geometry problems from Finnish National High School Mathematics Competition (FHSMC).
with aops links
1997 - 2019
1997 Finnish HSMC p2
Circles with radii $R$ and $r$ ($R > r$) are externally tangent. Another common tangent of the circles in drawn. This tangent and the circles bound a region inside which a circle as large as possible is drawn. What is the radius of this circle?
1998 Finnish HSMC p1
Show that points $A, B, C$ and $D$ can be placed on the plane in such a way that the quadrilateral $ABCD$ has an area which is twice the area of the quadrilateral $ADBC.$
1999 Finnish HSMC p4
Three unit circles have a common point $O.$ The other points of (pairwise) intersection are $A, B$ and $C$. Show that the points $A, B$ and $C$ are located on some unit circle.
2000 Finnish HSMC p1
Two circles are externally tangent at the point $A$. A common tangent of the circles meets one circle at the point $B$ and another at the point $C$ ($B \ne C)$. Line segments $BD$ and $CE$ are diameters of the circles. Prove that the points $D, A$ and $C$ are collinear.
2001 Finnish HSMC p1
In the right triangle $ABC,$ $CF$ is the altitude based on the hypotenuse $AB.$ The circle centered at $B$ and passing through $F$ and the circle with centre $A$ and the same radius intersect at a point of $CB.$ Determine the ratio $FB : BC.$
2002 Finnish HSMC p4
Convex figure $\mathcal{K}$ has the following property:
if one looks at $\mathcal{K}$ from any point of the certain circle $\mathcal{Y}$, then $\mathcal{K}$ is seen in the right angle.
Show that the figure is symmetric with respect to the centre of $\mathcal{Y.}$
2003 Finnish HSMC p1
The incentre of the triangle $ABC$ is $I.$ The rays $AI, BI$ and $CI$ intersect the circumcircle of the triangle $ABC$ at the points $D, E$ and $F,$ respectively. Prove that $AD$ and $EF$ are perpendicular.
2004 Finnish HSMC p3
Two circles with radii $r$ and $R$ are externally tangent. Determine the length of the segment cut from the common tangent of the circles by the other common tangents.
2005 Finnish HSMC p1
In the figure below, the centres of four squares have been connected by two line segments. Prove that these line segments are perpendicular.
2006 Finnish HSMC p4
Two medians of a triangle are perpendicular. Prove that the medians of the triangle are the sides of a right-angled triangle.
2007 Finnish HSMC p3
2008 Finnish HSMC p2
The incentre of the triangle $ABC$ is $I.$ The lines $AI, BI$ and $CI$ meet the circumcircle of the triangle $ABC$ also at points $D, E$ and $F,$ respectively. Prove that $AD$ and $EF$ are perpendicular.
2009 Finnish HSMC p3
The circles $\mathcal{Y}_0$ and $\mathcal{Y}_1$ lies outside each other. Let $O_0$ be the center of $\mathcal{Y}_0$ and $O_1$ be the center of $\mathcal{Y}_1$. From $O_0$, draw the rays which are tangents to $\mathcal{Y}_1$ and similarty from $O_1$, draw the rays which are tangents to $\mathcal{Y}_0$. Let the intersection points of rays and circle $\mathcal{Y}_i$ be $A_i$ and $B_i$. Show that the line segments $A_0B_0$ and $A_1B_1$ have equal lengths.
2009 Finnish HSMC p5
2010 Finnish HSMC p1
Let $ABC$ be right angled triangle with sides $s_1,s_2,s_3$ medians $m_1,m_2,m_3$. Prove that $m_1^2+m_2^2+m_3^2=\frac{3}{4}(s_1^2+s_2^2+s_3^2)$.
2010 Finnish HSMC p5
Let $S$ be a non-empty subset of a plane. We say that the point $P$ can be seen from $A$ if every point from the line segment $AP$ belongs to $S$. Further, the set $S$ can be seen from $A$ if every point of $S$ can be seen from $A$. Suppose that $S$ can be seen from $A$, $B$ and $C$ where $ABC$ is a triangle. Prove that $S$ can also be seen from any other point of the triangle $ABC$.
2011 Finnish HSMC p1
An equilateral triangle has been drawn inside the circle. Split the triangle to two parts with equal area by a line segment parallel to the triangle side. Draw an inscribed circle inside this smaller triangle. What is the ratio of the area of this circle compared to the area of original circle.
2011 Finnish HSMC p3
Points $D$ and $E$ divides the base $BC$ of an isosceles triangle $ABC$ into three equal parts and $D$ is between $B$ and $E.$ Show that $\angle BAD<\angle DAE.$
2012 Finnish HSMC p1
A secant line splits a circle into two segments. Inside those segments, one draws two squares such that both squares has two corners on a secant line and two on the circumference. The ratio of the square's side lengths is $5:9$. Compute the ratio of the secant line versus circle radius.
2013 Finnish HSMC p3
The points $A,B,$ and $C$ lies on the circumference of the unit circle. Furthermore, it is known that $AB$ is a diameter of the circle and $\frac{|AC|}{|CB|}=\frac{3}{4}.$ The bisector of $ABC$ intersects the circumference at the point $D$. Determine the length of the $AD$.
