geometry problems
from Venezuelan Mathematical Olympiads (OJM)
a) Make a drawing that shows the area where the dog can move.
The base of the house of the dog Nero is shaped like a regular hexagon with a side of $1$ m. Nero is tied to the house in one of the vertices of the hexagon with a rope that measures $2$ m. What is the area of the region outside the house that Neon can reach?
The three sides of triangle $ABC$ extend a distance equal to their lengths, as seen in the drawing. If the area of the triangle $ABC$ is $2$ cm$^2$. What is the area of triangle $XYZ$?
from Venezuelan Mathematical Olympiads (OJM)
OLIMPÍADA JUVENIL DE MATEMÁTICA
2004 - 2018
under construction
under construction
(so far 2004 - 2006)
years 10-11 (IV-V)
$ABCDEFGH$ is a cube of side $3$.
Point $X$ lies on the side $AB$ such that $AX= \frac{1}{3} AB$.
Point $Y$ lies on the side $GH$ such that one $GY = \frac{1}{3} GH$ .
Point $Z$ lies on the side $DE$ such that one $DZ= \frac{2}{3} DE$.
Find the area of the triangle $XYZ$.
A right triangle has legs of lengths $a$ and $b$ . A circle of radius $r$ is tangent to the two legs and has its center on the hypotenuse of the right triangle. Show that $\frac{1}{a}+\frac{1}{b}=\frac{1}{r}$
years 7-9 (I-III)
The Pinto will leave outside the weekend and to have them keep their house they have left in charge from this to your trusted dog. Knowing that the house has a square floor of $10$ meters side and they have tied the dog with a chain of $20$ meters attached to a corner of the house.$ABCDEFGH$ is a cube of side $3$.
Point $X$ lies on the side $AB$ such that $AX= \frac{1}{3} AB$.
Point $Y$ lies on the side $GH$ such that one $GY = \frac{1}{3} GH$ .
Point $Z$ lies on the side $DE$ such that one $DZ= \frac{2}{3} DE$.
Find the area of the triangle $XYZ$.
A right triangle has legs of lengths $a$ and $b$ . A circle of radius $r$ is tangent to the two legs and has its center on the hypotenuse of the right triangle. Show that $\frac{1}{a}+\frac{1}{b}=\frac{1}{r}$
2005 Venezuela OJM IV p1
Let $ABC$ be a right triangle in $A$. Consider the altitude $AD$ corresponding to the hypotenuse of $ABC$ and the altitude $DE$ corresponding to the hypotenuse of $ABD$. If $BD = 13$ and $DE = 12$, Calculate all the trigonometric numbers of the angles of the triangle $\vartriangle ADC$ .
2005 Venezuela OJM V p3
In the figure, $BCD$ is the one fourth part of a circle of radius $1$. The measure of angle $BCA$ is $60^o$ and $X$ is a point in the segment $CD$ . If the area of the region shaded is half the area of the quarter-circle $BCD$, how much is the segment $CX$ ?
Let $ABC$ be a right triangle in $A$. Consider the altitude $AD$ corresponding to the hypotenuse of $ABC$ and the altitude $DE$ corresponding to the hypotenuse of $ABD$. If $BD = 13$ and $DE = 12$, Calculate all the trigonometric numbers of the angles of the triangle $\vartriangle ADC$ .
In the figure, $BCD$ is the one fourth part of a circle of radius $1$. The measure of angle $BCA$ is $60^o$ and $X$ is a point in the segment $CD$ . If the area of the region shaded is half the area of the quarter-circle $BCD$, how much is the segment $CX$ ?
Consider a trapezoid $ABCD$ with bases $AB$ and $CD$ of lengths $5$ cm and $1$ cm respectively. Let $M$ be a point of $AD$ and $N$ a point of $BC$, such that the segment $MN$ is parallel to the side $AB$ and the area of the quadrilateral $ABNM$ is double of the area of the quadrilateral $CDMN$. Calculate the length of the segment $MN$ .
2006 Venezuela OJM V p5
Consider an isosceles triangle $ABC$, with $\angle B = \angle C = 72^o$ .
Let $CD$ be the angle bisector of the angle $\angle ACB$ .
Find the value of $\frac{BC}{AB-BC}$.
2007 Venezuela OJM IV p
Consider an isosceles triangle $ABC$, with $\angle B = \angle C = 72^o$ .
Let $CD$ be the angle bisector of the angle $\angle ACB$ .
Find the value of $\frac{BC}{AB-BC}$.
2007 Venezuela OJM IV p
2007 Venezuela OJM V p
2008 Venezuela OJM IV p
2008 Venezuela OJM V p
2009 Venezuela OJM IV p
2009 Venezuela OJM V p
2010 Venezuela OJM IV p
2010 Venezuela OJM V p
2011 Venezuela OJM IV p
2011 Venezuela OJM V p
2012 Venezuela OJM IV p
2012 Venezuela OJM V p
2013 Venezuela OJM IV p
2013 Venezuela OJM V p
2014 Venezuela OJM IV p
2014 Venezuela OJM V p
2015 Venezuela OJM IV p
2015 Venezuela OJM V p
2017 Venezuela OJM IV-V p4
$A, B, C, D$ and $E$ are collinear points such that $AB = BC = CD = DE = 1$ cm. $C_1$ and $C_2$ are semicircles of radius $1$ cm and centers $B$ and $D$, respectively. $C_3$ and $C_4$ are circles of radius $4$ cm and centers $A$ and $E$, respectively, that intersect at $F$. Determine the radius of the circle $C_5$ that is tangent externally to $C_1$ and $C_2$ and internally to $C_3$ and $C_4$.
