $d_1$ and $d_2$ are tangent on the outside to two circles with distinct radii, while $d$ their is tangent on the inside. The line $d$ is tangent to the circles at $A$ and $B$, the line $d_1$ at $A_1$ and $B_1$ and the line $d_2$ at $A_2$ and $B_2$ . The line $d$ cuts $d_1$ at $X$ and $d_2$ at $Y$.

(a) Suppose first that, further, the two circles are tangent to each other.

(i) Adapt the figure to this particular case.

(ii) Then express $|A_1B_1|$ in terms of the radii of the two circles.

(b) In the general case, prove that

(i) $|A_1B_1| =|A_2B_2|$

(ii) $|A_1B_1| =|XY|$

2015 Belgium OMB Juniors p1

Let $WXYZ$ be a square. Three parallel lines $d,d'$ and $d''$ pass through $X,Y$ and $Z$ respectively . The distance between $d$ and $d'$ is $5$ units while the distance between $d$ and $d''$ is $7$ units. What is the area of the square?

2015 Belgium OMB Juniors p3

The lines $BC$ and $AD$ intersect at $O$ , with $B$ and $C$ on the same side of $O$ , as well as $A$ and $D$. In addition $|BC|=|AD|$, $2|OC|=3|OB|$ and $|OD|=2|OA|$. The points $M$ and $N$ are the midpoints of $[AB]$ and $[CD]$ respectively . The quadrilaterals $ADPM$ and $BCQM$ are parallelograms. The line $CQ$ intersects $MN$ and $MP$ at $X$ and $Y$ respectively . Prove that the triangles $MXY$ and $QXN$ have the same area.

2016 Belgium OMB Juniors p4

Let $c$ be a circle with center $O$ and diameter $[AB]$ , and $t$ its tangent at $B$. Let $M$ be a point of $c$, distinct from $B$. The line $OM$ cuts $t$ at $E$. The tangent to $c$ at $M$ cuts $AB$ at $P$. Prove that $AM$ is perpendicular to $EP$.

2017 Belgium OMB Juniors p2

The rose window below, on the left, is drawn by bringing together copies of the motif shown on the right, which is constructed using two semicircles centered at $C$ and $E$, to which $FG$ is tangent.

a) What is the measure of the angle $\angle EAG$?

b) Show that $|AD|=|DE|$.

c) Calculate the ratio $|DE|:|BC|$ .

d) If the circle with center $C$ has radius $1$, what is the area of the shaded part of the rose window?

2018 Belgium OMB Juniors p2

A tunnel, the cross section of which is shown opposite, has two traffic lanes represented by half-discs of the same radius and a gas discharge duct represented by a smaller disc. The two half-discs and the small disc are tangent in pairs and tangent internally to a large disc of diameter equal to $12$ m. What is the diameter of the gas discharge pipe?

2019 Belgium OMB Juniors p3

The dimensions of a rectangular billiard table $ABCD$ are $|AB|=|CD| =1,5$ and $|AD|=|BC|=3$ . A ball starts from the corner $A$ and touches successively, in this order, the sides $]BC[,]CD[$ and $]AD[$ at points $M,N$  and $P$ respectively . It bounces there, forming at each bounce two equal angles with the affected band. The ball crosses its trajectory at a point $Q$ of $]AM[$ having passed through the sides of the quadrilateral $MNPQ$. The ball is here assimilated to a point: its diameter is considered to be zero.

a) For which values of $|BM|$, are the three sides $]BC[,]CD[$ and $]AD[$ they touched in that order?

b) What is the area of the quadrilateral $MNPQ$ in terms of  $|BM|$?

c) Can the quadrilateral $MNPQ$ be a rhombus? If so, for which values of $|BM|$ ?

2020 Belgium OMB Juniors p4

On a circle, the points $A,B,C,P$ are placed in this order, with $A$ different than $C$ and $|AB|=|BC|$ . The point $Q$ is the foot of the perpendicular drawn from $B$ on $AP$.

a) Let $A'$ be the symmetric of $A$ with respect to $Q$. Prove that   $\angle  BA'P=  \angle BCP.$

b) Is the angle $\angle BPC$ acute or obtuse? And the angle $\angle BPA$?

c) Prove that $|AQ| = |PC| + |PQ|$.

2021 Belgium OMB Juniors p2

As in the figure below, the point $F$ is inside the parallelogram $ABCD$, on its diagonal $[AC]$, and the point $E$ is the intersection of $BF$ with $AD$. The areas of the shaded triangles are $(BCF)=9$ and $(AEF)=1$.

(a) Find the ratio $\frac{|AF|}{|AC|}$.

(b) Find the area $(CDEF)$.

source: http://omb.sbpm.be/