2014 Finnish HSMC p2
The center of the circumcircle of the acute triangle $ABC$ is $M$, and the circumcircle of $ABM$ meets $BC$ and $AC$ at $P$ and $Q$ ($P\ne B$). Show that the extension of the line segment $CM$ is perpendicular to $PQ$.
2015 Finnish HSMC p2
The lateral edges of a right square pyramid are of length $a$.
Let $ABCD$ be the base of the pyramid, $E$ its top vertex and $F$ the midpoint of $CE$.
Assuming that $BDF$ is an equilateral triangle, compute the volume of the pyramid.
with aops links
(problems in English pdf)
Circles with radii $R$ and $r$ ($R > r$) are externally tangent. Another common tangent of the circles in drawn. This tangent and the circles bound a region inside which a circle as large as possible is drawn. What is the radius of this circle?
1998 Finnish HSMC p1
Show that points $A, B, C$ and $D$ can be placed on the plane in such a way that the quadrilateral $ABCD$ has an area which is twice the area of the quadrilateral $ADBC.$
1999 Finnish HSMC p4
Three unit circles have a common point $O.$ The other points of (pairwise) intersection are $A, B$ and $C$. Show that the points $A, B$ and $C$ are located on some unit circle.
2000 Finnish HSMC p1
Two circles are externally tangent at the point $A$. A common tangent of the circles meets one circle at the point $B$ and another at the point $C$ ($B \ne C)$. Line segments $BD$ and $CE$ are diameters of the circles. Prove that the points $D, A$ and $C$ are collinear.
2001 Finnish HSMC p1
In the right triangle $ABC,$ $CF$ is the altitude based on the hypotenuse $AB.$ The circle centered at $B$ and passing through $F$ and the circle with centre $A$ and the same radius intersect at a point of $CB.$ Determine the ratio $FB : BC.$
2002 Finnish HSMC p4
Convex figure $\mathcal{K}$ has the following property:
if one looks at $\mathcal{K}$ from any point of the certain circle $\mathcal{Y}$, then $\mathcal{K}$ is seen in the right angle.
Show that the figure is symmetric with respect to the centre of $\mathcal{Y.}$
2003 Finnish HSMC p1
The incentre of the triangle $ABC$ is $I.$ The rays $AI, BI$ and $CI$ intersect the circumcircle of the triangle $ABC$ at the points $D, E$ and $F,$ respectively. Prove that $AD$ and $EF$ are perpendicular.
2004 Finnish HSMC p3
Two circles with radii $r$ and $R$ are externally tangent. Determine the length of the segment cut from the common tangent of the circles by the other common tangents.
2005 Finnish HSMC p1
In the figure below, the centres of four squares have been connected by two line segments. Prove that these line segments are perpendicular.
Two medians of a triangle are perpendicular. Prove that the medians of the triangle are the sides of a right-angled triangle.
There are five points in the plane, no three of which are collinear. Show that some four of these points are the vertices of a convex quadrilateral.
The incentre of the triangle $ABC$ is $I.$ The lines $AI, BI$ and $CI$ meet the circumcircle of the triangle $ABC$ also at points $D, E$ and $F,$ respectively. Prove that $AD$ and $EF$ are perpendicular.
2009 Finnish HSMC p3
The circles $\mathcal{Y}_0$ and $\mathcal{Y}_1$ lies outside each other. Let $O_0$ be the center of $\mathcal{Y}_0$ and $O_1$ be the center of $\mathcal{Y}_1$. From $O_0$, draw the rays which are tangents to $\mathcal{Y}_1$ and similarty from $O_1$, draw the rays which are tangents to $\mathcal{Y}_0$. Let the intersection points of rays and circle $\mathcal{Y}_i$ be $A_i$ and $B_i$. Show that the line segments $A_0B_0$ and $A_1B_1$ have equal lengths.
2009 Finnish HSMC p5
As in the picture below, the rectangle on the left hand side has been divided into four parts by line segments which are parallel to a side of the rectangle. The areas of the small rectangles are $A,B,C$ and $D$. Similarly, the small rectangles on the right hand side have areas $A^\prime,B^\prime,C^\prime$ and $D^\prime$. It is known that $A\leq A^\prime$, $B\leq B^\prime$, $C\leq C^\prime$ but $D\leq B^\prime$.
Prove that the big rectangle on the left hand side has area smaller or equal to the area of the big rectangle on the right hand side, i.e. $A+B+C+D\leq A^\prime+B^\prime+C^\prime+D^\prime$.
Let $ABC$ be right angled triangle with sides $s_1,s_2,s_3$ medians $m_1,m_2,m_3$. Prove that $m_1^2+m_2^2+m_3^2=\frac{3}{4}(s_1^2+s_2^2+s_3^2)$.
2010 Finnish HSMC p5
Let $S$ be a non-empty subset of a plane. We say that the point $P$ can be seen from $A$ if every point from the line segment $AP$ belongs to $S$. Further, the set $S$ can be seen from $A$ if every point of $S$ can be seen from $A$. Suppose that $S$ can be seen from $A$, $B$ and $C$ where $ABC$ is a triangle. Prove that $S$ can also be seen from any other point of the triangle $ABC$.