2018 Venezuela OJM IV p
2016 Venezuela OJM IV p
2016 Venezuela OJM V p2017 Venezuela OJM IV-V p4
$A, B, C, D$ and $E$ are collinear points such that $AB = BC = CD = DE = 1$ cm. $C_1$ and $C_2$ are semicircles of radius $1$ cm and centers $B$ and $D$, respectively. $C_3$ and $C_4$ are circles of radius $4$ cm and centers $A$ and $E$, respectively, that intersect at $F$. Determine the radius of the circle $C_5$ that is tangent externally to $C_1$ and $C_2$ and internally to $C_3$ and $C_4$.
2018 Venezuela OJM V p
a) Make a drawing that shows the area where the dog can move.
b) Calculate the area of the area through which the dog can move.
2005 Venezuela OJM II p3
2005 Venezuela OJM III p1
A flag consists of a white cross with gray background. The white stripes have the same width, the gray rectangles at the corners are congruent and the flag measures $3 \times 4$ m. If the area of the cross is equal to the area of the gray region, what is the width of the cross?
2006 Venezuela OJM I p2
2007 Venezuela OJM I p
Four points $A, B, C$ and $D$ are placed consecutively on a straight line. If $E$ and $F$ are the midpoints of AB and CD, respectively, $AC = 26$ cm and $BD = 44$ cm, find the measure of segment $EF$ .
A flag consists of a white cross with gray background. The white stripes have the same width, the gray rectangles at the corners are congruent and the flag measures $3 \times 4$ m. If the area of the cross is equal to the area of the gray region, what is the width of the cross?
Let $D$ and G be the midpoints of the sides $AB$ and $AC$, respectively, of a triangle $ABC$. Sean $E$ and $F$ points on the side $BC$ such that $BE = EF = FC$. If the area of the triangle $ABC$ is $84$, find the area of the pentagon $ADEF G$.
2006 Venezuela OJM II p1
In the figure, $AXZY$ is a square, $ABCD$ and $PQRB$ are rectangles and $AR = RD$. Calculate the ratio of areas $\frac{(PQRB)}{(ABCD)}$
2006 Venezuela OJM III p3
Let $ABC$ be a triangle, $D$ the midpoint of $BC$. Consider a point $E$ on $AC$ such that $BE = 2AD$. Let $F$ be the intersection point of $AD$ and $BE$. Calculate the measures of the angles of the triangle $FEA$, if we know that $\angle DAC = 60^o$2In the figure, $AXZY$ is a square, $ABCD$ and $PQRB$ are rectangles and $AR = RD$. Calculate the ratio of areas $\frac{(PQRB)}{(ABCD)}$
Consider a trapezoid $ABCD$ with $AB$ its major base and $CD$ its minor base. Let $P$ be the point median $CD$ . On the side $AB$ take two points $R$ and $S$, such that $AR = RS = SB$ and the segments $CS$ and $AD$ are parallel. Let $O$ be the midpoint of $CS$. The line $AO$ intersects $DR$ at $M$, $PS$ at $N$ and $CB$ at $Q$. If the segment $AO$ has lenth $5$ cm, what id the length of the segment $MN$ ?
2007 Venezuela OJM II p
2007 Venezuela OJM III p
2008 Venezuela OJM I p
2008 Venezuela OJM I p
2008 Venezuela OJM II p
2008 Venezuela OJM III p
2009 Venezuela OJM I p
2009 Venezuela OJM II p
2009 Venezuela OJM III p
2010 Venezuela OJM I p
2010 Venezuela OJM II p
2010 Venezuela OJM III p
2011 Venezuela OJM I p
2011 Venezuela OJM II p
2011 Venezuela OJM III p
2012 Venezuela OJM I p
2012 Venezuela OJM II p
2012 Venezuela OJM III p
2013 Venezuela OJM I p
2013 Venezuela OJM II p
2013 Venezuela OJM III p
2014 Venezuela OJM I p
2014 Venezuela OJM II p
2014 Venezuela OJM III p
2015 Venezuela OJM I p
2015 Venezuela OJM II p
2015 Venezuela OJM III p
2016 Venezuela OJM I p
2016 Venezuela OJM II p
2016 Venezuela OJM III p
2017 Venezuela OJM I- III p4
In the figure, $P A_1QA_3R$ is a regular pentagon and $P A_2R$ is an equilateral triangle.
(a) Calculate the measure of the angle $\angle A_1A_2A_3$.
(b) If $A_4$ is taken so that $A_3A_4 = A_2A_3$ and $\angle A_2A_3A_4 = \angle A_1A_2A_3$, and then $A_5$ (not visible in the figure) so that $A_4A_5 = A_3A_4$ and $\angle A_3A_4A_5 = \angle A_2A_3A_4$, and so on, a regular polygon $A_1A_2A_3A_4...A_n$ is formed that ends up closing in $A_1$. Determine the value of $n$, that is, the number of vertices of that polygon.
2018 Venezuela OJM I p
(a) Calculate the measure of the angle $\angle A_1A_2A_3$.
(b) If $A_4$ is taken so that $A_3A_4 = A_2A_3$ and $\angle A_2A_3A_4 = \angle A_1A_2A_3$, and then $A_5$ (not visible in the figure) so that $A_4A_5 = A_3A_4$ and $\angle A_3A_4A_5 = \angle A_2A_3A_4$, and so on, a regular polygon $A_1A_2A_3A_4...A_n$ is formed that ends up closing in $A_1$. Determine the value of $n$, that is, the number of vertices of that polygon.
2018 Venezuela OJM I p
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