2011 Finnish HSMC p1
An equilateral triangle has been drawn inside the circle. Split the triangle to two parts with equal area by a line segment parallel to the triangle side. Draw an inscribed circle inside this smaller triangle. What is the ratio of the area of this circle compared to the area of original circle.
2011 Finnish HSMC p3
Points $D$ and $E$ divides the base $BC$ of an isosceles triangle $ABC$ into three equal parts and $D$ is between $B$ and $E.$ Show that $\angle BAD<\angle DAE.$
A secant line splits a circle into two segments. Inside those segments, one draws two squares such that both squares has two corners on a secant line and two on the circumference. The ratio of the square's side lengths is $5:9$. Compute the ratio of the secant line versus circle radius.
2013 Finnish HSMC p3
The points $A,B,$ and $C$ lies on the circumference of the unit circle. Furthermore, it is known that $AB$ is a diameter of the circle and $\frac{|AC|}{|CB|}=\frac{3}{4}.$ The bisector of $ABC$ intersects the circumference at the point $D$. Determine the length of the $AD$.
2014 Finnish HSMC p2
The center of the circumcircle of the acute triangle $ABC$ is $M$, and the circumcircle of $ABM$ meets $BC$ and $AC$ at $P$ and $Q$ ($P\ne B$). Show that the extension of the line segment $CM$ is perpendicular to $PQ$.
2015 Finnish HSMC p2
The lateral edges of a right square pyramid are of length $a$.
Let $ABCD$ be the base of the pyramid, $E$ its top vertex and $F$ the midpoint of $CE$.
Assuming that $BDF$ is an equilateral triangle, compute the volume of the pyramid.
2016 Finnish HSMC p3
From the foot of one altitude of the acute triangle, perpendiculars are drawn on the other two sides, that meet the other sides at $P$ and $Q$. Show that the length of $PQ$ does not depend on which of the three altitudes is selected.
From the foot of one altitude of the acute triangle, perpendiculars are drawn on the other two sides, that meet the other sides at $P$ and $Q$. Show that the length of $PQ$ does not depend on which of the three altitudes is selected.
2017 Finnish HSMC p5
Let $A$ and $B$ be two arbitrary points on the circumference of the circle such that $AB$ is not the diameter of the circle. The tangents to the circle drawn at points $A$ and $B$ meet at $T$. Next, choose the diameter $XY$ so that the segments $AX$ and $BY$ intersect. Let this be the intersection of $Q$. Prove that the points $A, B$, and $Q$ lie on a circle with center $T$.
Let $A$ and $B$ be two arbitrary points on the circumference of the circle such that $AB$ is not the diameter of the circle. The tangents to the circle drawn at points $A$ and $B$ meet at $T$. Next, choose the diameter $XY$ so that the segments $AX$ and $BY$ intersect. Let this be the intersection of $Q$. Prove that the points $A, B$, and $Q$ lie on a circle with center $T$.
2018 Finnish HSMC p2
The sides of triangle ABC are $a = | BC |, b = | CA |$ and $c = | AB |$. Points $D, E$ and $F$ are the points on the sides $BC, CA$ and $AB$ such that $AD, BE$ and $CF$ are the angle bisectors of the triangle $ABC$. Determine the lengths of the segments $AD, BE$, and $CF$ in terms of $a, b$, and $c$.
2018 Finnish HSMC p3
The sides of triangle ABC are $a = | BC |, b = | CA |$ and $c = | AB |$. Points $D, E$ and $F$ are the points on the sides $BC, CA$ and $AB$ such that $AD, BE$ and $CF$ are the angle bisectors of the triangle $ABC$. Determine the lengths of the segments $AD, BE$, and $CF$ in terms of $a, b$, and $c$.
2018 Finnish HSMC p3
The chords $AB$ and $CD$ of a circle intersect at $M$, which is the midpoint of the chord $PQ$. The points $X$ and $Y$ are the intersections of the segments $AD$ and $PQ$, respectively, and $BC$ and $PQ$, respectively. Show that $M$ is the midpoint of $XY$.
2019 Finnish HSMC p3
Let $ABCD$ be a cyclic quadrilateral whose side $AB$ is at the same time the diameter of the circle. The lines $AC$ and $BD$ intersect at point $E$ and the extensions of lines $AD$ and $BC$ intersect at point $F$. Segment $EF$ intersects the circle at $G$ and the extension of segment $EF$ intersects $AB$ at $H$. Show that if $G$ is the midpoint of $FH$, then $E$ is the midpoint of $GH$.
Let $ABCD$ be a cyclic quadrilateral whose side $AB$ is at the same time the diameter of the circle. The lines $AC$ and $BD$ intersect at point $E$ and the extensions of lines $AD$ and $BC$ intersect at point $F$. Segment $EF$ intersects the circle at $G$ and the extension of segment $EF$ intersects $AB$ at $H$. Show that if $G$ is the midpoint of $FH$, then $E$ is the midpoint of $GH$.
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